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Spring 5-31-2005
Algorithms and complexity analyses for some combinational optimization problems
Hairong Zhao New Jersey Institute of Technology
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ALGORITHMS AND COMPLEXITY ANALYSES FOR SOME COM I A O IA O IMΙΖΑΤΙOΝ O EMS
by a i g C ao
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design. In the area of scheduling, the main interest is in problems in the master-slave model.
In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and preprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios. In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, F), a nonnegative cost for each edge, and a nonnegative connectivity requirement u for every (unordered) pair of vertices u, v. The goal is to find a minimum-cost subgraph in which each pair of vertices u, v is joined by at least u edge (vertex)-disjoint paths. A Polynomial Time Approxi- mation Scheme (PTASs) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTAs or Quasi-PTASs are also de- signed for 2-edge-connectivity problem and connectivity problem and their variations in weighted or weighted planar graphs. e e o em o co s uc i g geome ic au - o e a s a e s wi ow cos a ou e ma imum eg ee is co si e e e i s esu s ows a e e is a g ee y a go i m w ic co s uc s au - o e a s a e s a i g asym o ica y o ima ou s o o e ma imum eg ee a e o a cos a e same ime e a e icie a go- i m is e e o e w ic i s au - o e a s a e s wi asym o ica y o ima ou o e ma imum eg ee a a mos o ima ou o e o a cos ALGORITHMS AND COMPLEXITY ANALYSES FOR SOME COM I A O IA Ο ΤΙΜΙΖΑΤΙΟΝ O EMS
by Haring Chaco
A Dissertation Submitted to the Faculty of New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computer Sciences
Department of Computer Science
May 2005 Co y ig © 5 y a i g C ao A RIGHTS RESERVED APPROVAL PAGE
ALGORITHMS AND COMPLEXITY ANALYSES FOR SOME COM I A O IA Ο ΤΙΜΙΖΑΤΙΟΝ O EMS
a i g ao
ose eu g isse a io Co-A iso a e is i guis e o esso o Com u e Scie ce ew e sey I s i u e o ec o ogy
A u C uma isse a io Co-A iso a e Associa e o esso o Com u e Scie ce ew e sey I s i u e o ec o ogy
e is O Commi ee Mem e a e o esso o Com u e Scie ce ew e sey I s i u e o ec o ogy
Wo ciec a e Commi ee Mem e a e o esso o Com u e Scie ce ew e sey I s i u e o ec o ogy
C i o S ei Commi ee Mem e a e o esso o I us ia E gi ee i g a O e a io s esea c Co um ia U i e si y BIOGRAPHICAL SKETCH
Author: a i g C ao Degree: oc o o i oso y Date: May 5
Undergraduate and Graduate Education:
• oc o o i oso y i Com u e Scie ce ew e sey I s i u e o ec o ogy ewa k 5
• Mas e o Com u e Scie ce ei i g U i e si y o os s e ecommu ica io s ei i g C i a 1997 • ac e o o Com u e Scie ce aiyua U i e si y o ec o ogy S a i C i a 199
Major: Com u e Scie ce
Presentations and Publications:
Y eu g a ao "Mi imi i g Mea ow ime a Makes a o Mas e -S a e Sys ems " Journal of Parallel and Distributed Computing, acce e o u ica io Y eu g a C ao "Mi imi i g Mea ow ime o Mas e -S a e Mac i es " Pro- ceedings of the 2004 International Conference on Parallel and Distributed Processing Techniques and Applications, o 939-9 5 A C uma M G ig i A Sissok o a C ao "A o ima io Sc emes o Mi imum -E ge-Co ec e a eco ec e Su g a s i a a G a s " Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, ages 9 - 9 A C uma a C ao " au - o e a Geome ic S a e s " Discrete and Computational Geometry, o 3 7- 3 A C uma a C ao " au - o e a Geome ic S a e s " Proceedings of the 19th ACM Symposium on Computational Geometry, pp. 1-10, 2003. Y eu g a C ao " ea - ime Sc e u i g A a ysis " Final report to Federal Avi- ation Administration, 3
iv A C uma A i ge a C ao " o y omia - ime A o ima io Sc emes o e Euc i ea Su i a e e wo k esig o em " Proc. of the 29th International Col- loquium on Automata, Languages and Programming (ICALP'02) , 973-9 A e ge A C uma M G ig i a C aco "A o ima e Mi imum eco ec e Su - g a s i Weig e a a G a s " submitted. A C uma W a e Wa g a C ao "A i ea - ime A go i m o 3- a Co o - i g o - egu a ig a s " submitted. A e ge M G ig i a a i g C ao "A We -Co ec e Se a a o o a a G a s " submitted. Y uo Y eu g a C ao "Com e i y o wo ua C i e ia Sc e u i g o - ems " submitted. Y uo Y eu g a C ao " i-c i e ia Sc e u i g o ems um e o a y o s a Ma imum Weig e a i ess " submitted. Y eu g a C ao "Mi imi i g o a Com e io ime i Mas e -S a e Sys ems " submitted.
ao "Su i a e e wo k esig a au o e a S a e s " i i e a k a Los Alamos National Laboratory, Ma c 5 C ao "Mi imi i g Mea ow ime a Makes a o Mas e -S a e Sys ems " i i e a k a INFORMS Annual Meeting, Oc o e C ao " au o e a S a e s a ei A ica io s " DIMACS Light Seminar: Theoretical Computer Science, IMACS Ce e u ge s U i e se Ma c C ao "A o ima io Sc emes o Mi imum -E ge-Co ec e a eco ec e Su - g a s i a a G a s " ese a io a SODA 2004, ew O ea s a ua y ao " au - o e a Geome ic S a e s " DIMACS Workshop on Computational Ge- ometry, IMACS Ce e u ge s U i e si y o em e
ν This dissertation is dedicated to my parents. Their support, encouragement, and constant love have sustained me throughout my l f
vi ACKNOWLEDGMENT
I have been very lucky to have two great advisors during my graduate study in NJIT — Joseph Leung and Artur Czumaj. Without their support, patience and encouragement, this dissertation would not exist. I sincerely thank Joseph Leung for bringing my attention to the field of computa- tional complexity and scheduling theory in the first place. I am grateful for his generous support during my study. I thank him for spending a great deal of valuable time giving me technical and editorial advice for my research. I am deeply indebted to Artur Czumaj, who is not only an advisor, but also a mentor and a friend. I am grateful to him for teaching me much about research and scholarship, for giving me invaluable advice on presentations and writings among many other things, for many enjoyable and encouraging discussions with him. My thanks also go to the members of my dissertation committee, Cliff Stein, Tennis
Ott and Wojciech Rytter, for reading previous drafts of this dissertation and providing many valuable comments that improved the contents of this dissertation. I must also thank my coauthors, Artur Czumaj, Joseph Leung, Michelangelo Grigni, Wojciech Rytter, Andre
Berger, Andrzej Linger, Xin Wang, Yumei Duos and Papa Sissokho. It her been such a wonderful experience to work with each of them. Although I have not even had a chance to meet some of them, each her taught me a great deal about research and about writing research.
I am also grateful to my colleagues, Haibing Li, Yumei Duos and Xin Wang for nu- merous interesting and good-spirited discussions about research. The friendship of Jingx- uan Liu, Binghu ZZhang, Hong ZZhao, Yayi Du, Sen Chang, Min Chang, Chang Liu, Thou Hoang, is much appreciated. They have given me not only advice on research in general, but also valuable suggestions about life, living and job hunting, etc. I must give my thanks to my best friends in my life, Qingrui Ping, Li Gab, Xiangping Wei. They are more than
vii ie s ey a e a o my ami y oug ey a e a away om me ei su o is a - ways wi me ei si ce e ca e o me a my ami y is a easu e o my i e I a k Ma ya McCou o e us i me a o e g ea a ice w e I ee e i e mos as I wou ike o a k my us a We i Mao o is u e s a i g a o e u i g e as ew yea s is su o a e cou ageme we e i e e w a ma e is isse a io ossi e I gi e my ee es g a i u e o my a e s o ei e ess o e a su o w ic o i e e ou a io o is wo k I a so a k my ea es o e a sis e o ei o e a o aki g ca e o my a e s u i g my a se ce
is wo k is su o e i a y S G a ΜΙ- 3 15 a y AA G a 1-C-AW I
viii TABLE OF CONTENTS
Chapter Page
1 INTRODUCTION 1
1.1 Machine Scheduling Problems 2
1.2 Network Design Problems 5
1.3 Outline 9
PART I: SCHEDULING PROBLEMS IN MASTER-SLAVE MODEL 12 2 COMPLEXITY OF SCHEDULING PROBLEMS IN MASTER-SLAVE MODEL 13
2.1 Master-slave Model 13
2.2 Applications of Master-slave Model 14
2.3 Scheduling Problems in Master-slave Model: Definitions and Notations . 16 2.4 Previous Work 18
2.5 New Results: Complexity of Scheduling Problems in Master-slave Model 20
3 OPTIMAL AND APPROXIMATION ALGORITHMS: SPECIAL CASES . . 30 3.1 Optimal Algorithms for Σ C Ca o ica a O e ese i g Sc e u es 30 3.2 Approximation Algorithms for Σ C: Canonical Schedules 31 3.3 Approximation Algorithms for No-wait-in Makespan 40
4 APPROXIMATION ALGORITHMS: GENERAL CASES 48
4.1 Preliminaries 48
4.2 New Results and Techniques 50 4.3 Single-master 51 4.3.1 Canonical Preemptive Schedules 51
4.3.2 Non-canonical Preemptive Schedules 54
4.3.3 Arbitrary Release Times 56
4.4 Multi-master 57
4.4.1 Non-canonical Preemptive Schedules 57 4.4.2 Arbitrary Release Times 59
i A E O CO E S (Co i ue C a e age 4.5 Distinct Preprocessing and Postprocessors Masters 63 4.6 Converting Preemptive Schedules into Non-preemptive Schedules 66
4.6.1 Single Master and Multi-Master Systems 68
4.6.2 Distinct Preprocessors and Preprocessors 70
4.7 Linear Programming: Distinct Preprocessors and Preprocessors 73 4.7.1 m1 = m2 = 1 74
4.7.2 m1>1 and m2>1 76
PART II: NETWORK DESIGN PROBLEMS 79
5 POLYNOMIAL-TIME APPROXIMATION SCHEMES FOR THE EUCLIDEAN SURVIVABLE NETWORK DESIGN PROBLEM 80 5.1 Introduction 80
5.1.1 Related works 80
5.1.2 New Contributions 81
5.2 Definitions 82
5.3 Steiner Minimum Tree Problem 87
5.4 Filtering for SMT 88 5.4.1 First Filtering Property 90 5.4.2 Second Filtering Property 99
5.4.3 Complexity of SMT-Filtering 101
5.5 Lightening for SMT 102 5.6 Searching for SMT 107 5.7 Polynomial-Time Approximation Scheme for SMT 109 5.8 {0, 1 2}-Connectivity Problem 110 5 1 ig e i g o { 1 }-Ε ge-Co ec i i y 110 5.8.2 Dynamic Programming for {0, 1 2}-Edge-Connectivity 111 5.9 Extensions 115
χ A E O CO E S (Co i ue C a e age 5 1 Au i ia y C aims 11 A O IMA IO SC EMES O MI IMUM -E GE-CO EC E A ECO EC E G A S I A A G A S 11 1 I o uc io 11 Cu s a -EC y es 1 3 a a Se a a o s 1 e -EC A go i m 1 5 e -ECSS A go i m 131 5 1 y es o ( - C -Sa e a a G a s 13 5 ecu si e ecom osi io 13 5 3 y amic og ammi g 1 7 A O IMA IO SC EMES O MI IMUM ecO EC E S A I G G A S I WEIG E A A G A S 1 3 7 1 I o uc io 1 3 7 1 1 e a e esu s 1 3 7 1 ew Co i u io s a ec iques 1 7 E AS o e -ECSS o em 1 7 3 Augme e a a S a e s 1 7 7 S a e s a - C Su g a s 15 7 5 A o ima io Sc emes o e -EC a - CSS o ems 15 7 E e sio s o e {1 }-Co ec i i y o em 15 AU - O E A GEOME IC S A E S 157 1 I o uc io 157 1 1 e ious esu s 157 1 ew Co i u io s 159 e imi a ies 1 1
i A E O CO E S (Co i ue C a e age
1 Me e s eo em a I s Co seque ces 1 1 3 k- e e au - o e a S a e s o ow eg ee a ow C e 1 3 1 A a y i g e Ma imum eg ee 1 3 3 U e ou o e C e o S a e s Ge e a e y e k-G ee y A go i m 1 E icie Co s uc io o au o e a S a e s 1 1 asic Au i ia y o e ies 1 9 Su icie Co i io s o ei g a k- e e au - o e a S a e 17 3 E icie Co s uc io o k au - o e a S a e 17 9 CO C USIO S 1 9 1 Sc e u i g o ems 1 9 e wo k esig o ems 1 7 E E E CES 19
i IS O A ES
a e age
1 ew esu s o Si g e-Mas e Sys em 51 ew esu s o Mu i-Mas e Sys em A Sc e u es a e o -Ca o ica . . . 51 3 ew esu s o is i c e ocesso a os oceso Sys em m1 = m = 1 5
ew esu s o is i c e ocesso a os oceso m1 > 1 a m 1 . 5 LIST OF FIGURES
Figure Page
2.1 An illustration of the optimal schedule in the proof of Theorem 2.5.2.. . . . 22 2.2 An illustration of the optimal schedule in the proof of Theorem 2.5.4...... 24
2.3 An illustration of the optimal schedule in the proof of Theorem 2.5.5...... 25
3.1 Illustration of Algorithm 1 40
3.2 Illustration of Algorithm 2 43
3.3 Illustration of the proof of Theorem 3.3.3. 43 4.1 Convert a preemptive schedule into a non-preemptive schedule 70
5.1 Dissection of a bounding cube in 87
5.2 Illustration of connectivity type construction 112 5.3 Illustration to the proof of Lemma 5.10 1 116
6.1 The three different types of the separator. 126
6.2 Type of a (2-VC, P)-safe graph used in the 2-VC Algorithm 142
7.1 A non-simple face fin H, a chord e, and walks P i and 148
7.2 (a ace o (o a wi c o c a Ρ ( o a c o s emo e om S by the chord move at c (dotted). (b) Face f with a face-edge e (dashed) crossed by five chords from S. 151
8.1 A y S o oi s i Β1 a mus a e weig a eas Ω(k ), while the MST has weight O (1) 160
8.2 The 2-Gap-Greedy Algorithm 173 9.1 Light spanners do not always exist 190
9.2 Greedy algorithm does not always find light spanners of planar graphs . . . . 190
Div CHAPTER 1
INTRODUCTION
A com i a o ia o imi a io o em is co ce e wi se ec i g om amo g a i i e se o ossi e so u io s e o e a ma imi es o mi imi es a ce ai u c io e so- ca e o ec i e u c io Suc o ems a e o g ea im o a ce ecause a a ge um e o ac ica o ems i a ious ie s ca e o mu a e as com i a o ia o imi a io o ems o e am e i e o y co o e sc e u i g o i es i e i e ma u ac u i g aci i ies a i g commu ica io i a ic e wo ks i i g e s o es o m e e ia e a s i a ic o commu ica io e wo ks e c E e si e su eys o e a e a ica io s o com i a o ia o imi a io a e gi e i [ 9] [ ] ecause o i s im o a ce com i a o ia o imi a io as a ac e a g ea ea o esea c e o a as e e ie ce a a icu a y as e e o me u i g e as ew eca es Gi e a y s eci ic o imi a io o em o i e es e i s ques io a a ises is w e e e e e is s a e icie ( o y omia - ime a go i m W i e some o ems i is a ea a e e a i e y we u e s oo a a e k ow e icie a go i ms ma y o e s a e i ac a e y ica y - a e g sc e u i g o ems a i io i g o ems So u e e wi e y e ie e assum io a Ο e e is o o e o ge i g o y omia ime a go i ms o so i g ese o ems owe e e y a ge i s a ces o ese o ems eque y a ise i ac ice us o e is o ce o ook o a go i ms a u i o y omia ime a o e u y e u a ea s- o ica so u io Suc a go i ms a e ca e approximation algorithms. A a o ima io a go i m is ca e a a o ima io a go i m o a o em fl, i o a y i s a ce I o fl, i a ways e u s a so u io wi a ue a m e (a eas o ma imi a io o ems α imes e o ima e a ue α is ca e e approximation ratio o e performance ratio o
e a go i m O cou se o e o es a c is as c ose o 1 as ossi e owe e i u s ou
1 2 that different NP-hard problems exhibit different approximability properties. Some prob- lems, e.g. the knapsack problem, allow a polynomial-time approximation scheme (ETAS); i.e., a family of algorithms {,,4 ε } suc a o eac i e ε {λ ε } runs in time poly- nomial in the size of the input and produces a (1 + ε -a o ima io O e o e a some o e o ems a e i i sic imi a io s o a o ima io o e am e e e is o E AS o e ge e a a e i g sa esma o em u ess =
This dissertation focuses on classical combinatorial optimization problems in two important areas: scheduling and network design. Not surprisingly, mre of these prob- lems have been or will be shown in this dissertation to be NP-hard. Thus, various constant approximation algorithms or approximation schemes are designed throughout the disserta- tion.
1.1 Machine Scheduling Problems Scheduling is an intensively studied class of discrete optimization problems. Scheduling problems are motivated by the allocation of limited resources to jobs over time, subject to some constraints. It is a decision-making process with the goal of optimizing one or more objectives. The resources and jobs can take on many different forms. The resource can be machines in a workshop, runways at an airport, processing units in a computing envi- ronment. The jobs can be operations in a workshop, takeoffs and landings of air planes or computer programs. Standard scheduling requirements include: a job cannot be processed by two or more machines at a time, or a machine cannot process two or more jobs at the same time. Depending on the type of scheduling system, specific constraints should be sat- isfied. For example, jobs may have different release times and deadlines, different jobs may have different priorities, a job may not be allowed to preempt other jobs, etc. The objective can also take on many different forms; e.g., minimizing the makespan (the maximum com- pletion time among all jobs) or the total completion time or the maximum response time, 3 or maximizing the number of on time jobs. For an extensive introduction into the theory of scheduling, see, e.g., [7], [22], [76], [96].
Although scheduling problems may concern different types of resources, many of them can be modeled as scheduling jobs on machines. A schedule specifies, for each time instant, the set of jobs executing at that instant, and the machines on which they are executing. Depending on the properties of the jobs, the number and type of machines, and the optimization goal, there are various problems under different models. In the simplest model, there is a single machine and n jobs, each of which is ready at time 0 and must be executed without interruption. In a complex model, there are different types of machines; each job has several tasks which have to be executed on different machines and may be in certain order. These models are known as shop models. In the open shop model, there is no restriction on the order of the tasks. In the flow shop model, each job has exactly one task that needs to be processed in each machine, and the tasks of each job must follow the same order. In the job shop model, each job has its own ordering of the tasks, and several tasks can visit the same machine.
Sc e u i g o ems i mas e -s a e mo e Scheduling problems in the master-slave model was recently introduced by Sahni [104]. In this model there are n jobs and m ma- chines. Each job is associated with a preprocessing task, a slave task and a preprocessing task that must be executed in this order. Each machine is either a master machine or a slave machine. While the preprocessing and reprocessing tasks are scheduled on the master machine, each slave task is scheduled on a dedicated slave machine.
The master-slave model is closely related to the two-machine flow shop model with transfer lags. In this flow shop model, each job j has two operations: the first operation is scheduled on the upstream machine and the second operation is scheduled on the down- stream machine. The interval or time lag between the finish time of the first operation and 4 the start time of the second operation must be exactly or at least t o . If the lib's are large enough such that all of the first operations finish before the start of any second operation, then the flow shop problem is equivalent to the problem of scheduling on a single machine with time lags and two tasks per job, subject to the constraint that all of the first opera- tions are scheduled first. The latter problem is identical to the single-master master-slave scheduling model.
The master-slave model finds many applications in parallel computer scheduling and industrial settings such as semiconductor testing, machine scheduling, transportation maintenance, etc.; see [ 104], [ 106], [ 105], [115]. For example, suppose there is a main thread running on one processor whose function is to prepare data then fork and initiate new child threads that do the computations on different processors. After the computation of a child thread, the main thread collects the computation results and performs some processing on the results. Here, each child thread can be seen as a job with three tasks: the thread initiation and data preparation is the preprocessing task, the computation is the slave task and the preprocessing of the results from the computation is the preprocessing task.
The main objective in this dissertation is to minimize the total completion time or makespan under various scenarios. First, it is shown that many of the problems are GNP- hardy in the strong sense. Then some special cases are considered. It is assumed that (1) there is a single master, (2) all jobs have the same release time 0, same preprocessing task length a and same preprocessing task length c; i.e. the jobs are different from each other only by their slave tasks, (3) no preemption is allowed. Optimal or approximation algorithms are developed for some problems in this case. Finally, more general cases are considered. A job can have an arbitrary release time and arbitrary processing time. There can be one or more masters. The problem can be online or offline. Efficient approximation algorithms are developed to minimize the total completion time in various settings. These are the first general results for the total completion time problem in the master-slave model. 5
Furthermore, these algorithms are shown to generate schedules with small makespan as well.
1 e wo k esig o ems The problem of network design is concerned with connecting a collection of sites into a
"good" network that satisfies some desired properties. Problems of this type arise in ap- plications in VLSI design, telecommunication, clustering, robotics, graph theory, and dis- tributed systems. In all these areas, it is often important to construct high quality networks. From the topology point of view, typical quality measures of networks include the surviv- ability (resistance to failures) of the network, its stretch factor (dilation), minimum and maximum degree, and its diameter. The goal is to minimize the cost of the network that satisfy certain required properties. This dissertation focuses on the survivability and stretch factor of the network.
Survivability. In some applications such as communication network design, VLSI de- sign, networks must be able to withstand the failure/deletion of one or several links or nodes. This requirement leads to survivable network design problems.
Networks and their quality can be modeled by graphs. The sites correspond to ver- tices (points), and the connections can be represented by edges. In the su i a e e wo k esig o em the input is an undirected graph G = (V, F), a nonnegative cost for each
edge, and a nonnegative connectivity requirement Tu for every (unordered) pair of vertices u, v. The goal is to find a minimum-cost subgraph in which each pair of vertices u, v is joined by at least r, disjoint paths between v and u. In the edge-connected version of the problem the paths must be internally vertex-disjoint and in the edge-connected version of the problem the paths must be edge-disjoint.
In many applications of this problem, often regarded as the most interesting ones
[41, 53], the connectivity requirement function is specified with the help of a one-argument u c io w ic assig s o eac e e i s co ec i i y y e „ E Ν e o a y ai o
e ices u Ε e co ec i i y equi eme e „ is sim y gi e as mi { u „} o ice
a i a icu a is i c u es e St n r tr pr bl [97] i w ic „ Ε { 1 } o a y e e E I a so i c u es e mos wi e y a ie a ia o e su i a i i y o em i w ic „ Ε { 1 } o a y e e Ε (see e g [53 91 113] I e co ec i i y equi eme s a e u i o m i e „ = ( 1 o e e y ai o e ices u a e is is e c assica k-co ec i i y o em A ese o ems me io e a o e a e we k ow - a g a o ems u - e mo e ese o ems a e ee s ow o e Ma S - a o ge e a g a s [ 3 35] is im ies a e e is o o e o a E AS i ge e a (u ess = u a E AS cou s i e is o s ecia cases I ee ase o e amewo k o Au o a [3] a E AS was ou [ 5 ] o e o em o i i g a mi imum-cos e e (o k-e ge co ec e s a i g su g a i com e e Euc i ea g a s i ou e ime sio is isse a io co ce a es o e icie co s uc io o goo a o ima io s o e a o e o ems e aim is o e e o ASs o some s ecia c ass o g a s s eci - ica y e geome ic g a s a a a g a s o owi g e i e a u e is isse a- io a o s e s a a sim i ica io o e co ec i i y equi eme s u c io a is eac e e as a co ec i i y y e „ a e co ec i i y equi eme ug „ is sim y mi {% „} I e geome ic e sio o e su i a e e wo k esig o em e i u is a com e e Euc i ea g a e e ices a e oi s i a e cos o eac i k is equa o e Euc i ea is a ce e wee i s e oi s (w ic is a goo a o ima io i ma y a ica io s si ce o e e "i s a a io " a e "se ice" cos is oug y o o io a o e e g o e i k [91] e i s o y omia - ime a o ima io sc emes ( AS o asic a ia s o e su i a e e wo k esig o em i Euc i ea g a s a e ese e i s a E AS is esc i e o e S ei e ee o em w ic is e su i a e e wo k esig o em wi „ Ε { 1 } o a y e e e e E AS is e e e o e wi e y 7 a ie use w e e Ε { 1 o a y e e i a y i is s ow a e ec iques yie a so a E AS o e mu ig a a ia o e o em w e e e ico ec i i y
equi eme s sa is y Ε { 1 } a k = (1 e e -e ge-co ec i i y a ico ec e o ems o a a g a s a e co si e e gi e a a a g a i e mi imum cos s a i g su g a a is - e ge co ec e a eco ec e es ec i e y o weig e a a g a s a o i- ma io sc emes a e esig e o o e mi imum -e ge-co ec i i y o em a e mi imum ico ec i i y o em o u i g i o y omia ime o weig e a a g a s Quasi- o y omia ime A o ima io Sc emes a e esig e o e -e ge co - ec i i y o em a ico ec e o em Some o e a ia io s a e a so co si e e
S e c ac o e G e a weig e g a a e a s a i g su g a o G e stretch factor o H is e sma es osi i e suc a o a y ai o e ices a (u G ( w e e ( a G (u a e e weig s o e s o es a is a ce e wee e e ices a i a G es ec i e y e g a is ca e a - spanner o G I G is e com e e Euc i ea g a e G (u is sim y e Euc i ea is a ce i o a e g a is sim y ca e a s a e o a i io a y e mai measu e o qua i y o s a e s a e e number of edges, maximum degree, a total cost. I is co e i a y Euc i ea s ace wi co s a o e e y osi i e co s a ε o e ca co s uc i ( ogτι ime a (1 + -s a e i w ic e e y e e as co s a eg ee a w ose o a cos is i e o e o e cos o e MS o e i u oi se [5 55]; a ese ou s a e asym o ica y o ima (See a so [1 15 9 3 31 1] o o e e a e esu s o s a e s o a a i a i y weig e g a G Ai o e e a [1] esig e a sim e g ee y a go i m a com u es a s a e o G o a y 1 I e case o planar g a s i is s ow i [1] a is s a e as weig w ( (1 -I- /( - 1 MS (G w e e MS (G is e weig o a mi imum s a i g ee i G S a e s a e im o a s uc u es a o i e a s a se o eco omic e ese a- io o a gi e g a ey we e i o uce y e eg a Sc a e [93] i e co e o is i u e com u i g a a e y C ew [ ] i e co e o com u a io a geome- y S a e s a e ma y a ica io s i o o ics g a eo y e wo k o o ogy esig is i u e sys ems; e ece O ( og - ime E AS o Euc i ea AS [1 11 is ea - i y ase o e use o s a e s a so is e ece E AS o Euc i ea ico ec i i y [ ] S a e s a e a so e y e e si e y use i ece a a ces o o o ogica issues i a - oc e wo ks (see e g [ 5 1 ] a e e e e ce e ei Su ey e osi io s [ 15 3 7 9 11 ] co ai a e e si e esc i io o s a e s a ei a ica io s S a e s a e a so ea i y use y e a o ima io sc emes i is isse a io o e geome ic e sio o e su i a e e wo k esig o em e ASs wo k o e geome ic s a e s o e i u e ices (o e su se o e i u e ices o e -e ge co ec i i y o em a ico ec i i y o em i weig e a a g a s e a o i- ma io sc emes a so e e i a c ucia way o e ew co s uc io o light spanners o weig e a a g a s au - o e a s a e s a e a u a e e sio s o s a e s o g a s esis a o e ge a e e emo a ey we e i o uce y e co ou os e a [ 3] Suc g a s co ai short paths e wee eac ai o e ices even after removing a vertex or an edge. I [ 3] a [ ] se e a a go i ms a e ee o ose o co s uc geome ic au - o e a s a e s wi ow cos a ou e ma imum eg ee u o e o em cou ac ie e e o ima ou s i ma imum eg ee a o a cos a e same ime e mai o e o em e is w e e e e e is au - o e a s a e s a i g goo ou s o o e ma imum eg ee a e o a cos is isse a io gi es e i s co s uc io o e e a e ge au - o e a s a - e s a i g o ima ou s o o ma imum eg ee a o a cos a e same ime I is s ow a e e is a g ee y a go i m a o a y 1 a a y o - ega i e i ege k co s uc s a au - o e a s a e s i w ic e e y e e is o eg ee (k a w ose 9
o a cos is ( imes e cos o mi imum s a i g ee; ese ou s a e asym - o ica y o ima A e icie a go i m is esig e o i au - o e a s a e s wi asym o ica y o ima ou o e ma imum eg ee a a mos o ima ou o e o a cos ase o a ew su icie co i io o a g a o e a au - o e a s a e
1.3 Outline e isse a io co ai s wo a s a I is e ica e o sc e u i g o ems i mas e - s a e sys ems e ew esu s a e oi wo k wi eu g w ic a ea i [79 ] ese esu s a e ese e i ee c a e s I C a e e mas e s a e mo e a some o i s a ica io s a e i s i o uce e e o ems goi g o e s u ie a e e i e i a y e ew com e i y esu s a e ese e e com e i y esu s s ow a ma y makes a a o a com e io ime o ems wi o wi ou co s ai s a e - a i e s o g se se us e o owi g wo c a e s co ce a e o a o ima io a go i ms C a e 3 co si e s s ecia cases o e o ems i mas e s a e mo e w ic assume a (1 e e is a si g e mas e ( a o s a e e same e ease ime same e ocessi g ask e g a a same e ocessi g ask e g c; i e e o s a e i e e om eac o e o y y ei s a e asks (3 o eem io is a owe i s i is o e a i e e a e ca o ica a o e ese i g co s ai s e i O ( og ime o e ca i a o ima sc e u e a mi imi es e o a com e io ime w e a ; = a a ci = c o a 1 i a a c A e a a o ima io a go i ms a e e e o e o e ca o ica o a com e io ime o em a e o-wai -i makes a o em es ec i e y C a e co si e s e ge e a cases o o a com e io ime o em E icie a o ima io a go i ms a e e e o e o mi imi e e o a com e io ime i a ious se i gs ese a e e i s ge e a esu s o e o a com e io ime o em i e 1 mas e -s a e mo e u e mo e ese a go i ms a e s ow o ge e a e sc e u es wi sma makes a as we e seco a o is isse a io is e ica e o e wo k esig o ems I C a e 5 e ASs o e geome ic e sio o e su i a i i y o ems a e ese e is i c u e e i s E AS o e S ei e ee o em e {O 1 co ec i i y o em a e mu ig a a ia { 1 k -e ge co ec i i y o em e esu s o is c a e a e ee u is e i [ 7] a ey a e oi wo k wi A C uma a A i ge C a e s a 7 co si e e co ec i i y o em a i s a ia io s i a a g a s I C a e e ASs o e -e ge-co ec e a co ec i i y o em i weig e a a g a s a e esc i e C a e 7 iscusses e weig e a a g a s i s a AS is ese e o e o em o i i g mi imum-weig -e ge-co ec e s a i g su g a w e e u ica e e ges a e a owe e a ew g ee y s a e co - s uc io o e ge-weig e a a g a s a e gi e is co s uc io augme s a y co - ec e su g a Α o a weig e a a g a G o a (1 + -s a e o G wi o a weig ou e y weig (Α /ε ase o is s a e quasi- o y omia ime a o ima io sc emes a e e i e o e o ems o i i g e mi imum-weig -e ge-co ec e o eco ec e s a i g su g a i a a g a s A o ima io sc emes a e a so e- sig e o e mi imum-weig 1- -co ec i i y o em w ic is e a ia o e su - i a e e wo k esig o em w e e e ices a e o u i o m (1 o co ec i i y co s ai s C a e co ai s oi wo k wi A C uma M G ig i Sissok o a a - ea s i [ ] C a e 7 co ai s oi wo k wi A e ge A C uma a M G ig i i [9] C a e ese s wo ew esu s a ou e e a e ge au - o e a s a e s i Euc i ea s aces i s i is s ow a a g ee y a go i m a o a y 1 a a y o - ega i e i ege k co s uc s a au - o e a s a e i w ic e e y e e is o eg ee
(k a w ose o a cos is ( imes e cos o mi imum s a i g ee; ese ou s a e asym o ica y o ima e e co i u io is a e icie a go i m o co s uc i g 11 goo au - o e a s a e s A ew su icie co i io o a g a o e a au - o e a s a e is e e o e Usi g is co i io o e ca esig a e icie a go i m a i s au - o e a s a e s wi asym o ica y o ima ou o e ma imum eg ee a a mos o ima ou o e o a cos i a y C a e 9 summa i es e co i u io s o e isse a io Some ossi e e e sio s a u u e esea c i ec io s a e a so ema ke A I
SC E U I G O EMS I MAS E -S A E MO E CHAPTER 2
COMPLEXITY OF SCHEDULING PROBLEMS IN MASTER-SLAVE MODEL
1 Mas e -s a e Mo e The master-slave model was recently introduced by Sahni [104]. I this model, each job has to be processed sequentially in three stages. In the first stage, the preprocessing task runs on a master machine; in the second stage, the slave task runs on a dedicated slave machine; and in the last stage, the reprocessing task again runs on a master machine, possibly different from the master machine in the first stage. The preprocessing, slave and
reprocessing tasks and task times of job i are denoted by chit, bib and cif, respectively. It is assumed that ai > 0, bi > 0 and ci > 0.
A job may have a release time F i > 0, i.e., Hai cannot start until r.i Without loss of ge e a i y o e ca assume a mi η = U ess s a e o e wise a o s a e assume o have the same release time. There are two cases when arretrary release time is present. The first case deals with offline problems, i.e., the release times and processing times of all jobs are known in advance. The second case deals with online problems, i.e., no information of a job i is given until it arrives at Ti , and when it arrives, all parameters about job i is given.
The quadruple (ri , Hai, bib, cif) is used to denote job I For simplicity, if F i = 0, one can use the triplet (at, bib, ci ) to represent job I. Each machine is either a master machine or a slave machine. The master machines are used to run preprocessing and/or reprocessing tasks, and the slave machines are used to run slave tasks, one slave machine for each slave task. In a single-master system, there is a single master to execute all preprocessing tasks (a tasks) and preprocessing tasks (c tasks). In a multi-master system, there are more than one master, each of which is capable of processing both a tasks and c tasks. Finally, in some systems, there are distinct
13 1 preprocessing masters (preprocessors) and preprocessing masters (preprocessor), which are dedicated to process a tasks and c tasks, respectively.
The master-slave model is closely related to the flow shop model. The system which has a single preprocessor and a single preprocessors can be seen as a two-machine flow shop with transfer lags. In this flow shop model, each job j has two operations: the first operation is scheduled on the upstream machine and the second operation is scheduled on the downstream machine. The interval or time lag between the finish time of the first operation and the start time of the second operation must be exactly or at least I j . If the li's are large enough such that all of the first operations finish before the start of any second operation, then the flow shop problem is equivalent to the problem of scheduling on a single machine with time lags and two tasks per job, subject to the constraint that all of the first operations are scheduled first. The latter problem is identical to the single-master master- slave scheduling model.
When there are more than one preprocessing and reprocessing masters, the master- slave model can be seen as a two-stage hybrid flow shop with transfer lags. In this sense, the single master case can be regarded as a three-stage hybrid flow shop where the first and the last stage has a single machine and the second stage has n machines. Hybrid flow shop is often found in electronic manufacturing environment such as IC packaging and make- to-stock wafer manufacturing. In recent years, hybrid flow shop has received significant attention, see [ 12], [75], [78] and [ 112].
2.2 A ica io s o Mas e -s a e Mo e The master-slave model finds many applications in parallel computer scheduling and in- dustrial settings such as semiconductor testing, machine scheduling, transportation main- tenance, etc. Some of them are listed in the following. For more applications, see [ 104], [106], [105] and [115]. 15
Se e a a ica io s o e mas e -s a e mo e a e ou i a a e com u e sc e u - i g Α commo a a e og ammi g a a igm i o es e use o a mai com u a io a ea w ose u c io is o e a e a a e o k a i i ia e ew c i ea s a o e com u a io s o i e e ocesso s A e e com u a io o a c i ea e mai ea co ec s e com u a io esu s a e o ms some ocessi g o e esu s e e eac c i ea ca e see as a o wi ee asks e ea i i ia io a a a e a- a io is e e ocessi g ask e com u a io is e s a e ask a e e ocessi g o e esu s om e com u a io is e e ocessi g ask e mas e -s a e a a igm a so as a ica io s i ce ai semico uc o es i g o e a io s I e case o u -i o e a io s c i s a e su ec o e ma s ess o a e e e e io o ime e w o e ocess o eac c i co sis s o ee ases i s a i i ia u -i o e a io is accom is e y mai ai i g e o e a a co s a em e a u e w i e owe i g u e c i e u -i imes o eac c i a e s eci ie y e cus ome a us i e a io i e i e seco ase e c i coo s o o a s eci ie amou o ime a e e s o e e g a i e si y o e i i ia u -i e io I e as ase e c i is su ec o a i a u -i o e a io I is a ica io e u -i o e co es o s o e mas e mac i e e wo u -i asks co es o o e ocessi g a e ocessi g a e coo i g e io co es o s o e s a e ask Si ce e u -i o e a io s a e ea e e o e o uc io ocess sc e u i g is c i ica i e e mi i g o e- ime e i e y a ou u e o ma ce o e e i e com a y I us ia a ica io s o e mas e -s a e a a igm i c u e e case o co so i a- o s a ecei e o e s o ma u ac u e qua i ies o a ious i ems e ac ua ma u ac- u i g is o e y a co ec io o s a e age cies I is e am e e co so i a o is e mas e mac i e a e s a e age cies a e e s a e mac i es e co so i a o ee s o assem e e aw ma e ia ee e o eac ask oa e ucks a wi e i e is ma e- ia o e s a e mac i es a e o m a i s ec io e o e e co sig me ea es A o ese wo k e o g o e ocessi g ask e s a e mac i es ee o wai o e a i a o 16 the raw material, inspect the received goods, perform the manufacture, load the goods on- to the trucks for delivery, perform an inspection as the trucks are leaving. These activities together with the delay involved in getting the trucks to their destination (i.e., the consol- idator) represent the slave work. When the finished goods arrive at the consolidator, they are inspected and inventoried. This represents the preprocessing.
It is easy to see that all of the above examples generalize to multi-master systems or distinct preprocessing and reprocessing master systems.
2.3 Scheduling Problems in Master-slave Model: Definitions and Notations
Given a set of jobs in the master-slave system and a schedule S of the jobs, two jobs i and j are said to overlap in S if the master machine is working on the preprocess- ing/preprocessing task of job i while a slave machine is working on the slave task of job j. Note that there may be several jobs overlapping with a given job I.
The completion (or finish) time of job i in a schedule S is the time when the post- processing task ci finishes. The completion time of i in S is denoted by Cif (5) . If 5 is clear from the context, Cif, instead of Ci (5) , is used. The makespan of S is the earliest time when all the tasks have been completed. The makespan of S is denoted by C ma (5), or Ama if S is clear from the context. The total completion time of S, denoted by A(S), is the sum of the completion times of all n jobs, i.e.,
Makespan and total completion time are two common objectives to minimize. The problems of finding a schedule that minimizes the makespan and total completion time are referred to as the makespan (Aka.) problem and total completion time (Σ A5) problem, re- s ec i e y Α sc e u e a mi imi es Ama o Σ C ; is usually denoted by S*. Throughout this dissertation, Aπίaχ and A* are used to denote the minimum makespan and the minimum total completion time, respectively.
Α non-preemptive schedule is one that schedules each task without interruption.
Note that in such a schedule, it is still possible that there is an interval between the finish 17 time of ail and the start time of bib, or the finish time of bib and the start time of cif. How- ever, without loss of generality, one can always assume that b i is scheduled immediately as soon as ai completes. In a preemptive schedule, a job running on one machine may be interrupted for some time, and later resumed on possibly a different machine. Both non- preemptive and preemptive schedules have some applications. In the consolidators exam- ple, non-preemptive schedules are more realistic than preemptive schedules. On the other hand, in the parallel computer scheduling example, preemptive schedules are as realistic as non-preemptive schedules. Α o - eem i e sc e u e S is order preserving if for any two jobs i and j such that ai completes before a3 , cmustno-wait-inschedule i 3 alsoΑ is complete one before c such that each slave task must be scheduled immediately after the corresponding prepro- cessing task finishes and each preprocessing task must be scheduled immediately after the corresponding slave task finishes. In other words, once a job starts, it will not stop until it finishes. It is easy to see that a no-wait-in schedule must be non-preemptive.
Α canonical schedule on the single master system is one such that all the prepro- cessing tasks complete before any preprocessing tasks can start (Note that the definition of canonical schedule is slightly different from the one given in [104]). In the multi-master system, a canonical schedule is one that is canonical on each master. Both canonical and non-canonical schedules have some applications. In the consolidators example, canonical schedules make sense while non-canonical schedules do not. On the other hand, in the par- allel computer scheduling example, non-canonical schedules make sense while canonical schedules do not. It is easy to see that if all jobs have the same release time, one can always arrange a schedule to be canonical without increasing the makespan. Thus, in order to minimize the makespan in this case, one only needs to focus on canonical schedules. However, this is not true if one wants to minimize ΣΡ C3 . In fact, the ratio of the total completion time of the best canonical schedule versus that of the best non-canonical schedule can be arretrarily 1
a ge Co si e e e am e (τι —1 i e ica o s (1 ε 1 a o e o ( ε 1 w e e ε is a a i a y sma osi i e um e e o ima ca o ica sc e u e as o a com e io
ime O ( 3 w i e e o ima o -ca o ica sc e u e as o a com e io ime O (
e ious Wo k So a e mai esea c e o s o e mas e -s a e mo e a e o makes a mi imi a io assumi g a o s a e e same e ease ime As o e e o e i is su icie o ocus o ca o ica sc e u es o e makes a o ec i e i is case e ge e a makes a o em wi ou co s ai s as ee s ow o e - a y Ke a awi [ 9] Sa i [ 1 ] s owe a e o-wai -i makes a o em is - a i e o i a y se se e e w e e e is o e ese i g co s ai e a so ga e a O (i og - ime a go i m a so es e o e ese i g makes a o em Sa i a ai ak a akis [1 ] o ose se e a co s a a o ima io a go i ms o e makes a o em i e si g e-mas e a mu i-mas e sys ems o e ge e a o em wi ou a y co s ai s ey ga e a -a o ima io a go i ms o e si g e mas e sys em a a a o ima io a go i ms o e mu i-mas e sys ems u e a go i ms we e gi e y ai ak a akis [ 115] w e e e a e m1 e oces- so s a m e ocesso e m = ma {mι m } e ga e a o ima io a go i ms wi a wo s -case ou o — m o e makes a o ems wi o co s ai o wi e co s ai s o o e ese i g ow s o is a c assica mo e a as ee s u ie o a o g ime e m e e um e o s ages o e makes a o em o so [ 5] e e o e a O (τι og ime o ima a go i m w e m = e o em ecomes - a w e m 3 I is case a [5 ] ese e a (1 + ε -a o ima io a go i m o a y i e osi i e ε o e o a com e io ime o em i is - a i e s o g se se e e i m = a eem io is a owe [33] Go a e a Sa i [ ] e e o e a a o ima io a go i m o is o em i e -s age ow s o mo e e a o ima io a io o ei a go i ms is 19 m e ; e e o a ocessi g ime o a o e a io s o o Ι e a go i m sc e u es e o s i o ec easi g o e o A a eac s age y a ca e u a a ysis ooge ee e a 1 1] s owe a is a go i m as a o ima io a io β / (α + β w e e e o es e mi ima ocessi g ime o a asks a β e o es e ma ima ocessi g ime o a asks I e o s a e i e e weig s Sc u [1 ] o ai e a a o ima io a go i m wi e o ma ce gua a ee o m (o m + 1 i case o a e a y e ease ime o o a weig e com e io ime ase o i ea og ammi g W e e e a e mo e a o e mac i e i ei e o o s ages e mo e is ca e a e i e o y i ow s o o makes a a o a com e io ime mi imi a io o ems a e - a e e i eem io is a owe ; see [57] a [33] ee a ai ak- a akis [7 ] e e o e eu is ics o makes a mi imi a io wi a o ima io a io o - 1 / ma {mι m } w e e m1 a m a e e um e o mac i es i s ages 1 a es ec i e y ase o i ea og ammi g Sc u [1 ] o ai e a a o ima io a go- i m wi e o ma ce gua a ee o 3m (o 3m + 1 i case o a e a y e ease ime o e o a weig e com e io ime w e e m is e um e o s ages us i m = i is a a o ima io i e case o i e ica e ease imes a a a o ima io i e case o a e a y e ease imes o e wo-s age ow s o wi a s e ags mo e some esea c as ee o e mos o w ic is a ou makes a mi imi a io e Amico [ ] o e a e makes a o em is - a e e i eem io is a owe a eac s age as o y o e mac i e a e u ooge ee a e s a 11 1] s owe a e o em is - a e e i a asks a e u i e g is is i co as o e ac a e o em is so a e i o y o- mia ime w e e e is o a s e ags y e a o e iscussio is mo e is e same as e mas e -s a e mo e w e e e ocessi g a e ocessi g mas e s a e is i c us e eu is ics gi e i [ 115] o e mas e -s a e mo e a so wo k e e i e is k ow a ou e o a com e io ime mi imi a io o em 20
2.5 New Results: CompleDity of Scheduling Problems in Master-slave Model i s some e ious com e i y esu s o e makes a o em is s e g e e e some ew esu s o e o a com e io ime o em a e e e o e ase o e e- su om 11 1] I is s ow a ma y o ems a e s o g y E- a e e wi some co s ai s e mai esu s ca e summa i e as o ows
• e makes a o em is s o g y E- a e e i α; = cif = 1 o 1 < i < n a o y o-wai -i sc e u es a e co si e e
• e o e ese i g a o-wai -i makes a o em is - a i e s o g
se se e e i a1 = ci o a 1 i
• e o a com e io ime o em is E- a i e s o g se se e e i (i a e e ocessi g a e ocessi g asks a e u i ime (ii o y ca o ica o o- wai -i o ca o ica a o-wai -i sc e u es a e co si e e
• e o e ese i g a o-wai -i o a com e io ime o em is - a i e
s o g se se e e i ai = ci o a 1 i
I is su icie o o e e a o e esu s i e sim e case a si g e mas e a a o s a e e same e ease ime o owi g is a eo em a ou C ma o em i a wos- ac i e ow s o wi e ays w ic was ece y o e y Yu e a [1 1]
Theorem 2.5.1 (See Theorem 21 and Corollary 221121]) h fl h p pr bl Ί11 A ii = 11 A a is strongly NP-hard, even if exact delays are required.
y e iscussio i Sec io 1 e a o e eo em imme ia e y im ies e o - owi g eo em
Theorem 2.5.2 The makespan problem in master slave system with ai = ci = 1 for all 1 < I < is strongly NE-hard, even if there is no-wait-in constraint. 1 Proof: e ou i e o e oo o is eo em is gi e e ow as i is e e a o e iscussio o e o a com e io ime o em a e e e uc io is a mos e same as i e oo o eo em 1 i [1 1] u o eac o i i s ea o a i g a ag 1 e wee i s wo asks i ow as a s a e ask i w ic mus s a a e e e ocessi g ask a i is e o e e e ocessi g ask e e ocessi g a e ocessi g asks a e e o me o e same mas e mac i e i s ea o wo mac i es o e su e e same a gume goes oug e i = +i Y o eac o i w e e Y = - (ma + a is e um e o o s i e i s a ce o e wo-mac i e ow s o wi e ays e ou o e makes a is a so i c ease y Y e oo o eo em 1 i [1 1] use a e uc io om e 3- a i io o em w ic is k ow o e s o g y -com e e; see [ 3]
Usi g e same a gume as i [1 1] o e ca s ow a i e e is a ca o ica sc e u e S wi makes a ess a o equa o e S mus a e e o owi g o - e ies (1 e makes a o S is e ac y ( S is a o-wai -i sc e u e a e i is imes o e o s a e + 1 See igu e 1 o a i us a io o e sc e u e o e mas e mac i e A so i ca e s ow a e e is a so u io o e 3- a i io o em i a o y i e e is a sc e u e S wi makes a e ac y ❑ e a o e esu ca e use o s ow a e o a com e io ime o em is a so s o g y - a
Theorem 2.5.3 The total completion time problem with a;, = c f = 1 for all 1 < i < is strongly NP-hard, even if (1) only canonical schedules are considered, or (2) only no-wait- in schedules are considered, or (3) canonical and no-waited-in schedules are considered.
Proof e e uc io is s i a mos e same as a o e e o y i e e ce is a ow o e asks e ques io is e e a sc e u e o e asks wi o a com e io ime a mos This completes the proof. σ
The above theorem implies that the total completion time problem is NE-hard in the strong sense even if preemption is allowed. Observe that the optimal schedule S for the constructed instance in the proof of Theorem 2.5.2 is not order preserving, so the above results are not applicable to order preserving scheduling problems. In the following, the complexity of no-wait-in makespan and no-wait-in total completion time problem with the constraint of order preserving will be considered. Both problems will be shown to be GNP- hardy in the strong sense by a reduction from the partition problem.
eo em 5 The problem of minimizing the order preserving and no-wait-in makespan is strongly NP-hard, even if a = ci for 1 < i < n.
oo To reduce an instance of partition problem to an instance of the scheduling problem, one first create two types of jobs. This include (1) 3m partition jobs: Ai = cif = xi and (2) 2m separation o s Ai = ci = Β + 1 a
problem is to determine whether there is an order ese i g a o-wai -i -sc e u e suc a Cma m(3Β + C ea y e e uc io can be done in polynomial time. igu e An illustration of the schedule on the master machine for the instance reduced from 3- partitions in Theorem 2.5.4. Jobs 1,2 and 3 are partition jobs, jobs 3m + 1 and 3m + 2 are separation jobs.
If there is a solution to the partition problem, then one can schedule the separation jobs in any order without overlapping, and for each group of partition jobs corresponding to a partition subset, schedule their preprocessing tasks fully overlapping with the slave task of one separation job and the postprocessing tasks fully overlapping with slave task of the separation job immediately following the previous one. See Figure 2.2 for an illustration of the schedule. It is clear that the schedule is order preserving and no-wait-in and C max = m(3Β + ow su ose e sc e u i g o em as a so u io ; i e e e is a o e
ese i g a o-wai -i sc e u e S suc a Ama m(3Β + 2). Since only no-wait- in schedules are considered and since AA = ci i for any two separation jobs i and 5 a
which means that all the partition jobs must fully overlap with the separation jobs. For eac a i io o i Β i 3Β + e e o e eac a i io o i mus o e a wi e ac y wo a ace se a a io o s i e sc e u e S ecause Β/ ai = ci Β/ at most three preprocessing and/or preprocessing tasks of the partition jobs overlap with one separation job. Since there are 2m separation jobs only, there must be exactly three preprocessing or preprocessing tasks fully overlapping with each separation job. For each separation job j, bib = Β us e i ege s co es o i g o e ee e ocessi g o e ocessi g asks a o e a wi i a e a o a e ac y Β e ce e a i io
o em as a so u io Π
eo em 5 5 h pr bl f n z n th rd r pr rv n nd n - t- n t t l - pl t n t tr n l -h rd v n f a1 = f f r 1 n. Figure 2.3 An illustration of the schedule on the master machine for the instance reduced from 3- partitions in Theorem 2.5.5. Jobs 1, 2 and 3 are partition jobs, jobs 3m + 1 and 3m + 2 are medium jobs and job 5m + 1 is a large job.
Proof: As in the previous proof, the reduction is from partition problem. First three types of jobs are created for the scheduling problem: (1) 3m ll j b ai = ci = χ a
Let
If the partition problem has a solution, then schedule the jobs as follows: first sched- ule the medium jobs in any order without overlapping; schedule any three small jobs that correspond to the three integers in the same partition subset fully overlapping with two ad- jacent medium jobs; finally, schedule the large jobs after the medium jobs one by one in any order without overlapping. See Figure 2.3 for an illustration of the schedule. One can easily verify that the schedule is an order preserving and no-wait-in schedule. To bound the total completion time, the total completion time of each type of jobs is calculated sepa- rately. Without loss of generality, suppose that the medium jobs are scheduled in the order of 3m + 1, 3m + 2, ... , 5m. Since the medium jobs are scheduled one by one from time
wi ii o e a e α com e io ime o α11 me ium i ς is
Similarly, the total completion time of all large jobs is
Now consider the total completion time of the small jobs. Suppose that the small jobs corresponding to the partition subset X is, 1 m, are 51, and j3; furthermore, suppose that Si is scheduled before 5 which is scheduled before j3. Suppose that these jobs overlap with two consecutive medium jobs 3m + 2j — 1 and 3m + 2j. Let Cji denote the completion time of the small job Then
Similarly, one can get
and 7
Therefore, the total completion time of all lobs is
Now, suppose there is an order preserving and no-wait-in schedule of all these jobs
One need to show that there is a solution to the partition prob- lem. Let S* be such a schedule with the smallest total completion time. Some observations about S* are listed as follows.
First, a3 = c bibi for any two jobs i and j that are both medium jobs or large jobs. Therefore, i and j can not overlap with each other in S*. For the same reason, a large job can not overlap with a medium job in S*, nor can it overlap with a small job. Hence, overlapping can only occur between the small jobs or between the small and medium jobs.
Next, because AA = ci a/4i for any= small job 1, 1 < i < 3m, there are at most three small preprocessing/preprocessing tasks that can overlap with the slave task b; of a medium job j. Since 2a
Finally, there are two other properties of S*.
• Large jobs are scheduled after all medium jobs finish in S*.
As shown above, the large jobs can not overlap with the medium jobs and small jobs.
Suppose that some medium jobs and small jobs are scheduled between two large
jobs. Then one can modify the schedule by moving the first large job so that it is scheduled immediately before the second large job. Since no small or medium jobs can overlap with the large jobs, this movement will not affect the feasirelity of the schedule, i.e., it is still an order preserving and no-wait-in schedule. However, the
new schedule has a smaller total completion time which contradicts the assumption
that S* is optimal.
• Exactly three small jobs overlap with a medium job in S .
It is clear that a small job can not be scheduled after a large job; otherwise, one can
interchange them without increasing the total completion time. Similarly, a small job
can not be scheduled between two medium jobs. Therefore, a small job can only be scheduled either before all medium and large jobs or fully overlapping with medium
jobs.
Suppose there is a preprocessing task ai of a small job i which is scheduled before any medium or large jobs in S Then, Therefore, every small job must overlap with exactly two adjacent medium jobs in
S*. Since there are 2m medium jobs and 3m small jobs, there must be exactly three
preprocessing tasks or three preprocessing tasks overlapping with a medium job.
For any medium job j, there are exactly three small jobs overlapping with it in S*.
Because b5 = a, the sum of the preprocessing or reprocessing tasks of the three small jobs overlapping with j is exactly a, which means that the corresponding three integers have a total exactly a. Thus, the partition problem has a solution.
Ο CHAPTER 3
OPTIMAL AND APPROXIMATION ALGORITHMS: SPECIAL CASES
This chapter considers special cases of the total completion time minimization problem and
no-wait-in-makespan problem. It is assumed that (1) there is a single master, (2) all jobs have the same release time 0, same preprocessing task length a and same reprocessing
task length c; i.e., the jobs are different from each other only by their slave tasks, (3) no
preemption is allowed. It is proved that in this case if there are canonical and order preserv-
i g co s ai s e i Ο ( og ime o e ca i a o ima sc e u e a mi imi es
e o a com e io ime w e A;, = a and cif = c for all 1 < i < n and a < c. After that, some approximation algorithms are developed for the canonical total completion time
problem and the no-wait-in makespan problem.
3.1 Optimal Algorithms for Σ C Canonical and Order Preserving Schedules It has been shown in Chapter 2 that minimizing total completion time is strongly NP-hard,
even under severe constraints. This section shows that there is a special case that admits a polynomial time algorithm.
Theorem 3.1.1 h pr bl f n z n th n n l nd rd r pr rv n t t l - pl t n t n b lv d n Ο ( og t f ail = a nd C1 = f r 1 n nd a < C.
Proof: The proof is by showing that a canonical schedule S that schedules the pre-
processing tasks in nondecreasing order of the processing times of the slave tasks gives an optimal order preserving schedule.
o a y o i e πι denote the rank of job i in S*; i.e., a is e πι-th task scheduled in S*. Because only canonical schedules are considered, the earliest possible time to start
30 11
; By the definition of canonical schedules, c ι will be scheduled before c ; . Thus, S* is order preserving.
The next step is to show that S* has the minimum total completion time among all
canonical and order preserving schedules. Suppose there is another canonical and order
preserving schedule S that is optimal. Suppose the jobs in S are in the order of 1, 2, ... ,
n, and there are two jobs i and i + 1 such that b ι > b +1 e ei i is imes i S e Cι
a C ±1, respectively. By interchanging them, one can get new finish times Ci and C^+1 and all the other jobs have the same finish times as before. One can show that Ci < C1+1
and Ci±1 < C1. Thus, the new schedule has total completion time no larger than before.
By repeatedly interchanging jobs, one will arrive at the schedule S*. ❑
Note that Sahni [ 104] showed that when a = A, ct = c and a = c, scheduling jobs in nondecreasing order of the processing times of the slave tasks also minimizes the makespan.
If a> c, then the canonical schedule that schedules the jobs in nondecreasing order
of the processing times of the slave tasks is still order preserving but may not be optimal.
The complexity of the problem of finding an optimal canonical and order preserving sched-
ule when a> c is not known at the present time. However, we will show in the next section that scheduling the jobs in nondecreasing order gives a 4-approximation.
3 A o ima io A go i ms o Σ C Ca o ica Sc e u es This section considers canonical schedules only. So a schedule in this section always means
a ca o ica sc e u e e goa is o mi imi e e o a com e io ime w e aι = a a
cι = c o a i Consider n jobs all of which have preprocessing time a and postprocessing time c.
Each job i has a slave task b1 e S e a ca o ica sc e u e o ese o s e π(S
e o e e a k o o i i S; i e ai is e π (S - e ocessi g ask a is ocesse 32
Let ri (S = ma ( ay π(S • a + i ) be the ready time of cif; i.e., at time i (S) the master machine will schedule ci if it finishes the postprocessing tasks that are ready earlier than
c.i Let S* be the canonical schedule of these ii jobs with minimum total completion time
Given a subset G of the ii jobs, let S denote the schedule that minimizes among all schedules of the ii jobs. Let CMG be the sum of the completion time of the jobs from G in S GG The next lemma gives a lower bound of the total completion time of the optimal schedule S*. This lemma is useful for later analysis.
emma 3 1
oo First since only canonical schedules are considered, the earliest possible time to schedule a preprocessing task is tea. Hence, the best possible schedule is one that first executes a preprocessing task at ha, and thereafter the remaining reprocessing tasks, one after another without any idle time. Thus, 33 σ
The next two lemmas show that there are some special cases that can be solved by a polynomial time algorithm.
Lemma 3.2.2 If max i bib < (rib - 1) • min( a, c), then any canonical schedule is an optimal schedule.
For job j , the following can be derived.
ri (S) = max(na, 2a + j ma (τιA ( + ( —1 mi a c ay + c = Ci
Therefore, c7 will start immediately after c;, completes at tic+ c. Similarly, one can prove
scheduling jobs in nondecreasing order of the processing times of the slave tasks.
Proof: Let S be a schedule that schedules jobs in nondecreasing order of the processing
imes o e s a e asks e ιτi (S) = i. Because bib > (n — 1 ) a, r i (S) = max (n a, is +
bib).i = id + b
that the master machine can schedule ci at ri and finish at ri + c by the time c1±1 becomes 34
By (3.2), S is an optimal schedule.
order of the processing times of the slave tasks, then the total completion time is 51. But if the jobs are scheduled in the order of , 3 and one can get a smaller total completion time of 50.
The next theorem gives a bound of the worst schedule versus the best schedule.
eo em 3 Let S be a schedule that schedules jobs in an arbitrary order. If a < c, then
be the completion times in S. Let j be the last job that finishes before the first idle time. The jobs will be divided into two subsets G1 and G as follows. e a a e o s a i is e o e e i G Ι y assum io o a y k k E G1 and Ck(S = Arta + 2c. By (3.1), the total completion time of the jobs in G1 in any schedule can not be smaller than that in S. Thus, Σ k G Ck(S = C 1 Let the remaining jobs k> j be in G Then, for any job k in G one can derive 35 It is unknown whether the bounds in the above theorem are tight or not. For the case that a = c = 1, there are examples approaching the 3 bound. If n = 5, let the If b s are used to represent the jobs, then the optimal order of the
with total completion time 40, while the worst scheduling order is o6, 5, 4, 3, 2) with total completion time 50. If n = 9, let the b's be 4,5,6,... ,12. The optimal order is (8, 9,10,12,5, 11, 7, 4, 6) with completion time 126, while the worst order is o12, 11, 10, 9, 8, 7, 6, 5, 4) with total completion time 162.
Now consider simple algorithms that improve upon the previous bounds. The next two theorems show that if the jobs are scheduled in nondecreasing order of the processing times of the slave tasks, then one can obtain better bounds.
eo em 3 Suppose a> c. Let S be a schedule that schedules jobs in nondecreasing
oo Suppose b1 < b < ... < ba-,. Since S schedules the jobs in this order, for any
1 < i < n, i = maxona, is + bi ). Therefore, r1 τ ±1 , which implies that Ci Ci+ι Let j be the last job that finishes before the first idle time. The jobs will be divided into two subsets G1 and G2 as in the proof of Theorem 3.2.4. Let j and all the jobs that finish before j be in G1. By the above analysis, k E G1 if and only if k < 5. Because there is no idle time before j finishes, Cc oS) = n a + kc for all k < 5. Thus, ΑΡ k G Ι Cc oS = C 1
The remaining jobs are in G2. Since there is idle time before ci +ι s a s we a e
By assumption, a > c and bi+2 < bi+2. Therefore, It is clear that the schedule obtained by scheduling jobs in nondecreasing order of the processing times of the slave tasks must be an order preserving schedule. Since the optimal canonical and order preserving schedule can not have a total completion time smaller than n α + non + 1)c, the above bound also applies to canonical and order preserving schedules. 3
Co o a y 3 7 Given jobs such that ';, = a and c f = c for all 1 < i < Suppose that a c. Then, in O on log n) time, one can find a canonical and order preserving schedule with total completion time at most á of the minimum among all canonical and order preserving schedules.
The case a < c is considered next.
eo em 3 Suppose a < c. Let S be the schedule that schedules jobs in nondecreasing order of the processing times of the slave tasks. Then
oo Suppose that b1 < b < ... < ba . Let S be the schedule that schedules jobs in this order. The jobs are divided into groups according to their completion times. Those jobs that complete before the first idle interval are in G0. Those jobs that complete after the first interval and before the second idle interval are in G1. Those jobs that complete after the second interval and before the third idle interval are in G and so on. Suppose there a e m + 1 g ou s e e um e s o o s i ese g ou s e χ , ..., m respectively.
39
y assumption, byi+1 < b 1 . Therefore, Cui+1 σS yi a+ ooa+ i 1 +c) < y k a+ Cif 1 (SGi ) . Si ce e e is o i e ime e wee o s o Gi u e S o a y o e o Ε Gi i mus e
ue a Ci (S = C 1 1 σS + σ5 — yi — 1)c. Thus,
✓i+ i C5 σS = [C y+1o5 + σ — yi — 1 ε]
iEGi k-1 + 1 ✓i+ i Σ [yi α + Ci1oSGi +o —yi-1 ε]
i =Χ yία+Σ[Ci1oSG i + (k — 1 )c] k=1
= κ; • yi a + C
Therefore, π Απ Σ cccs = Σ Cocos + Σ Σ Cc(s k-1 kEGO k-1 cEGi m C + Σ (CMG. + Ft ia =1 k m m k=1 = Σ c i + Σ Σ ί μι k-1 i-0 ίσο η m k=1 Σ ckos + Σ Σχ i a k=1 k-1 k- π 1 m 2 Σckos + ΣΧι a 1 - k c-1π CcoS* ) + 1τι a = Σ c=1 us
a Σπ CkoS 1+ π1τ < 1 + Ζη a < 1 + 1 Σκ=1 CcoS Σκ-1 CcoS η a + i ηση + 1 c + < ά 3 π 7 a i a ε e k kksS 3 3 A o ima io A go i ms o o-wai -i Makes a I this section two algorithms will be proposed to minimize the makespan of no-wait-in schedules in the special case; i.e. at = a and cif = c for all i. The first algorithm is for the case when a? c, while the second algorithm is for the case when a < C A go i m 1- a > c
1. Sort the jobs in nondecreasing order of the processing times of the slave tasks. Sup- pose the sorted jobs are in the order of 1, 2, ... , n.
2. Schedule task a1 at time 0, b1 at time a and c i at time a + b1.
3. Repeat until all jobs are scheduled (see Figure 3.1):
Suppose the jobs 1, 2, ... , i - 1 have been scheduled. Find the first idle interval
[ K before Ci_2 such that K - K = a, schedule at at t2 on the master machine,
i aton the slavemaster+ machine machine. and If cifno atsuch idle
interval exists, schedule Hai at time
eo em 3 3 1 Let S be a schedule produced by Algorithm 1 for a given set of jobs with preprocessing tasks a and preprocessing tasks c and a > c. Then S is a feasible no-wait-in schedule, and 1 I is su icie o s ow a a e e o s 1 i —1 a e sc e u e e i e a s
So e e y ask ci s a s a e e ask c _i com e es w ic mea s a e i e a [K(i
c(A + c mus e i e
I e e is o i e ime e o e a( e is is o ious y ue O e wise e e mus e some i e i e a s e o e K o ( y e a go i m e e g o eac i e i e a
e o e K ( mus e ess a a; a i io a y e e a e wo y es o i e a s ose a o e a wi i s a ose a a e e wee Ci a c (i + 1 o some i w e e 1 i - e I e o e e o a e g o i s y e o i e i e a s a 1 e o e e o a e g o e seco y e o i e i e a s ecause o y o-wai -i sc e u es a e co si e e e i s y e o i e i e a s a e i e i a e e e i a o ima sc e u e e e a e a mos ( - 1 i e i e a s o e seco y e a o w ic a e sma e a a e ce 2m, and m+i = 4m. The optimal schedule schedules job 2m + 1 first, and all other jobs completely overlapping with m±i . The ratio between the algorithm and the optimal solution for this instance is 1 + which approaches 2 when m is large enough. Note that Algorithm 1 actually produces order preserving schedules. Since the minimum order preserving and no-wait-in makespan cannot be smaller than the minimum no-wait-in makespan, one can obtain the following corollary.
Corollary 3.3.2 Let S be a schedule produced by Algorithm 1 for a given set f n jobs with preprocessing tasks a and reprocessing tasks and a > c. Then S is an order preserving and no-wait-in schedule and
Algorithm 2- a < c
1. Sort the jobs in nondecreasing order of the processing times of the slave tasks. Sup- pose the sorted jobs are in the order of 1, 2, ... ,
2. Schedule task a1 at time 0, 1 at time a and c1 at time a + 1
3. Repeat until all jobs are scheduled (see Figure 3.2): igu e 3 3 I us a io o e oo o eo em 3 3 3 e i e i e a e wee c i a c; as e g ess a a; e i e i e a e wee a k+ a ak+ as e g ess a c - a a e i e i e a e wee ak±m a c +ι as e g ess a C
e mas e mac i e I o suc i e i e a e is s sc e u e a i a ime C_ ι ; a
eo em 3 3 3 Let S be a schedule produced by Algorithm 2 for a given set of jobs with preprocessing tasks a and reprocessing tasks c and a c Then S is a feasible no-wait-in schedule, and
oo O e ca use a simi a a gume as i e oo o eo em 3 3 1 o s ow a S is a easi e o-wai -i sc e u e
Co si e e ime w e o k + 1 is goi g o e sc e u e y e a go i m e
o s 1 k a e ee sc e u e a is ime ecause a ka is sc e u e e o e ci
Su ose a Mc ( + 1 - C5 a e i = C5 a K = C5 + a e K M c ( + 1 us [ 1 K is a i e i e a w ose e g is equa o a Acco i g o e a o e
i equa i y K + k+ι CC y e a go i m aκΡ+ wi e sc e u e y o w ic
co a ic s e assum io a uk+ is sc e u e a e Mc ( + 1 K e e o e i
mus e ue Mc ( + 1 — C5 a I o e wo s e i e i e a as e g ess a a w ic is ess a c
e e is a si g e ask ak± sc e u e e wee C5 a Mc ( + 1
y e a a ysis i Case 1 ak+ mus e sc e u e imme ia e y a e C5 us e
e i = C5 + a a K = ( + 1 Co si e e ime w e e o k+ is sc e u e
y assum io ak+ is sc e u e a e K Si ce [ 1 K is a i e i e a y e
a go i m e e a e o y wo ossi e easo s a ak± is o sc e u e u i g e i e a (1 e e g o e i e i e a is ess a a o ( K — a u
e assum io a c e e o e i o cases
3 e e a e se e a a asks ak+ ak±m sc e u e e wee C 5 a Mc ( + 1
As i Γαse _ e ask fl mιιs e sc e u e imme ia e y a e G _ A e e a e 45
i s co si e e e g o e i e i e a e wee a (k + + a a o (k + + 1 w e e 1 m-1 Wi ou oss o ge e a i y su ose a is i e a as e g
us y e a go i m e ask ak+ + mus e sc e u e a o e o e 1; a is
w ic mea s a
o e i e a [( ack + m + a a( + 1 o e ca use e same a gume as i Case o s ow a e e g o e i e ime u i g is e io is ess a c
ow o e ca ou e e g o e i e a [Mc (1 o ( e i e a co sis s o ee y es o sma e i e a s (i e i e a s occu ie y a asks (ii e i e a s occu ie y c asks a (iii i e i e a s
Su ose a ai a q a e e a asks sc e u e e o e c (1 e e e a e
( —1 — q a asks sc e u e u i g e i e a [M c (1 oe ( e e o e e o a e g o e y e (i i e a s is ( — 1 — q a eca a i is assume a C i o ( C1± 1
So e um e o c asks u i g e i e a [M c (1 a is i a e o a e g o e y e (ii i e a s is ic y Cases 1 a 3 e e g o e i e i e a a e eac a ask is a mos c
Si ce e e a e ( - 1 - q asks u i g e i e a [Mc (1 a ( e o a e g o e i e i e a s a e ese a asks is a mos ( — 1 — q c I e e is o a ask e wee C; and C5±1 for 1 < n i-i then there is an idle interval between Cj and Cj +2 with length at most a. Since there are i jobs that finish before t α ( e o a e g o e i e i e a s
e wee e o s C a C + wi o a ask e wee em a 1 5 i — 1 is at most 7
τ τ 1 Α1 Αι Γ• ι ι
ratio between the algorithm and the optimal solution can approach 2 arbitrarily closely.
Note that Algorithm 2 produces schedules that are order preserving. Since the min- imum order preserving and no-wait-in makespan cannot be smaller than the minimum no- wait-in makespan, Theorem 3.3.3 implies the following corollary.
Co o a y 3 3 Let S be a schedule produced by Algorithm 2 fora given set of n jobs with preprocessing tasks a and preprocessing tasks c and a c Then S is an order preserving and no-wait-in schedule and CHAPTER 4
APPROXIMATION ALGORITHMS: GENERAL CASES
In this chapter, the total completion time minimization problem will be considered under various scenarios. This includes single-master systems, multi-master systems, and distinct preprocessing and reprocessing master systems. Since the problem is strongly NP-hard, the focus will be on approximation algorithms. For each type of system, both the case when all jobs have the same release time and the case when the jobs have different release times will be considered. Algorithms are designed to approximate the best preemptive or non-preemptive schedules in these cases. The organization of this chapter is as follows. In
Sections 4.3 - 4.5, algorithms are designed to minimize total completion times of preemp- tive schedules. Section 4.3 is devoted to the single-master case, Section 4.4 is devoted to the multi-master case, and Section 4.5 is devoted to the case of distinct preprocessor and
preprocessors. In Section 4.6, conversions from preemptive schedules into non-preemptive schedules will be considered. In section 4.7, linear programming relaxation approach is applied to systems with distinct preprocessor and preprocessors.
4.1 Preliminaries An ι - ppr x t n algorithm for makespan (or total completion time) is an algorithm that for any set of jobs generates a schedule S whose makespan (or total completion time) is at most x times the optimal makespan (or total completion time). It is an (α β - approximation algorithm if it is an approximation algorithm for makespan and at the same time a approximation algorithm for total completion time. For a schedule S, if
then S is said to be an (cc, β -sc e u e The Shortest-Processing-Time (SPAT) rule, which always runs the job with the least processing time, and the Shortest-Remaining-Processing-Time (SRPT) rule [107, 111],
48 9 which always runs the job with the least remaining processing time, are two well-known algorithms for minimizing total completion time. Usually, the SPT rule is used to generate non-preemptive schedules, while the SPT rule is used to generate preemptive schedules.
Suppose each job consists of a single task. If all jobs are available at time 0, then the SPT rule is optimal for total completion time in the single-machine or multi-machine environ- ment. If the release times are arretrary, then the SPT rule is optimal for a single machine and it is a 2-approximation (see 195]) in the multi-machine environment.
In this chapter, both rules will be adapted for the problems in master slave model.
Both are applied to generate preemptive schedules. A scheduling decision is made when an a task or a c task completes so that a master machine becomes free, or when a new a task or a c task becomes available. At any such time instant, the SPAT rule schedules, from the set of available tasks (including those that have been preempted but have not yet completed), the one with the smallest processing time. Depending on how one chooses from the set of available jobs, one can obtain the SPT rule and the SPC C rule. Specifically, in the SPTQ rule, preemption occurs only among the a tasks and the preemption is based on the length of AA. In the SPTc rule, preemption occurs only among the c tasks and the preemption is based on the length of ci On the other hand, the SPT rule schedules, from the set of available tasks, the one with the smallest remaining processing time. Similarly, one can define the SRPTa rule and the SRPTa rule. Both the SPAT rule and the SRPTa rule may generate schedules with migration when there are multiple machines, i.e., after interrupted on one machine, a task is resumed on a different machine. In this chapter, the jobs are frequently sorted in certain nondecreasing order of some parameters xi of job i e.g. xi = cif. For convenience, in this chapter xi < x) is used as long as Fib comes before x in the sorted order, even though it may be the case that xi = x. As in the previous chapters, one can always assume that the b tasks are scheduled immediately when they become available. Thus one can focus on the schedule of a tasks and c tasks only. 5
ew esu s a ec iques eem i e e a a io a i ea og ammi g e a a io a e wo im o a ec iques o ge i g co s a - ac o a o ima io s o o a com e io ime o o - eem i e sc e - u es; see [95 59 1 5 1 9 1 ] Mos o ese a go i ms wo k y i s co s uc i g a e a e so u io ei e a eem i e sc e u e o a i ea og ammi g e a a io ese e a a io s a e e use o o ai a o e i g o e o s a e o s a e is sc e u e (i e o u o ce i e ime i is o e Mo e a ou ese me o s wi e iscusse i Sec io e a a age o e eem i e e a a io is a usua y e e a e e y e icie a go i ms o ge e a e o ima o ea -o ima sc e u es I mos cases ese a go i ms ( o eem i e a o - eem i e ca e im eme e o u i a o i e as io see [95] a [59] e i ea og ammi g e a a io owe e is mo e ime co sumi g a ca o y e im eme e o u i a o i e as io O e o e a i o i es e e a o ima io gua a ies i some cases u e mo e e me o usua y wo ks e e i o e wa s o o imi e e o a weig e com e io ime; see [59 1 5 1 9 1 ] o a oac es wi e use i is c a e I a cases e icie a go i ms a e i s e e o e o ge e a e eem i e o i e o o i e sc e u es wi goo a o ima io ese sc e u es a e s ow o a e sma makes a as we e y a yi g e i eas i [95] a [1 ] o e mas e s a e mo e s ese eem i e sc e u es ca e co e e i - o o - eem i e sc e u es wi ce ai eg a a io i e qua i y o a o ima io I e case w e e e a e is i c e ocessi g mas e s a e ocessi g mas e s i ea og ammi g e a a io is a so use I is s ow a o - eem i e sc e u es o ai e i is way some imes a e e e e o ma ce gua a ies a ose o ai e y e e- em i e e a a io a oac e esu s a e summa i e i a es 1 3 a w e e e is e ase o a u a oga i m 51
Table 4.1 ew esu s o Si g e-Mas e Sys em
Table 4.2 ew esu s o Mu i-Mas e Sys em A Sc e u es a e o -Ca o ica
4.3 Single-master is sec io assumes a e e is a si g e mas e Ca o ica sc e u es a e s u ie i s a e o -ca o ica sc e u es i a y e case w e o s a e i e e e ease imes
4.3.1 Canonical Preemptive Schedules i s wo owe ou s o e o a com e io ime i ca o ica sc e u es a e e e o e Su ose e e a e o s 1 o ca o ica sc e u es w e e eem io is a owe o o o e ca e i e e o owi g owe ou 5
a e 3 ew esu s o is i c e ocesso a e oceso Sys em ma = m = 1
a e ew esu s o is i c e ocesso a e ocesso m1 1 a m 1
w ic assumes a e e is o i e ime i e sc e u e a a e c asks a e sc e u e i asce i g o e o ei e g s A o e owe ou is
w ic assumes a o s a e sc e u e i i c easi g o e o e Ai a a e asks a e c asks a e sc e u e imme ia e y a e ey a e a ai a e i a y e o owi g i ia owe ou w ic is sim y e summa io o a e ocessi g imes o s o a y sc e u e
is o ows om e o se a io a C ; c ; + ; + c Summi g ; om 1 o gi es e esu i ( 3 53
In canonical schedules, all the a tasks are scheduled first. Since all the a tasks are a ai a e a ime Ο a e c asks ca o s a u i a e a tasks finish, there is no need to preempt the a tasks. Hence, only a c task can be preempted by another c task in this case.
A go i m Ca o ica s PTA : Schedule the a tasks in an arretrary order without preemp- tion. After all the a tasks finish, schedule the available c tasks by the SPA rule.
eo em 3 1 Al r th C n n l-SΡ Τ generates a (2,2) canonical preemptive sched- ule in O (n log n) time.
oo Let Ca denote the time a; completes. Then at time t) = max(A, (C a + b)), all the a tasks finish and the c; is available to be scheduled. Since Cai < A, it must be
ue a Α + . According to the algorithm, if there is another available task cif that hasn't finished at time ti and cif < c;, then c; has to wait until cif finishes. Also, during the execution of c;, if there is another task cif < c; that becomes available, then cif preempts C .
In both cases, it is said that c5 is delayed by cif. Let C; be the completion time of c; in the
sc e u e ge e a e y A go i m Ca o ica -SΡ Τ . Then,
where the last inequality comes from the lower bounds (4.1) and (4.3). The above armroximatioii ratio is tight. Consider this example: for 1 < i < n — 1,
The optimal canonical schedule schedules a1 a a _ι first and then followed by a . The total completion time is about (n + 1) a However, if one schedules aa1 first followed by a1 a _ ι then the total completion time is about Ana. (For this example, the optimal preemptive schedule has the same total completion time as the optimal non-preemptive schedule.) 5 I as ee s ow i [ 1 ] a a y ca o ica sc e u e wi ou eem io is a - a o ima io o makes a Si ce eem io amo g e c asks ca o i c ease e makes ań e sc e u e ge e a e y A go i m Ca o ica s Τc as makes ań a mos wo imes e o ima
Su ose e o s i Se a e a a ge i i c easi g o e o e s a e o s i 5 a e a a ge i ec easi g o e o e s I [ 1 ] i was s ow a e ca o ica sc e u e i w ic e a asks o S ι a e sc e u e e o e e A asks o S as makes ań a mos 3/ imes e o ima sc e u e I e a
asks a e sc e u e i is o e i A go i m Ca ο ica -S Τ c e o e s i ge s a 3/ - a o ima io o makes ań si ce eem io o e a ai a e c asks wi o i c ease e makes ań
Co o a y 3 There is an O ( og time algorithm that generates a (3/ canonical preemptive schedule.
o e a w e eem io occu s a go i m Ca o ica -S Τc uses S A u e i - s ea o e S u e is is o e u ose o a a ysis o y I ac ice o e ca use e S u e o ge a e e a o ima io o o a com e io ime
3 o -ca o ica s eem i e Sc e u es I is mo e e A asks a e c asks ca e sc e u e a e a i e y A owe ou o e C ca e o ai e y assumi g a e asks a e e g
A go i m o -ca o ica s Q+c o a y wo o s i ( a + c; (c i -1- c1 e
o c a a yci ii ime c; a e i sai A e o mas e a e ig e io i y a a ocesso is ee o assig me assig e a ai a e ask wi e ig es io i y I a ew ask ecomes a ai a e a as ig e io i y a e cu e y u i g ask e ew ask eem s e cu e y u i g ask oo Si ce a e a asks a e ea y a ime e e is o ee o a a ask o eem a o e ask us eem io occu s o y e wee a ig e io i y c ask a a
owe io i y c ask o a owe io i y a ask e C a s e o e e com e io ime o a i e sc e u e ge e a e y A go i m o -ca o ica s a +c I o e o e a asks is
eem e y a c ask e Ca wou e ++c ai a ; Σαi+ci α ecause o eem io
Cca o ya si a e e ay e aye i y some ig e io i y c asks I o e wo s c a + ci a + c =+ ecomes3oa A e ime ask a ai a e Cc Acco i g o e a go i m co ca o y e e aye y a ask ci suc a ai + ci a + c o e a i ci
e ays a i wi o e ay co agai us o e ca ou e com e io ime C
o ou e makes a o e sc e u e ick e as o suc a co is sc e u e imme ia e y w e i is ea y Suc a o a ways e is s si ce e o i wi e ig es
io i y a ways sa is y is c i e io e e i e a Ι ι = [ Ca a e i e a a e 3 3 A i a y e ease imes When arbitrary release times are present, it is not meaningful to have canonical schedules any more. Thus, only non-canonical schedules will be considered. The lower bound (4.4) still holds for this case. Let R = Σ r1 . Then, a trivial lower bound for the minimum total completion time of any schedule is
U Gι G 1υι1 i-uguiiiiiiii 1 U1- ιιυιιι 1-3 C I a±c ιιυικ s IΗU υιιιν11UI ιAUU1 111C C1C 1 C time of a job. So, one can still apply Algorithm non-canonical-SPT a±c when jobs have arretrary release times. Unlike the case when all jobs have the same release times, in this case, a higher priority a task may also delay a lower priority a task or c task.
eo em 3 Algorithm Non-canonical-SPT ± generates a (2,2) preemptive online sched- ule for a single-master system even when the jobs have arbitrary release times.
oo One can first bound the total completion time of the schedule generated by
Algorithm Non-canonical-SPT a±c . Let Ca s be defined as before. Then, 57
where the last inequality comes from the fact that the two sets of tasks delaying a and c;
are disjoint, and they all have higher priority than a ; and c;. Similarly, one can bound the total completion time
To bound the makespan, one picks the last job 5 such that c5 starts immediately after it is
To conclude the proof, note that Algorithm Non-canonicalsPTa+c schedules jobs in an online fashion. ❑
4.4 Mu i-mas e This section assumes that there are m? 2 masters, each of which is capable of processing both the a tasks and the c tasks.
1 o -ca o ica eem i e Sc e u es 58
Algorithm FAM-SPTa+c (1) Assign the jobs in order of n, n — 1, ... , 1 using FAME
(First Available Machine, see [ 106]) rule: associate each machine i with a variable A i .
Initially all machine are available and AA = Ο Pick the next unassigned job n and assign it to the machine i with the smallest T and after the assignment Ti is increased by an amount of ail + ci Repeat this procedure until all jobs are assigned. (2) Apply Algorithm Non- canonical-SPTa+c to each master machine to schedule the jobs assigned to it
Theorem 4.4.1 Algorithm FAM-SPTa+c generates a (2,2) preemptive schedule for multi- master systems without migration when all jobs have release time Ο
Proof: Without loss of generality, we may assume that n = mk for some integer 2.
Otherwise, one can add dummy jobs with a = bi = ci = Ο For convenience, one can reindex the jobs assigned to each machine p in the form of (p, q) such that c( ,q) + c( , q ) <
c( ,q) + C( , q+l) . A lower bound of the total completion time comes from the fact that
Algorithm FAME ΡΤα+c is optimal if b ( , q = for every job (p, q) . 59
Suppose the job with the maximum completion time among all jobs is assigned to machine p. Let t be the last job assigned to p. If there is no idle time on machine p, then the makesuan is
4.4.2 Arbitrary Release Times Offline schedule without migration In this case, one can still apply Algorithm
FAM-SPTa+c. However, note that to assign jobs to machines, Algorithm FAM-SPT a+c requires that one has full knowledge of all the jobs at the beginning. Thus, Algorithm FAM-SPT a+c is an offline algorithm. One can comrene the arguments in Sections 4.3.3 and 4.4.1 to get the following theorem. 60
Theorem 4.4.2 Algorithm ΑΜ-S ±, generates a o2, 2) offline preemptive schedule with- out migration for multi-master systems when jobs have arbitrary release times.
Now consider the makespan. Suppose the job with the maximum completion time among all jobs is assigned to machine p. Let l' be the last job on machine p so that Al is scheduled immediately after it is ready. Define C a , as before. Then the machine p is busy Online schedule with migration I is case o e ca a y A go i m o -ca o ica -
S a+c o mu i-mas e sys ems o e a i a ew ask ecomes a ai a e a o e o mo e cu e y u i g asks a e owe io i y a e ew ask e e ask wi e owes io i y wi e eem e
Theorem 4.4.3 Algorithm non-canonical-SPT + generates a (3 online preemptive sched- ule with migration on multi-master systems when jobs have arbitrary release times.
Proof : e oo ee s a o e owe ou o mu i-mas e sys ems Co si e e
o m mas e s e S e a o ima sc e u e o ese o s e s a i g om ime
o eac ime u i i ai o Ci is sc e u e i is u i o some mac i e e o e ca sc e u e 1 /m o e ask o e si g e mas e i e same ime u i I is easy o see a e co s ai o e ease ime a e ese e a a e i e a e wee e i is ime o ai a e s a ime o ci is ei e e same o i c ease us o e o ai s a a i sc e u e o e i s a ce I o e si g e-mas e sys em e e o e
w ic is a owe ou o e o igi a i s a ce e i e y ( i ai i i
e Ca e o e e com e io ime o i i e sc e u e ge e a e y A go i m
o -ca o ica -S a+c A e α is e ease a i i may e e aye y o e asks wi
ig e io i y e o e i com e es a is a i o i ca e ay a o y i ai + i a5 + C5 I is easy to see that there is no idle time on any machine during the interval
cj) ); otherwise, ail would have completed earlier. Furthermore, only tasks with higher priority than n can be scheduled during this interval. At time = C a + 5 the task c5 becomes ready. Similarly, there is no idle time on any machine during the interval and only tasks with higher priority than n can be scheduled u i g is i e a o e a e ask se s i e i e a s 1ι a Ι a e is oi e 11i ί and 11 1 e o e e e g s o Ιι a 1 es ec i e y e i mus e ue a 1Ι1( + (Ι 1
Σaί+cί ιι;+ci (a + ci /m Thus, one can bound the completion time C 5 as follows:
For the makespan of the schedule, let k be the job with the maximum completion time among all jobs. Let 1 be the last job among all jobs so that A 1 is scheduled immediately after it is ready. Define C a, as before. Then, all machines must be busy during the intervals The total length of the two intervals 63
4.5 Distinct Preprocessing and Preprocessing Masters Ι is sec io e mo e wi is i c e ocessi g mas e s a e ocessi g mas e s wi e iscusse i e e om e e ious cases e e a ci ask ca o y e eem e y a o e a ask a a c ask ca o y e eem e y a o e c ask I a cases
A go i m S Τa SΡ c wi e a ie o o ai a eem i e sc e u e
Algorithm SRPTa SPTa : Sc e u e e a ai a e a asks usi g e SPTa u e o e e-
ocessi g mas e Sc e u e e a ai a e c asks usi g e S A u e o e e ocessi g mas e e ma a m e o e e um e s o e ocessi g mas e s a e ocessi g mas e s es ec i e y i s e sim e case m 1 = m = 1 wi e s u ie
Theorem 4.5.1 Al r th S A -S Τς n r t (3, 2) nl n pr pt v h d l h n
= l = 1 nd e f r ll n.
Proof: i s co si e e o a com e io ime e C e e ime di i is es i a o ima sc e u e e
e e e u e ιι ι es i e scπeUυ e υ [ 1 eυ Dye gu i m K α 1 a iI ce ai 1a
e SPTa u e is o ima i = c = A go i m S ΡΤα SPA mus a e e mi imum
C a; amo g a ossi e sc e u es a is
us e o a com e io ime is a mos
The length of each interval is at most Cm . Therefore, the makespan is
eo em 5 Algorithm S a S A generates a (4,2) preemptive schedule with mi- gration when m> > 1, ml > 1 and η = for all n .
oo Since all a tasks are available at time 0, then the S a rule is the same as the
SPA rule. As mentioned before, the SPA rule is optimal when the b tasks and the c tasks have zero length; that is, it minimizes the total completion time of the a tasks. Let C 1 be the finish time of ai in an optimal schedule, and let C a; be the finish time of ai in the schedule generated by Algorithm S S A Then, as in the case of in = m2 = 1, a owe ou o e o a com e io ime is 65
Theorem 4.5.3 Algorithm SRPTa -SPTa generates a ( 3 preemptive schedule with mi- gration when m> > 1, ml > 1 and η > 0 for all 5.
Proof: e C e e com e io ime o ι i a o ima sc e u e As i e as sec io o e ca ge
e Ca e e com e io ime o a; i e sc e u e ge e a e y A go i m S S A
As me io e e o e e S u e is a -a o ima io w e ; = c; = Ο us i mus ue a 66
4.6 Converting Preemptive Schedules into Non-preemptive Schedules As mentioned before, non-preemptive schedules can be obtained by converting preemptive schedules. Our approach is based on the techniques introduced by Phillips et al. in [95], and improved by Checker et al. in [16]. For completeness, their approaches are described in the following. The model studied in [95] consists of one or more identical machines and n jobs. For this model, a general approach of converting a preemptive schedule S into a none-
presumptive schedule S' has been given in [95]. The idea is to form a list of jobs in in- creasing order of their completion times in S and then list schedule the jobs in this list one by one, respecting their release times. Let 2Cj and C; denote the completion time of job n in S and S', respectively. Sup- pose the jobs are indexed such that Ci < C1±1. Then they showed that C ' < 2C for a single machine environment and C) < 3Ci for a multi-machine environment. Let
OmO. i ii The = result maxi is based on the observations that (1) Cif > rm, (2)
Σi-1Σi -1C>in i/mi 1 the i single-machine e mu i-mac i e case, case and C w e emachine i is the processing time of job n, (3) C) < Om + Σi-1 Ai if there is one if there are two or more machines. These results im-
y a i 5 is a β-a o ima io o o a com e io ime e S is a β-a o ima io
o o a com e io ime i e si g e-mac i e e i o me a a 3 β-a o ima io o total completion time in the multi-machine environment. In addition, this conversion also yields an online non-preemptive algorithm if the preemptive schedule can be generated on- line: simply simulate the algorithm for preemptive schedule, start a job n if n completes in the preemptive schedule or put it in the waiting queue if the machine is busy. 7
Later, Chekuri et al. [ 16] improved the above results for total completion time. They designed a deterministic O on ) time offline algorithm such that the schedule obtained has total completion time at most e ti 1.58 times that of the preemptive schedule. They also gave a randomized online algorithm which generates schedules having expected completion time e times that of the preemptive schedule. The difference between the two approaches i [1 ] a [95] is ow o o ai e is o o s Gi e S a a a ame e Ε o 1] e
A (S e λ- οi o e e ime w e i (a λ- ac io o o is com e e I s ea of forming a list based on A i o5 ow o m a is ase o A oS Α -sc e u e is a o e- esum i e sc e u e o ai e y is sc e u i g o s i i c easi g o e o A e oS ossi y introducing idle time to account for the release times. It is easy to see that the algorithm given by [95] is a b-scheduler with b = 1. It is also clear that a b-schedule can be made to be an on-line algorithm if the underlying preemptive algorithm is an on-line algorithm.
e mai esu i [ 1 ] is a o eac gi e i s a ce e e e is s a e es λ such that the b-scheduler has total completion time at most e times that of S. and has makespan at most o1 + b) times that of S. One can obtain such a b-scheduler by finding the best b in O (n ) time offline deterministically; or one can obtain, through a randomized on- line algorithm, a schedule whose expected total completion time is at most e e times that of S and whose makespan is at most o1 + b) times that of S. In the multi-machine case, Chakrabarti et al. [ 14] showed that the convert procedure given in [95] has a bound of 7/3 for total completion time, instead of 3 times that of S. The following sections describe how to convert preemptive schedules generated in
Sections 4.3-4.5 to non-preemptive schedules. The difficulty of the conversion in the master slave model is that one needs to respect not only the release time of A 1, 1 < i n, but also respect the constraint that the interval between the finish time of ail and the start time of i has length at least bi. 68
4.6.1 Single Master and Multi-Master Systems First consider the single master systems.
Theorem 4.6.1 In O (n ) time, one can obtain a o2 , ee^) non-preemptive canonical sched- ule when there is a single master and η = 0 for all 5.
Proof: Let S be a preemptive canonical schedule of jobs obtained by applying
Corollarybeaca the partial the 4.3.2. schedule Let S of S during the interval (0, A], and S partial schedule of S a during the interval oA, A ka,] .
Clearly Sac contains all a tasks only. By the Algorithm Canonical-SPT a, there is no preemption in Sac. Let C a e e com e io ime o α; I is easy o see a e a ia schedule S a contains all c tasks only and it can be seen as a preemptive schedule of n tasks on a single machine where each task 5 has a "release time" maxoA, C a s +b) and processing time c.
Theorem 4.6.2 In O o log n) time, one can obtain a (3, 4) online non-preemptive sched- ule when there is a single master and r ; > 0 for all job 5
oo Let S be a non-canonical preemptive schedule S. One can get a reschedule S', b = 1 similarly as [95] : (1) Sort all tasks (both a tasks and c tasks) in increasing order of their completion times. (2) List schedule these tasks on the master machine, with the constraint that each task a; must start after 3 a e i e a e wee e ime α; i is es and the time c; starts is at least b). 9
For the purpose of analysis, one can visualize the above procedure as follows (see
Figure 4.1): For each task, a; or c , remove all but the last scheduled piece of the task.
Suppose the last piece of a and c ; have length Ba and ka respectively. Now process the tasks one by one in the scheduling order of their last pieces in S. If the current task is a,
complete its last piece by inserting an extra piece with length (a5 - ka immediately after
the last piece of a5 ; at the same time push backward in time all the last pieces of the tasks
which finish after a; in S by an amount of (a5 - a I the task is c;, complete its last
piece by inserting an extra piece with length (c; — Ca ) immediately after the last piece of
c5 ; at the same time push backward in time all the last pieces of the tasks which finish after
c; in S by an amount of (C ; — C 3 ). Let the schedule be S". It is easy to see that during
this process, the two constraints, (a) each task A M is scheduled after η and (b) the interval
between the time AM finishes and the time c ; starts is at least b;, are not violated. Thus S"
is still a feasible schedule.
Now one pushes all jobs forward in time as much as possible without changing the order of the tasks, or violating the constraints (a) and (b) mentioned above. The result
is exactly the schedule S'. Since a task c ; can only be moved back by processing times
associated with tasks finished earlier than c; in S, we have A; < 2A 5 and A) < A; +
Σ (a1 + i) < A5 -1- Ak , where A5 is the completion time of c; in S'. This implies that if one takes the (2, 2) non-canonical schedule in Theorem 4.3.4, then one can get a (3, 4) non-preemptive schedule. Furthermore, it can be implemented online if the preemptive schedule is online. ❑
The following theorem shows how to get offline non-preemptive schedules for
multi-masters systems. It is not known how to obtain an online non-preemptive schedule.
eo em 3 When there are multi-masters, one can obtain a (4,4) non-preemptive of- fline schedule.
oo e S e e ( -sc e u e ge e a e y A go i m ΑΜ-S αΡ+a. Then S has no migration. One can obtain the non-preemptive schedule S' by converting the schedule 7
o eac mac i e se a a e y i e same way as esc i e i e oo o eo em us e o a com e io ime o S is a mos wo imes a o S Si ce S is a - a o ima io o o a com e io ime S is a a o ima io o o a com e io ime
o e makes a C; C5 + maim ι Δ1 w e e Δί is e o a e g o e a asks a e c asks assig e o mac i e i I Sec io i as ee s ow a Δία C us
C; Co + maim Ai C5 + Aka Si ce S is a o ima io o makes a S is a a o ima io o e makes a is co c u es e oo ❑
is i c e ocesso s a e ocesso s is su sec io co si e s e case w e e e a e ma e ocesso s a m e ocesso
oo e S e e (3 -sc e u e ge e a e y A go i m S a SΡAa Si ce =o Ο
o a e e is o eem io o e si g e e ocesso e Ca ; e e com e io ime o αα i S o ge a o - eem i e sc e u e o e ca i e sc e u e o e a asks a o e co e sio sim y o e si g e e ocesso s usi g e ac y e a oac gi e i [ 1 ] 71
es ec i g e " e ease ime" o ask c (C a + y o owi g e ac y e same a gume σ
W e e e ease imes a e a e a y o e ee s o o e co e sio ca e u y so as o make su e a e i e e ce e wee e i is ime o a a e s a ime o c3 is a
eas 3
pr p ιv n tn n t r n n tn non-pr p v r t n xp p r r-
oo e S e e (3 -sc e u e ge e a e y A go i m S ΡA α SΡAa e Ci e e com e io ime o o i S e C a e e com e io ime o ai i S e co e sio co sis s o wo s e s i s o e ca emo e e eem io s amo g e a asks o ge e es (o i e esc e u e o e a asks o e si g e e ocesso y usi g e a oac o [1 ] e C e e ew com e io ime o a e o e mus a e
C i e e Ca; ow i e sc e u e o a asks a emo e e eem io s amo g e c asks o ge a esc e u e o e c asks w e e = 1 a make su e e i e a e wee C a e s a ime o c; is a eas ; o eac o i 7
us S is a ( + o - eem i e o i e sc e u e Simi a y as i [161, o e ca a so ge a o i e sc e u e w ose e ec e e o ma ce is ( + i is co c u es e oo O
eo em In O ( og time, one can obtain a ( 1 /3 non-preemptive schedule when r b = 0 for all n, m1 1 and m ? 1
oo e S e e ( -sc e u e ge e a e y A go i m S ΡAα SΡAa Si ce a o s a e e same e ease ime o eem io occu s o e e ocesso s ow i e sc e u e o e a asks e C a e e com e io ime o ai i S e ask c ca e see as a ask wi e ease ime (C + e co e sio is e o me o e e ocesso usi g e a oac gi e i [95] su ec o e co s ai a cc ca o s a ea ie a i s " e ease ime" (C a; + e e y [ 1 ] e o a com e io ime o S is a mos
imes a o S i e S is a 3 -a o ima io o o a com e io ime
eo em 7 In O ( og time, one can obtain a (5 13 non-preemptive online sched- ule when m> > 1 and ml > 1
oo e S e a ( 3 -sc e u e ge e a e y A go i m S ΡΤα SΡAa e co e sio co sis s o wo s e s i s we use e a oac o [95] o emo e eem io s amo g e a asks o ge a = 1 sc e u e o e a asks es ec i g e e ease imes o e a asks e
C e e ew com e io ime o a y e esu o [ 1 ] o e mus a e ow i e sc e u e o e a asks Eac ask cc ca e see as a ask wi e o e ca emo e eem io s amo g e c asks o ge a esc e u e o e 73 c asks w e e = 1, a make su e a e i e a e wee A a e s a ime o c5 is a eas 5 o eac o 5
4.7 Linear Programming: Distinct Preprocessors and Preprocessor As me io e e o e a o e im o a a oac o - a sc e u i g o ems is o o mu a e e o em as a i ea og ammi g o em is me o as e a a age a i a so wo ks o o a weig e com e io ime owe e i s isa a age is a i akes e a i e y o g ime o o ai a so u io I is sec io a o ima io a go i ms a e ese e o e case w e e e a e is i c e ocessi g a e ocessi g ocesso s Eac o is a owe o a e a weig we e asis o e a o ima io a go i ms i is sec io is a i ea og ammi g e a a io a uses as a ia es e com e io imes o e A asks a e c asks o eac ask e i e Ca ; a Ci o e e com e io ime o Ai a ci es ec i e y e o a weig e com e io ime mi imi a io o em ca e o mu a e as o ows e i icu y wi e a o e c a ac e i a io is e so-ca e " is u c i e" co s ai s ( 1 -( 13 w ic a e o i ea i equa i ies a ca o e mo e e usi g i ea i equa - i ies I s ea o e ca use a c ass o a i i equa i ies o a y easi e sc e u es (may e eem i e i o uce y Quey a e [9 ] a Woo sey [119] Su ose a se o asks
e o e y 1 a e sc e u e o a si g e mac i e e 3 e e ocessi g ime o a A; e e com e io ime o i a y easi e sc e u e e e o owi g i - equa i y is a i o a y su se C Ν
e key o e qua i y o e a o ima io e i i g om e a o e e a a io s is e o owi g emma Quey a e [9 ] as s ow a e i ea og am wi co s ai s ( 1 is so - a e i o y omia ime ia e e i soi a go i m; e key o se a io is a e e is a o y omia ime se a a io a go i m o e e o e ia y a ge c ass o co s ai s gi e y ( 1 I ou mo e o e ca a y e a o e co s ai s o e e ocessi g mas e a e e ocessi g mas e es ec i e y
Algorithm List-Schedule-Guided-by-LP-Single-Master: i s o ai a o ima so u- io o e i ea og am o me y ( 9 ( 1 ( 11 ( 15 a ( 1 e o e e com e io ime o e o s y ςι C ti e o e ca o m a sc e u e y sc e u i g o e a asks a e c asks i i c easi g o e o C u e e co i io a A ca o s a u i a e i e a e wee e s a ime o c; a e i is ime o a is a eas ; i e c; ca o s a u i ; i is es
Theorem 4.7.2 Suppose that m1 = m = 1 Then Algorithm List-Schedule-Guided-by- LP-Single-Master produces a (3,5)-schedule when all jobs have the same release time and ( 6)-schedule when each job has an arbitrary release time. Furthermore, the a tasks and the c tasks complete in the same order.
Proof : e S e e sc e u e o ai e y A go i m is -Sc e u e-Gui e - y- -
Si g e-Mas e e Ca1 e e com e io ime o a; i S y e way e a asks a e 7
'1k since the latest time the preprocessor be- comes oust' is ma k k ι eτ e me com e io ime o o 5 i S ecause e c asks are scheduled in the same order as the a tasks, it is possible that c; can not start even af- ter b; finishes at C a; + b; . This is because c;_ι may o a e com e e ye owe e e
ime a c ; needs to wait after it is ready is at most max k
Thus, if all jobs have the same release times, then C ; < 6 C; ; otherwise, C, < 6 C; . This implies that S is a 6-approximation for total completion time when r ; = Ο o a 5 a a 6-approximation when η Ο is is so e e i eac o 5 as a weig For the makespan, A go i m is -Sc e u e-Gui e - y- -Mu i-Mas e First obtain an optimal solu- tion to the linear program given by (4.9), (4.10), (4.11), (4.18) and (4.19). Denote the completion time of the jobs by A ι , ... , A a. Schedule both the a tasks and the c tasks one by one in nondecreasing order of j under the condition that: a can not start until Ti , and the interval between the finish time of a and the start time of c; is at least bj. In other words, c; can not start until ; finishes.
eo em 7 Suppose that ma > 1 and ml? 1 Then Algorithm List-Schedule-Guided- by-LP-Multi-Master produces a (4, 6)-schedule when all jobs have the same release time and (6, 7)-schedule when each job has an arbitrary release time.
oo Let S be the schedule obtained by Algorithm List-Schedule-Guided-by-LP-Multi-
Master. Let Ca5 be the completion time of task aj in S. By the way the a tasks are 7
e c asks a e sc e u e i e same o e as e a asks i is ossi e a c; ca o s a e e a e ; i is es a CQ5 + ; ecause c5_1 may o a e com e e ye owe e e
ime a c; ee s o wai a e i is ea y is a mos ma k 5{ k + Σk 5_ι ck/m us A II
E WO K ESIG O EMS CHAPTER 5
POLYNOMIAL-TIME APPROXIMATION SCHEMES FOR THE EUCLIDEAN
SURVIVABLE NETWORK DESIGN PROBLEM
5.1 Introduction is c a e co si e s e geome ic e sio o e survivable network design problem. e i u e ices a e assume o e oi s i W a e cos o eac i k is equa o e Euc i ea is a ce e wee i s e oi s (w ic is a goo a o ima io i ma y a ica io s si ce o e e "i s a a io " a e "se ice" cos is oug y o o io a o e e g o e i k [91] e ocus is o wo mos asic a ia s o e geome ic su i a e e wo k esig o em (1 SM o em i w ic „ Ε { 1 } o a y Ε a ( { 1 - e e - a
-e ge-co ec i i y o em i w ic Ε { 1 o a E o e a SM o em is a s ecia case o { - e e - a -e ge-co ec i i y o em e a gume s o i e y G sc e eta!. [53] (see a so [91 113] sugges a e seco s ecia case i e a o e mo e s we ma y a ica io s o e su i a e i y o em e g e o em o esig i g su i a e i e e e o e e wo ks [91 113] I e case o i e commu ica io e wo ks o e e o e com a ies e wo k o o ogies wi co ec i i y equi eme s i { 1 o i e a a equa e e e o su i a e i y o e is i guis e ce a o es o co ec i i y y e Sim y mos ai u es usua y ca e e ai e e a i e y quick y a as s a is ica s u ies a e e ea e i is u ike y a a seco ai u e wi occu o ei u a io
5.1.1 Related Works e e as ee a o o esea c o e su i a e e wo k esig o em y ica y e esea c a esses ei e ac ica eu is ics a a go i ms (see e g [ 1 5 51 53 53 91 113] o e ge e a o em o a e a y e wo ks (see e g [3 37 1
80 81 117] o e o em es ic e o e y s eci ic e wo ks I a icu a e esu ue o ai [ ] gi es a o y omia - ime a o ima io a go i m o e e ge-co ec e su i a e e wo k esig o em ( o arbitrary connectivity requirements). A so
o y omia - ime a o ima io a go i ms o a e a y e wo ks i e case τ u Ε { 1 o e e y u a e ee ece y ese e [3 37] e e is o o e goo o y omia - ime a o ima io a go i m s ecia i e o e geome ic e sio o e su i a i i y o em e ce e case w e Ε { 1 } o e e y [97] I „ E { 1 } o e e y a U = { Ε = 11 e o e ca easi y s ow a a mi imum s a i g ee o U gua a ees e a o ima io a io o i a y me ic s ace (a us i a icu a i a y Euc i ea s ace Im o a y i is case e geome ic su i a e i y o em is a ge e a i a io o e c assica Euclidean (complete) Steiner tree problem (see e g [ 3 97 11 ] e Euc i ea (com e e S ei e ee o em o a i i e se o oi s S i Ι is o co s uc a mi imum e g ee w ose e e se co sis s o a oi s i S a ossi y some o e oi s i us e a e o em ca e ega e as a su i a e e wo k esig o em o a i i i e e e omai i e = a = 1
o a y Ε S a = o e wise y e ce e a e esu s ue o Au o a [3] a Mi c e [9 ] (see a so [1 1] e Euc i ea (com e e S ei e ee o em a mi s a o y omia - ime a o ima io sc eme o a y co s a
5.1.2 New Contributions
e f r t p l n l-t ppr x t n schemes (PTAs) a e esig e o e wo a o e- me io e asic a ia s o e geome ic su i a e e wo k esig o em
i s e sim es case i w ic Ε { 1 } o a y e e Ε V is co si e e a is e S ei e ee o em A a go i m is esig e suc a a o a y co s a a a y co s a ε i e u s a S ei e ee w ose cos is a mos (1 + imes a ge a e mi imum e a go i m u s i ime O ( og n) . o ge e a a ε i s u i g ime is 82
e e case w e „ Ε { 1 } o a y e e E is co si e e ; is is e c assica o em i es iga e o oug y y G ii sc e a Mo ma t l [5 51 5 53 91 113] e a go i m o e S ei e ee o em is e e e o esig a a go i m a o a y co s a a a y co s a ε e u s a g a sa is yi g a e e e (o e ge es ec i e y co ec i i y equi eme s a a i g e cos a mos (1 -1- imes
e mi imum e a go i m u s i ime O ( og ii W e a ε a e a owe o a y a e a i y i s u i g ime is ( ogi ( /ε °( + ( ( /ε ί νε °(11 i a y o se e a e ec iques yie a so a ASs o e mu ig a a ia w e e e e ge-co ec i i y equi eme s sa is y Ε { 1 k a k = (1 A e o y omia - ime a o ima io sc emes i is c ea e o ow a a oac simi a o ose use i e ece As o i i g AS (com e e S ei e ees a mi imum-cos ico ec e s a i g su g a i Euc i ea g a s see [3 1 1] ow- e e e e a e ma y im o a i e e ces a make e ew esu s sig i ica y mo e com ica e i s o a o e as o ea wi e es ic io o e S ei e oi s o e se gi e pr r (u ike i e mi imum-cos Euc i ea (com e e S ei e ees o em i w ic S ei e oi s a e a owe o e a y oi s i u e mo e o e as o ea wi o u i o m co ec i i y equi eme s e su s a ia i e e ces a com ica io s oc- cu i e so ca e i e i g ase a sea c i g ase ( y amic og ammi g A e ASs a e a omi e a ac ie e e omise a o ima io gua a ees a u i g ime o e a e age owe e a ese a go i ms ca e r nd z d i a way simi a o a use y ao a Smi i [1 1] e e a omi a io ese es e u i g ime o ( og o co s a a ε
5 Definitions o owi g a e some o a io s a e i i io s o Euc i ea (geome ic g a s a wi e use i is c a e 3 A Euclidean g a w ic eque y wi e ca e i is c a e a graph, is a ai G = (P, E), w e e P is a se o oi s i a Euc i ea s ace i a Σ is a su se o e ai s o oi s i P. E e y Euc i ea g a is weig e a e cre o e ge ( y is equa o e Euc i ea is a ce e wee oi s a y e cre of the graph is e sum o e cos s o i s e ges A i io a y Euc i ea mu ig a w ic a e as Euc i ea g a s u may co ai a a e e ges a e a so co si e e Co sis e y wi e e i i io e e ges o a Euc i ea g a o mu ig a G = ( Σ a e i o e- o-o e co es o e ce wi e s aig - i e segme s (i co ec i g e i ci e e ices (Suc g a s a e eque y ca e s aig - i e g a s i e i e a u e Some imes o ec ica easo s i is a so a owe o bend some e ges A e e ge e wee a ai o oi s i P wi e i e i ie wi a straight-line path (a a co sis i g o s aig - i e segme s co ec i g e oi s S a e s o ge e a g a s a e ee e i e i C a e 1 I is c a e geo- me ic s a e s o a se o oi s a e co si e e
Definition 5.2.1 (Geometric spanners) Let P be a set of points in A graph G on P is called a geometric s a e of ? 1, if for any pair of points q E there exists a path in G from p to q of length at mre t times the Euclidean distance between p and q
Si ce o y geome ic s a e s a e co si e e o sim ici y a geome ic s a e is sim y e e e o as a s a e e o owi g esu is o e i [5 1 1
Lemma 5.2.2 [56] Let P be a set of points in Ι ί where d n t nt t ε b n positive constant. Then, there exists an O ( og )-time algorithm that finds a (1 + t - spanner of having constant maximum degree and the total cre of order of the minimum spanning tree of P.
1 o ice a e a go i ms use i emma 5 a e i e so-ca e ea AM mo e w ic is e a ge aic ecisio ee mo e e e e wi i i ec e a essi g [5 ] I o e assumes e a ge aic ecisio ee mo e e e u i g ime wi e y a ac o o og / og og a ge [5 ]
For arbitrary nd ε th r nn n t f th l r th U
nd th r lt n (1 + ε - p nn r f Ρ h x d r pp r b nd d b ε ε and the total cre upper bounded by d,ε times the cre of the minimum spanning tree of P, where
Τ ε = ( / ° and ε ε = ( / °
I o ma y is e i i io says a i e e e is s a e ge e wee υο a o e a y a o i c u i g (υ ν mus a e e g g ea e a • 1w 1
emma 5 ([3 ] If a set of line segments Σ in d-dimensional space satisfies the ( - leapfrog property where t > 1 th n th ht f Σ w (MS where w (MS th r f n p nn n tr nn t n th ndp nt f Σ h n t nt pl t in the 0- notation depends on and
e i i io 5 5 (S ei e ees Let P ι be a set of points in Rd . A Euclidean tree is called a S ei e ee of Ρ1 if its vertex set includes all the points Ρ1 All the vertices of a Steiner tree of Ρ 1 outside P are called its S ei e oi s If the Steiner points are restricted to a point set P , the tree is called a S ei e ee o Ρ a wi es ec o Ρ and the points in P are called S ei e oi ca i a es
e i i io 5 (Euc i ea (com e e S ei e ees Let be a set of points in The Euc i ea (com e e S ei e mi ima ee of Ρ is a Steiner tree of P having the minimum cre. 85
Definition 5.2.7 (Steiner minimum trees (SMT)) Let Ρ = Ρο U Ρι be a point set in the
Euclidean space A St n r tr f Αι with respect to o having the minimum cre will be called a S ei e mi imum ee (SM of P with respect to o
O se e e i e e ce e wee e e i i io o Euc i ea (com e e S ei e ee a S ei e mi imum ee; i is c a e e a e ia io SMU is use o y o e o e a S ei e mi imum ee As me io e e o e e ocus i is c a e is o e a ia o e o em w e S Ε { 1 } o a y Ε Α is o em is ca e e {0, 1, vertex- or -edged- biconnectivity o em e e i g o w e e e e ge-co ec e o e e ge-co ec i i y e sio o e o em is co si e e ( o ice a e { 1, e e - a - ico ec i i y
o em i c u es e SM o em i w ic S Ε { 1 } o a y Ε (see e i i io 5 7 u e mo e o e ca e ea e a gume s use i [ ] (w ic we e a so use ea ie i [3 Sec io 3] o s ow a i me ic s aces { 1 wo-e ge-co ec i i y a { 1 e ge-co ec i i y a e esse ia y equi a e ( e a gume s i [ ] a [3 ] we e gi e o y o ico ec i i y s wo-e ge-co ec e
Lemma 5.2.8 Let o Pι Ρ be any three sets of points in a metric space. Let Η be a multigraph with the vertex set Po U ι U Α such that for any pair of vertices E i and
E there are at least mi {i i edge-disjoint paths from Z to n Η h n n l n r t n n tr n f r Η into a graph G without increasing the total cre such that for any pair of vertices Ε i and Ε Α there are at least mi {i internally vertex-disjoint paths form Z to in G
Proof : I i ia y se G = Η a a seque ce o mo i ica io wi e e o me o a s o m G o ge e equi e o e ies I is easy o see a i Ε o U Ρ1 e g a G a ea y as e equi e o e ies wi es ec o e e o e o e o y as o ea wi oi s u Ε Ρ a o w ic a e i e same wo-e ge-co ec e com o e o G I is si g e com o e is a so a ock i is o e; o e wise ick a y wo e ges ( a ( w suc 86 a a w a e i i e e ocks i o e wo s is a a icu a io oi e ace e ges ( a ( w y ( w ow a w wi e i e same ock y e ia g e i equa i y e cos o e ew ock wi o e a ge a e sum o e o e wo ocks O e oes is e aceme u i a oi s o Ρ a e i e same ock e emo e a a a e e ges om e g a a G ecomes a g a as a cos o g ea e a a o
H, a o a y ai o e ices Ε ; a Ε j e e a e a eas mi i } i e a y e e - is oi a s o m o i G Usi g s a a ec iques is a s o ma io ca e e o me i ime i ea i e um e o e ges i ❑
is emma a ows o e o co ce a e o y o e 1 }-e ge-co ec i i y o - em a o a ow e ou u o e gi e i a o m o a mu ig a
Partitioning the space. A im o a com o e o e a o ima io a go i ms i is c a e is a a i io i g sc eme i o uce y Au o a i [3] a a e e e e i [ 5 ]
Definition 5.2.9 (Dissection, 2 ary tree) Given a set S of points in 1" d, a ou i g o of S is the smallest d-dimensional axis-parallel cube A d containing the points in S A ( a y S[3](seeis the recursive issec io Figure 5.1) partitioning of of the cube into smaller sub-cubes, called egio s Each egio A of volume > 1 is recursively partitioned into
regions (U/2)d . A 2 ary tree (for a given 2 d-αr d t n tr h r t corresponds to A and whose other non-leaf nodes correspond to the regions containing at least two points from S (see Figure 5.1). For a non-leaf node of the tree corresponding to a region R, its children in the tree correspond to the 2d regions that partition R in the dissection.
For any d-vector a = ( ii αά h r ll α r nt r a1 < A, the
s i e issec io [3, 25] of a set of points in the cube A d in Ii is the dissection of
the set in the cube ( A) in obtained from b tr n f r n h p nt Ε to
D + a. A a om s i e issec io of a set of points in a cube A d in is an shifted Figure 5.1 issec io o a ou i g cu e i ( e a e co es o i g -a y ee ( ig I e ee e c i e o eac o e a e o e e om e o ig o / e squa e o om/ e squa e o om/ ig squa e a o / ig squa e
I is c a e a s ecia c ass o geome ic g a s wi es ec o a gi e issec io [ ] wi a so e s u ie
Definition 5.2.10 (r-locally-light graphs) A graph is - oca y- ig [26] with respect to a shifted dissection if for each region in the dissection there are at mre edges having exactly one endpoint in the region. The crossings on the border of the region generated by these edges are called relevant crossings.
e mai easo o i o uci g is c ass o g a s is a w i e i is λίΡ- a o so e e { 1 }- e e - o -e ge-co ec i i y o em (o e e e mi imum-cos S ei e ee o em o a i a y Euc i ea g a s o e ca s ow ow o so e is o em e icie y o - oca y- ig g a s i Sec io 5
5.3 Steiner Minimum Tree Problem I e a em o o i e a e icie a o ima io sc eme o e { 1, e ge-co ec i i y o em i Euc i ea g a s e SMU o em wi e co si e e i s e ew a oac ca e see as a com i a io o e a oac o Au o a [3] a ao a Smi [1 1] e e - o e o esig a E AS o e S o em wi e a oac o C uma a i ge [ ] developed to design a BETAS for k-connectivity problems. The new algorithms is based on efficient implementations of the following three procedures.
i e i g Let Ρ and Ρ1 be sets of points in and let be any positive real number. Find
a subset of o that satisfies the following two properties:
• The cost of the SMUT of P, with respect to is at most 1 + times the cost of
th SMU f Ρ 1 with respect to Ρ
• The cost of the minimum spanning tree o U Ρ 1 is upper bounded by times
the cost of the minimum spanning tree of Ρ1 where is a function of d and
t only will be set to 20(d4)/t014)).
ig e i g Let Ρ and Ρ 1 be sets of points in and let be any positive number. Let G
be any (1 + t - p nn r of Ρ U Ρ 1 satisfying the so called ( 1 + t -l pfr property
f56] for 1 < 1 + Modify G to obtain an r-locally-light graph with the vertex
set Ρ U ι th t h t b r ph St n r tr f Ρ 1 with respect to Ρ whose cost
is at most (1 + 2 times the cost of the SMUT of Ρ 1 with respect to Ρ
Sea c i g Let Ρ and Ρ1 be sets of points in and let r be any positive integer. Let G
b n r-l ll -l ht r ph n Ρο U Ρ1 Find a minimum-cost Steiner tree of Ρ 1 with
respect to Ρ that is a subgraph of G.
5 i e i g o SM
I this section it will be shown how to perform the filtering phase efficiently. First it will be droved that the following algorithm finds a subset X of Ρ that satisfies the required filtering 89
o eac e ge e o e s a e w ose cos e cee s e i 1 - ac io o e cos o mi imum s a i g ee o 13 1 ci cumsc i e a - ime sio a a a(e wi e
ce e a e mi e o e a o a ius (e equa o ρ /ε imes e e g o e
e ge w e e Ρ is a u c io e e i g o y o Ρ =
3 e Y e a su se o o a i c u es a oi s co ai e i e co s uc e a s a ossi y some o e oi s i Ρ ο a is a ce a mos (e om e ce e o suc a a (e
o eac a (e e i e a ( ec i i ea - ime sio a cu e Coe o si e e g 8 oe a is co ce ic wi e a o e Wi i eac cu e Coe i o uce a g i
wi i e aci g e ε /( Δ A w e e Δ is e ou o ma imum eg ee o a y MS o oi s i ime sio e se e i i ia y em y e ea e y i e i c easi g e g o e o e e ges e o S assig eac oi Ε Y associa e wi a a oe o e c oses oi o e g i Coe o eac g i oi i Coe i e e is a eas o e oi Ε Y assig e o i a o e suc a oi o
Lemma 5.4.1 (SMT Filtering) For any point sets o nd ι n and any positive real n b r ε th b t of o satisfies the filtering property.
I o e o o e emma 5 1 O e as o o e a e se ς o o ai e i e a o e co s uc io sa is ies e o owi g wo o e ies
1 e cos o e SM o Ρ ι wi es ec o is a mos 1 + ε imes e cos o e SM o Ρ 1 wi es ec o Ρ
e cos o e mi imum s a i g ee o U Αι is u e ou e y °Ν /εoς
imes e cos o e mi imum s a i g ee o Ρ1
I e o owi g e se is i s s ow o sa is y e i s o e y a e i is s ow o sa is y e seco o e y e oo is a e o g a wi e ese e i a ge e a o m a ca e e ua y e use i some u e a ica io s 9 5 1 i s i e i g o e y e oo ee s a cou e o au i ia y emmas o owi g is a s a a esu a ou e u e ou o e cos o e mi imum s a i g ees co ai e i a ime sio a a ( o a oo see e g [77]
emma 5 Let P be a set of points in contained in a dndimensional ball of ra- dius Then, the minimum spanning tree of P has cre upper bounded by - 1-1/ ( -1 (1_ _1/ In particular, if and then the cre of the MST of P is upper bounded b Τn1-1 /d. ❑
e seco emma gi es a u e ou o e ma imum eg ee i a mi imum s a i g ee (a a SM o a oi se i
emma 5 3 [1 ] Let P be a set of i points in Rd . Then, any minimum spanning tree of
P has maximum degree Δ upper bounded by A = ( where c is an absolute constant.
Ο
om ow o Δ wi e use o e o e e u e ou o e ma imum eg ee o a y mi imum s a i g ee o a se o oi s i ( oug we wi o use i e o a io e e e e ce o ow a sim e emma wi e o e w ic s ows some asic o e y o SMU (w ic o s a so o MS a a s i s a e s a i ge e a i a i a y co ec e g a s
emma 5 Let Ρ and P 1 be disjoint point sets in Rd . Let G be any graph connected with respect to P1 (for example, a spanner on Ρ 1 ) and let be an SMUT of P1 with respect t Ρσ . Let and be any two points in 1 and let A be the shortest path in G between
Z and Let e be any edge in T of l n th £. If after removal of e from T, points Z and v are in different connected components of the resulting forest, then at least one edge on the p th Ρ ν is not shorter than f. 91 oo e oo is y co a ic io Su ose a e emo a o e om A wo su ees
A1 a Al wi Ε A1 a Ε A2 a e o ai e a su ose a a e ges o /3 i G a e s o e a Si ce a a e i i e e su ees e e e mus e is a e ge
Cy - suc eΤ υν o assum io a e C y a Ε Τ1 A ιCy1 y Ε e e o e
Τι + Al + y o ms a S ei e ee o Ρ 1 wi es ec o Ρ w ose cos is sma e a e cos o A u is co a ic s e assum io a Τ is a SM o Ρ1 wi es ec o o ❑
e emma is co ce e wi a s co ai i g o oi s om Ρ1 Ι wi e s ow a i a case i a y SM a y co ce ic a o a s ig y sma e a ius as e um e o S ei e oi s u e ou e y { o ice a a simi a emma is use y ao a Smi i [1 1] a e oo is is c a e uses ma y i eas om [1 1] e mai i e e ces a e cause y e ac a i is c a e e S ei e ees co si e e a e e e e se i a su se o o U P 1 w i e e oo i [1 1] wo ke o o = Rd. is im ies amo g o e a ao a Smi cou use ma y we k ow o e ies o suc S ei e ees ( o e am e i a case e ma imum eg ee i a mi imum-cos S ei e ee is 3
emma 5 5 Let o nd ι b n d j nt p nt t n R . t Ι nd Ι b n t concentric d-dimensional balls in d of radius 1 and S 1, respectively. Then for any SMUT of P1 with respect to o if Idο1containsth n Ιι nt n no points t r from P
(16 e Δ = 2 ( Steiner points, i.e., SMUT points from Ρ
oo i s i a SM o Ρ 1 wi es ec o o a e o e i y A e Δ e o e e ma imum eg ee o S ei e oi s i e SM ( o ice a emma 5 3 e su es a
Δ C o ce ai co s a A Ι ca e assume wi ou oss o ge e a i y a e e y S ei e oi is o eg ee a eas 3 e ce a so e um e o oi s i ι w ic wi e e o e y n, is igge a e um e o S ei e oi s e s e o e e um e o
S ei e oi s co ai e i Ι ; ι ; e goa is o o e a s < (16 e Δ 9 of places where A touches the border of i.e., the number of edges that have exactly one endpoint contained in a Next, let S denote the number of Steiner points contained in
and let denote the portion of Τ contained in Note that by assumption, for any
Pick any R and R* such that S < R < R* < 1 o ice a i Ν e
Ν a e ce e emma is o e So su ose a Ν , and thus
2d• Observe that since co ai s o oi s om Ρ 1 , contains S Steiner points, and all Steiner points a e o eg ee a eas 3 us Ν S + 2 > S (This follows from the fact that co ai s a eas S "internal" nodes of degree at least 3 and
"leaves" which are of degree 1.) Furthermore, notice that the length of is at least
(R* — R) • S On the other hand, co ai s a eas S Steiner points and additional N points on its border. Therefore, by Lemma 5.4.2,
Recall that Ν o = Ν 1 e e o e o e ca com i e e i equa i y Δ • S that holds for all 0 < R < 1 with a e i equa i ies a o e o o ai
e e o e si ce ι o e ca ge
A more general form o is esu is s a e as o ows 93
Since Δ - S for any 0 < R < 1, one can apply the claim above to obtain the following bound
o ice a i S ι _ ι δ Δ o a y i ege 1 1 1 e ( y emma 5 3 e emma is o e e e o e o e ca assume a S ι ι ό Δ a e ce e u e ou a o e ca e sim i ie o ge
First filtering property of set Y Now Y will be proved to satisfy the first property. Sim- ilarly as in the proof of Lemma 5.4.5, some of the arguments are similar to those used by Rao and Smith [101] in their ETAS for the Euclidean (complete) Steiner tree problem. This time, however, the differences caused by the fact that Rao and Smith worked with Ρρ = are even bigger. Therefore, most of the details of the following proof are completely new. 94
h n th r b r ph Η f G th t nn t d th r p t t v rt n
nd h r pp r b nd d b (1 + ε t th n n r St n r tr f ι th r p t t Ρ
Proof: e e e SMU o Ρ 1 wi es ec o Ρ I ca e assume wi ou oss o ge e a i y a e e y S ei e oi i is o eg ee g ea e a e a oac is o co s uc e su g a Η o G usi g e o owi g wo s e oce u e
1 o e e y e ge i wi Ε Y U i Η co ai s e s o es a i G e wee a
A e S e 1 Η may e isco ec e wi es ec o Ρ 1 ; make Η co ec e y a i g a i io a e ges a i g o a mi imum weig
I is easy o see a e o ai e g a Η is co ec e wi es ec o Y U i (a so wi es ec o e ices i Ρ 1 oo W a wi e o e e ow is a i s cos is u e ou e y (1 + ε imes e cos o e oo co sis s o wo a s ecause o e s a e o e y e g a Η o ai e i S e 1 as i s cos u e ou e y (1 + ε imes e cos o e e ges use i is s e e e o e w a ea y as o e o e is a e seco a o g a Η as a u e ou suc a e o a cos o Η is ou e y (1 + ε imes e cos o 95
es ec i e y I e o owi g i wi e s ow a (assumi g u ςΙ Y U Ρ1 e e mus e a e ge Cy i a eas o e o u a w ose e g is g ea e a o equa o /k
• i s i wi e s ow a i is im ossi e a a e same ime a e ges i u a e
o e g sma e a /k S SM u co ai s a oi C Ε Ρ 1 a is co ai e i B2 a S M co ai s a oi y Ε i a is co ai e i B2 (A equi a e case is w e is e c a ge wi a is w e a e ges i a e o e g sma e a /k S ΜΤΣ co ai s a oi C Ε i a is co ai e i B2 . a S M co ai s a oi y E ι a is co ai e i
I a e ges i u a e o e g sma e a /k e u is co ai e i B2 . Si ce a y a e isco ec e a e emo a o e ge emma 5 im ies a S as a eas o e e ge e o a s o es a e wee a y w ose e g e is a eas ε
ow i wi e s ow a a A χ y ca o a e a e e ou si e ±ε • I ee i
a we e o co ai e i 1( ± Y e e e we e a oi i A χ y a is o
co ai e i Β ±ε • e e g o e a A χ ± is o s o e a I I + i u is co a ic s e e i i io o e s a e ecause e e g o y is assume o e u e ou e y 1 + ε/ imes e e g o Cy a o e ca easi y s ow (see e g emma 5 1 1 a
So ow i is k ow a a A χ y mus e co ai e i ( +ε r • is i a icu a
mea s a e ge e is co ai e i Β ( ±ε . • o ice a e e oi s o e ge e e o g
o i e e o e y e co s uc io o se Y a e oi s om Ρ a a e co ai e i e a o e ( a i g e ce e a e mi oi o e a a ius A • ιeΙ/ε e o g o Y ow si ce e e g o e is g ea e a o equa o £ e a ius o oe is a
eas A Εεε Si ce A Επε = (3 + o se e a o e co ai s e e i e a Β 9
Therefore points Z a mus e o g o Y U Ρ 1 is is a contradiction, because it is
assumed that Z Y U P i .
• Otherwise, there either mus e e ges i eac o u a of lengths greater than or
equa o εk o S SM u mus co ai o oi om 1 a is co ai e i o
SM a 1 is mus co ai e co ai i o oi om Ρ
Co si e e case w e S ΜΤΣ co ai s o oi om Ρ 1 a is co ai e i
Then, either A is co ai e i Β o e e is a e ge i u o e g g ea e a
o equa o εk i s s ow a u ca o e co ai e i Β I ee i A we e
co ai e i A e wou =S ei e A Ek o1 eco ai mo e a ΙΤυ
oi s O e o e a i is k ow a Β co ai s o oi s om Ρ1 i 5ΜA υ
y emma 5 5 is yie s a co a ic io e ce u ca o e co ai e i Β
is i u im ies a o e o u a say u mus co ai a e ge o e g g ea e a o equa o /k
us o e ca co c u e a e e is a e ge y i u w ose e g is g ea e a e cos o e ge wi e "c a ge " o Cy i S e 1 o e co s uc io o
e Κ1 e e se o e ges i Τ a i g o e oi s i Y U ι a e Κ e o e e se o e emai i g e ges i Τ Eick a y e ge e i a e o e i s cos y A O se e a e may e c a ge o y a most 2 (2 Lk — 1) <4 k "s o " e ges o ice u e a e cos o eac suc a s o e ge is upper bounded y A εε o1 k (I ee y co s uc io a s o e ge o e g Ε is charged to edge e only i A 1 Ε kεε e e o e e o a cost of all edges in A charged to e is upper bounded by ε A. Furthermore, notice that only the edges in Κ a e c a ge i this way. This im ies a e o a e g o a e ges i Κ is u e ou e y ε imes e o a cos o Τ
e Η 1 e o e e subgraph of Η obtained in Step 1 and Η e o e e su g a o Η o ai e i S e o the construction. Notice that the subgraph of A i uce y e 97 e ges i Κ is a o es ick a y ee τ i is o es a e e e se o e ices i τ e o gi g o C Y U 1 C ea y τ is a S ei e ee o I is we k ow a e cos o a mi imum s a i g s a i g ee o a y se o oi s (i a y me ic s ace is u e ou e y wice e cos o e SMU e e o e e cos o a mi imum s a i g ee o
is u e ou e y wice e cos o τ e ce si ce e cos o Η is u e ou e y
e sum o e cos s o mi imum s a i g ees o o e a ees τ i Κ e cos o Η is u e ou e y wice e cos o e e ges i Κ y e iscussio a o e e o a
e g o a e ges i Κ is u e ou e y ε imes e o a cos o us e cos o is u e ou e y ε imes e o a cos o
o summa i e e g a co sis s o g a Η 1 a y e s a e o e y e cos o Η1 is u e ou e y o1 + imes e o a cos o y ou a gume s a o e e cos o Η is u e ou e y ε imes e o a cos o is im ies a e o a cos o Η is u e ou e y o1 + ε imes e o a cos o A a e ce yie s
e oo o e emma ❑
First filtering property of se emma 5 gi es "a mos " a su se o o we wa o o ai i e i e i g ase owe e e so o ai e se Y oes o a e o sa is y e seco i e i g o e y a is a e cos o a mi imum s a i g ee o Y U i is o o io a o e cos o a mi imum s a i g ee o ι e o em is a i e e a e oo ma y oi s i Y e e e cos o e mi imum s a i g ee o Y ca e a i a y a ge e e o e o i es iga io s o e ca c oose a su se o Y a is o ai e y a "s a si ka io " o Y O e ca s ig y mo i y e oo o emma 5 o o ai e o owi g esu
Lemma 5.4.7 Let Αρ and P be two point sets in Let S be an (1 + spanner of Ρ1 . For each edge e of the spanner whose cre exceeds the Α 1 fraction of the cre of minimum spanning tree of Α 1 l t Coe be the ball with the center at the midpoint of e and with the radius equal to ρ /ε t n the length of the edge e where A = 0( 3) • For each 98 ball Β oe let Ye be the subset of o consisting of all points that are contained in it. Let e
Then, there is a subgraph H of G that is connected with respect to vertices in i and whose cost is upper bounded by (1 + 2 ε + (1 times the minimum-cost Steiner tree of P1 with respect to o.
Proof: e oo is a s ig mo i ica io o e oo o emma 5 e e o e o y e aces w e e a oo as o e mo i ie wi e e e e Su ose i s a e e o e e y s o e ges e o S e a s o e a e se s
Υλι ed i uce y em a e ee c ea e O e o ows e a gume s om e oo o emma 5 O e ee s o mo i y
em o y w e e e is a a A a is co ai e i C +ε a e e o e e e is a e ge e o e g g ea e a o equa o £ a is co ai e i Ι +ε Τ a as i s o e oi s i 13 1 Ι is case e co a ic io wi o e o ai e as i e oo o emma 5 owe e o se e a si ce e e g o e is ess a o equa o ( + S a si ce e a o e co ai s a (a e ce o oi s a e i Ye e e mus
ei e oi i w a es ec i e y I is easy o see a eac suc a mo i ka io may i c ease e cos o e g a y a mos
e e o e e co s uc io o ca e mo i ie as o ows
1. o e e y e ge i wi u Ε Χ U 1 co ai s e s o es a i G e wee a
O e wise o e e y o e e ge i i e e is a e ge e i S (a e ce wi
o e oi s i Ρ 1 suc a a e o g o Y e e ick e c oses oi s 99
W Ε Χ o a es ec i e y e mo i y y mo i g a e ges i ci e o a o a e ei e oi i w a es ec i e y I a y ew e ge as o e oi s i U i e o as i S e 1
3 A e wa s Η may e s i isco ec e wi es ec o 1 ; make Η co ec e y a i g a i io a e ges a i g o a mi imum weig
O e ca a y e same a gume s as i e oo o emma 5 o s ow a e cos o e e ges i Η c ea e i S e s 1 a 3 is u e ou e y (1 + imes e mi imum- cos S ei e ee o 1 wi es ec o Α O e o e a y e a gume s a o e e o a cos o e mo i ka io s e o me i S e is u e ou e y w ic is i ε imes e mi imum-cos S ei e ee o ι wi es ec o A o o ai e c aime esu i emai s o mo e a e oi s i Χ o e e ges e w ose cos s o o e cee 1 I- o e cos o e mi imum s a i g ee o i o a e oi o e Suc mo eme s ca i c ease e o a cos o G y (si ce e o a um e o e ges i e s a e is ( /ε °( ime mum s a i g ee w ic wi i c easi g 1 Ι is a i a i y sma i com a iso o e cos o a mi imum S ei e ee o ι wi es ec o A ❑
e o owi g is imme ia e co o a y o emma 5 7
5 Seco i e i g o e y I o e o o e a se Χ sa is ies e seco i e i g o e y e o owi g emma wi e o e i s
emma 5 9 Let A and ι be two point sets in Let S be an (1 + spanner of ι r h d e of the spanner whose cost exceeds the 113, ( = fraction of the cost of 1 minimum spanning tree of P 1 , let B oe) and oe be the balls with the center at the midpoint
of e and with the radius equal to le E /ε and • e • E /ε respectively, where A = ( 3 For each ball oe let YS be the subset of consisting of all points that are contained in it, and let Y be any subset of o containing all points in Y and possibly some other points contained in the ball oe Let Bed be any subset of Y D such that (i) if y E YS — e then that there is C E e such that 1y e ε /( E and (ii) if C, y E e then
Icy > Ie ε /( A ‚/ Let B = USES BSI Then, there is a spanning tree of B U ι whose total cost is upper bounded by _ (A / • (ο ( Ρ /ε3 -1 ° /ε° times the cost of the spanner S
oo i s a s a i g g a Τ o U ι wi e co s uc e i s o a Τ co ai s a mi imum s a i g ee o 1 e o e e y e ge e E S we i a mi imum s a i g
ee ea o e ( o co e io i SI is u e i e i wi e ea e as a em y se co ec i o a y o e e oi s o e ge e a a e o ai e ee o I is easy o see a so
e i e g a is a s a i g ee o U 1 e e o e e cos o e mi imum s a i g ee o U ι is u e ou e y e cos o Τ a e e o e e a e io wi e ocuse o es ima i g e cos o Τ
i a a i a y e geg i S y o e a S i Ε y1 e - 11 /( e A u e mo e a oi s i SI a e co ai e i a a o a ius u e ou e y Ice ρ /ε is imme ia e y im ies a e si e o SI is ( (ν Η Eá/ε3 e ce y emma 1 5 e cos o e is ( 1e E /ε • (ο ( Eá/ε 3 e e o e i ΜSΤ(ι e o es e cos o e mi imum s a i g ee o ι a y cοsτ (S e cos o e s a e S o e ca o ai e o owi g u e ou o e o a cos o d-1 MS (ι + Σ( e A / • (ο ) d-1 < ( A /ε • (οο COS S ES 1 1
Si ce sa is ies e wo o e ies o e = a a y o e a o e emma
e e e is a s a i g ee o U 1 w ose o a cos is u e ou e y Ι imes e cos o e s a e S eca a e is uu u o me s a e ose cos is ou e y ε ε = ( / imes ΜSΤ(ι o e ca easi y see a e cos o Mi imum S a i g ee o U i is ou e y a ε = (2°(ί1 /ε°( ) • ( / °( = Ο( °(ι1 /ε°( imes e cos o Mi imum S a i g ee o
ι Μsτ(ι
5 3 Com e i y o SMU i e i g is sec io s ows ow o im eme SM i e i g ase e co s uc io o e se esc i e a o e as ee s ecia y u e o e a e a e kie im eme a io e o owi g emma wi e o e i s
emma 5 1 h f lt r n l r th f r SMU n b pl nt d t d t r n th t n t (d/ε ° • n oge h r n = 1Α U i
oo y emma 5 S e 1 ca e im eme e i ime ( / °( ι 1+ ( • ι og ii o im eme S e s 3 i e o e e mi e e se Y o e ca use a co e a a s uc-
u e o ppr x t p nt l t n n l b ll ue o Ι yk a Mo wa i [ 7] o a gi e a o ima io ac o c 1 a a gi e ea ey esig e a s a ic a a s uc u e o a se o oi s S C ca e (c - E suc a o a y oi q Ε R', i S co ai s a oi wi i is a ce om q e (c τ E E ou u s a oi q E S a is omise o e wi i a is a ce a mos c om q I yk a Mo wa i [ 7 eo em 3] s ow a e e is a a go i m o ( τ E E (i e Euc i ea - ime sio a me ic a a e a O(ISI ° - ime e ocessi g ac ie es ( que y ime o e a i e a go i m a o e e cos s o e s a e e ges om w ic e a s o igi a e a i o a oga i - mic um e o i e a s o e o m [ o k± a w e e o is ei mi imum cos (w ic is owe ou e y e cos o e mi imum s a i g ee o 1 o e i 1 o e e - 1 3
ema k 5 5 h pr p rt f th n tr t n n 5 5 1 th t th bt n d r ph Η has Αρ U Ρι as its vertex set. This distinguishes it from previous constructions, e.g., (3, 101), where
r l t d r ph Η ll d t rb tr r p nt t d ΑΒ U ι h pr p rt r t l n th approximation of SMUT and other connectivity problems (in contrast to, e.g., ASP approximation).
Ind d pp th t Η h v rt x p nt q Ρσ U ι h h tnv rt x n Η nd th t Η
nt n tr t b pr n d t St n r tr f r Ρ ( r t b t h n f th d r f q in
T is very high, it might be impossible to remove q from T to obtain a tree of the cost as good (or
l t d th t f ( n ntr t AS n p r t f Ρ n b l d f d t bt n
S n Ρ th t n t n r Ob rv th t th n tr t n ll t e the edges, that
n d b t n t p nt x nd n Ρ p th f tr ht-l n nt b t n x nd the discussion at the beginning of Section 5.2.
ow e mai i eas e i e oo emma 5 5 1 wi e iscusse As me io e is emma uses (a mos i ec y esu s a ea y e e o e i [ ] e e o e e esu s ese e e e a e gi e wi ou oo s eo em 3 1 om [ ] is e key ec ica eo em i a a e
emma 5 5 3 [ ] Let and be any positive but otherwise arbitrary realms (that is, they may depend also on other parameters). Let k be an arbitrary positive integer. Let P be an arbitrary point set in Rd. Let G be a spanner for P that has n ( o °( edges, has total cost COS (G and that satisfies the ( 1 + t nleapfrog property, where 1 < t < 1 + t. Choose a shifted dissection uniformly at random. Then, one can modify G to r ph Η with vertex set P such that
• there exists a knedgenconnected multigraph Μ h h p nn n b r ph f Η with possible parallel edges (of multiplicity at most k whose expected cost (over the 102
o a e ce e oi s o s a e e ges a i g cos i e i e a [ 1 o i+1 o a e que y i wi a e oi s i Ρ e ime ee e o e co s uc io o e oga i mic um e o PLEB a a s uc u es is ( og o imes e um e o s a e e ges (w ic is
ε ε = ιι ( o e a e ime ee e o e o que ies is (Ιοg 11 o1 (
1 7] e ce e o a ime equi e i S e s a 3 is ( ( o ( og τι O se e a u i g e co s uc io o Y o e ca a so i se eac oi accou e o Y i o a is co es o i g o e sma es a i e o gs o (a o ima e y y ocessi g e ( og ii E que ies wi ou i c easi g e asym o ic ime e o ma ce ese is s a e use u i e im eme a io o S e Sim y o eac o e a s oe a o eac oi i e co es o i g is Aoe o e c ecks w e e o o e oi a e ou i g o o e ea es oi o e g i Coe is a ea y ma ke If o o e ma ks e g i oi a a o I is easy o see a i is way S e ca e im eme e i
ime (( o °ί • τι ❑
5.5 Lightening for SMUT o e ig e i g ase e amewo k e e o e i [ ] wi e use o a s o m s a - e s i o - oca y- ig g a s mai ai i g co ec i i y o e ies the minimum-cost k-edge-connected multigraph spanning where MST (P) denotes
the cost of the minimum spanning tree of P
Moreover, the modification can be performed in time 0(d 3 • • og + • ( °c +
( / °k ❑
Remark 5.5.4 Notice that the original theorem in [26] the soncalled "isolation property" of the spanner was required. However, the proof of that theorem can be easily modified to spanners satisfying the "leapfrog property" (see [56] for a precise definition). The reason of this change is that in [26] the authors were using the spanner construction due to Aryan e a [5]. Unfortunately, it has been shown (see, e.g., [56]) that there is a serious flaw in the construction due to Aryan e a The only known existing construction of "optimal" spanners is due to Gudmundsson e a [56], see Lemma 5.2.2. This construction satisfies the leapfrog property and therefore we gave a modified version of Theorem 3.1 from [26].
Remark 5.5.5 Some brief intuitions why the leapfrog property is required (and why it can replace the isolation property) in the proof of Lemma 5.5.3will be given here . As it is shown in [25, Lemma 2.3] (a predecessor paper of [26]), with a very little cost increase one can transform any spanner (or any k-edge-connected multigraph) into a graph that is "almost" r-locally-light. By "almost" it is meant that the number of "short" relevant crossings is at most / where a "short" crossing of a facet of side length A is of the length O(Va • A However, the number of longer crossings may be significantly larger. But if the input spanner satisfies the leapfrog property then one can show that the number of "long" relevant crossings is small. This allows to prove that the construction leads to an r-locally-light graph. 105
Remark 5.5.6 Actually, Lemma 5.5.3 requires that the input points are slightly "perturbed" and are soncalled "well-rounded." This modification of the input points is used in all pa- pers following Aroma's framework (see, e.g., [3, 26, 101]). It requires to move all input points (by a very tiny vector) to a certain grid in Rd. Although formally this step is very
Remark 5.5.7 Finally notice that the key feature of the construction in Lemma 5.5.3 is that th bt n d r ph Η h t v rt x t h th pr p rt th t d t n h t from previous construction, for example, as in [3, 101]. This property is not required if one wants to find an approximation, for example, for ASP, but it is critical when dealing with the SMUT problem and other connectivity problems. Indeed, suppose th t Η n new point q P and this vertex is a cut-vertex in Η and suppose that Η contains a certain tree T that we would like to use as a Steiner tree for P (or its subset). Then, if the degree of q in this tree is very high, then it might be impossible to remove q from T to obtain a tree of the cost as good (or almost as good) as the cost of T . (Notice that this is easy to be done in the case of TSP, as in [3, 101], because ASP on a superset of P can be easily modified to obtain a TSP on P without any cost increase.) Additionally, this construction allows to e the edges, that is, an edge between two points C and y in P is a path of straight-line segments between C and y, see the discussion at the beginning of Section 5.2.
Kee i g i mi e ema ks a o e o ice a e oo o emma 5 5 3 is ac ua y muc s o ge e oo o emma 5 5 3 is e o me y a seque ce o emo a s o ce ai e ges om a i se i g ew e ges o G e i ea e i i se i g e e ges is a ey a e equi e o kee e same co ec i i y o e o ai e g a as o G is oes o mea o y g o a co ec i i y u a so e oca o e (u o co ec i i y k e e o e e oo (wi ou a y mo i ica io o emma 5 5 3 ea s o e o owi g muc s o ge esu ( a is s a e e e wi ou a oo 106
Lemma 5.5.8 t ε nd b be any positive but otherwise arbitrary realms (that is, they may depend also on other parameters). Let k be an arbitrary positive integer. Let Α be an arbitrary point set in Rd. Let G be a (1 + 2 t)-spanner for P that has ( / °ί edges, has total cost COST (G), and that satisfies the (1 + 2 t, t) -leapfrog property, where 1 < t < 1 + t. Choose a shifted dissection uniformly at random. Then, one can modify G to a graph Η with vertex set P such that
• there exists a multigraph Μ h h p nn n b r ph f Η th p bl p r ll l
edges (of multiplicity at most k) such that
1. for every pair of points x, y in P, if there are r edge-disjoint paths between C
and y in G then there are at least minfr y , kl edge-disjoint path between C and y in Μ
2. the expected cost of M (over the random choice of the shifted dissection) is
) times the minimum-cost multigraph span-
ning satisfying th b v property 1, where MST(P) denotes the cost of the
minimum spanning tr f 107
5.6 Searching for SMUT e a oac i e Sea c i g ase is o a y y amic og ammi g o i a o ima S ei e ee i a - oca y- ig g a
Lemma 5.6.1 Let Α and ι be sets of jointly n points in d and let r be any integer. Let
G be an r-locally-light (with respect to a certain given shifted dissection) graph on P U ι
Then, in time Οίτι • r° (ld Τ n n f nd n - t St n r tr A of P 1 with respect t Ρσ that is a subgraph of G
is emma is o e y s owi g ow o use y amic og ammi g o i a mi imum-cos S ei e ee Τ o ι wi es ec o Α I wi e s ow a e u i g ime o e co s uc io is O( i • ( άΓ e y amic og ammi g oce u e is esse - ia y e same as e o e use i E AS a go i ms o S a e Euc i ea com e e S ei e ee o ems ue o Au o a [3 101], u o e sake o com e e ess i is ese e e e i e ai s I s ou e me io e e e a ike [3 5 1 1] i e y amic og ammi g oce u e e i u is assume o e a we - ou e oi se w ic is o e y e u a- io o e o igi a oi s ike me io e e o e o sim ici y o ese a io i wi e eg ec e i s e i e e su o em i a egio R o e issec io Let S be a set containing m relevant crossings on the facets of R. Given a partition (Si 5 Ski 1 k m of S, find a minimum-cost steiner forest that (i) consist of k trees, (ii) the i-th tree - ; of the forest contains all the crossings (called portals in [3, 101]) in S as their leaves, and (iii) all the trees of the forest collectively contain all points Ε ι in this region. I o suc o es e is s o is a i io is a i io is sai o e invalid. e ee A is ou i a o om-u as io i s a mi imum-cos S ei e o es o eac ea egio is ou e e mi imum cos o es s o eac o - ea egio y com i i g e o ima so u io s o i s c i egio s e A we a e ooki g o is e mi imum-cos o es o e oo egio 108
1 o e su o ems o a ea egio e e is a mos o e oi i e egio e com u a io is i ia o e ca easi y com u e e o ima so u io i ( e e a e ee cases e e i g o e um e a y e o e oi
• o oi e e o e o e e a c ossi gs a a e o es is em y a e cos is a ways Ο
• O e oi Ε Α ι e e a e o y se e a ways o a i io o S a a e a i a mus e co ai e i o e o e ee w ic makes e o a cos o e o es o e a i io mi imum
• O e oi Ε Α A mos e same as e seco case e ce a e e is a S ei e ca i a e so is o equi e o e co ai e i e o es S eci ka y w e k = m e o ima so u io is a o es wi m ees eac co ai i g o y o e c ossi g a e o a cos is
o eac ma ima seque ce o egio s 1 e suc a o i = 1 q - 1,
R±1 is e o y c i egio o i a as oi s i i e mi imum cos o es wi i
1 ca e easi y com u e om a wi i e us y e e i g e a e a o g e
e ges a c oss e ace s o e
3 o e su o ems a isi g om a o - ea egio wi mo e a o e oi i i i wi e s ow a o e ca com u e a i s o ima so u io s eac wi es ec o a
a icu a a i io o e c ossi gs i ime ( ο(
Eac o es i a e egio is a com i a io o o es s om i s c i egio s
1 Rl,... Red es ec i e y So e mi imum-cos o es i ca e ou y e u- me a i g a com i a io s o o es s om i s c i egio e com i a io is o e oug o e a i g c ossi gs o e s a e ace s o c i egio s I e e is o misma c e c ossi gs a o es i e a e egio is o ai e is o es a so e- i es e c ossi gs se S o e o e o a a a i io o S o a s eci k se S 1 9
a a s eci k a i io o S e e may e ma y com i a io s co es o o i o e ee s o i e o e a gi es e mi imum cos o es
o e a i a c ossi g s a e y wo c i egio s is e e a o o e c i u o o a o e a is e e is a e ge w ic comes om a oi ou si e e egio ge s oug o e o i s c i a e s i a o e c i e com i a io is s i a i e segme o e e ge i e o me c i is o i c u e i e com u a io o e c i u i ee s o e a e o e cos o e a e egio
e e a e ( o es s o eac c i egio so e ime com e i y o com i a io
is Ο( (
o a s i e issec io i as O( ea egio s wi a oi i i a as o y ( o - ea egio s wi mo e a o e oi i em so e ime com e i y o e w o e o em is O ( • °1
5.7 Polynomial-Time ApproDimation Scheme for SMT ow i wi e s ow ow o com i e a e a gume s om Sec io s 5 — 5 o o ai a E AS o e Euc i ea SMU o em e i u o e o em co sis s o wo se s Ρ
wi csρec ιο o wιιυSC 1S 1 55 iimIi o equa ιο 1 -1- ε imes me mi imum
i s a y emma 5 1 o i a su se o o a i g e omise o e ies
(wi t = á ε e ake a (1 + -s a e G o U i a a y emma 5 5 1 o mo i y G i o e o o ai a - oca y- ig g a Η a as as a su g a a S ei e ee o 1 wi es ec o w ose cos is u e ou e y (1 - - imes e cos o e SM o ι wi es ec o i a y a y emma 5 1 o i a mi imum-cos S ei e ee o ι wi es ec o a is a su g a o Η a ou u i is ea s o e o owi g eo em 11
eo em 5 7 1 There is a polynoiiiial-time approximation scheme for the Steiner Mini- mum Problem in Euclidean space In particular, for any sets of points nd Ρ1 n ‚
one can find a Steiner tree of P th r p t t Ρο h t t t 1 -1- ε t th n For constant nd t th r ιτn η t f th l r th O ( og Ο
5 1 co ec i i y o em O e ca e e e a go i m om e e ious sec io o o ai a o y omia - ime a - o ima io sc eme o e { -Co ec i i y o em i Euc i ea g a s Ac ua y s ecia a e io as ee ai o ese e a go i m o e SM o em i a o m e e a e o i c u e e 1 co ec i i y o em ue o emma 5 o e may co si e o y e 1 -e ge-co ec i i y o em a a ow e ou u o e gi e i a o m o a mu ig a e a go i m uses simi a ee ases as e a go i m o e SMU o em e i e i g ase is esse ia y e same as o e SMU o em e ce a o e wo ks o e s a e o Ρ ι U Ρ ow i o e o i a Y Simi a y e ig i g ase is esse ia y e same as o e SM o em see Sec io 5 1 Sea c i g ase is e o y ase a is com e e y i e e a a e icky u s i o e ca im eme is ase e kie y y usi g e i ea o connectivity type o e mu ig a wi i a egio o e issec io e e ai e esc i io o e y amic oce u e a e oo o i s co ec ess is i Sec io 5
5 1 ig e i g o 1 -E ge-Co ec i i y o 1 -e ge-co ec i i y o e ca a gue a a ogous y as i e oo o e ig e i g emma o SM Remark 5.8.2 The arguments above can be also easily modified to include the Lightening
Lemma for arbitrary f0,1, ... , 21-edge-connected in the same time complexity.
5.8.2 Dynamic Programming for {0, 1,2)-Edge-Connectivity The goal of searching ase o 1 1-e ge-co ec e is usi g y amic og ammi g
o so e e o owi g o em e = Ρ U ι U e a we - ou e oi se i Ι ; where P1 is e se o oi s w ose co ec i i y equi eme is i Gi e a a i a y s i e issec io e G e a Euc i ea g a o a is - oca y- ig wi es ec o is issec io i e mi imum-cos { 1 1-e ge-co ec e mu ig a Η o Ρ o w ic e i uce g a is a su g a o G e o e esc i i g e oce u e o y amic og ammi g some e i i io s wi e i o uce i s o a y egio o e issec io a e u ica i g some o e e ges o G a multigraph within is e a o e mu ig a co ai e wi i esu i g om e emo a o a e ges a c oss u a e o e oi i si e (see igu e 5 (a- e mu ig a wi i as wo y es o oi s ose om w ic wi e ca e e input Figure 5.2 I us a io o co ec i i y y e co s uc io (a Α a o g a G i si e a egio R, (b) a multigraph M within R, (c) contraction of two-edge-connected components in M and (d) contraction of paths in M p nt a ose e i e y e c ossi gs o e e ges wi e o e o P, w ic wi e ca e e b rd r p nt e i ea o connectivity type i o uce i 1 5 ] wi e use e e u i e e a mo e su e c a ac e i a io s a e ee e e connectivity type o a mu ig a is e (" seu o"- o es o ai e y e o owi g s e s (See igu e 5 o a i us a io
• Co ac i g eac ma ima wo-e ge-co ec e com o e i e mu ig a i uce y i s o - ea e ices o a si g e e e a associa e a co ec i i y equi eme wi is e e w ic equa s o e ma imum co ec i i y equi eme amo g a e ices o e com o e
• Co ac i g eac ma ima a com ose o i u a /o co ac e oi s o e- g ee wo i o e si g e edge with e e oi s ei g e i s a e as e e a e a a associa i g with eac e oi o e ew e ge e co ec i i y e- qui eme equa o e ma imum co ec i i y equi eme amo g a e oi s o e a
I ca a e a a "leaf" in such a seu o- o es a e wo e ges co ec i g i wi i s a e s( us e e a e ossi e cyc es o e g wo is e ai s e ame " seu - o es s" is issue s a e ignored and e o a io s a e s ig y a use y ca i g suc " seu o- o es s" as o es i e co i ua io 113
om e e i i io a o e i is k ow a a co ec i i y y e is a o es a wo mu ig a a e sai o a e e same co ec i i y y e i e o es s a e isomo ic o a egio wi a mos e e a c ossi gs eac o es o ai e om a mu ig a w ose i uce g a is e g a wi i e egio as a mos ea es us ess a ( e ices so e e a e a mos (11 suc o es s a e e o e e e a e a mos ( co - ec i i y y es
Dynamic programming. e y amic og ammi g oce u e wi e e mi e o eac egio a o eac ossi e co ec i i y y e e mi imum-cos mu ig a o is y e wi i e egio
1 o eac ea egio eac ossi e mu ise S o a mos c ossi gs o e ace s o e egio si ce e e is a mos o e oi i e egio e mi imum-cos mu i- g a o eac co ec i i y y e co es o i g o S ca e easi y com u e i ime
ike y amic og ammi g i SMU o eac ma ima seque ce o egio s 1
suc is a q i+1i e as a o y oi s o c i i = egio 1 q o — 1 i i e mi imum cos mu ig a o eac co ec i i y y e wi i 1 ca e easi y
com u e om ose wi i e i co s a ime Si ce e e a e (1 co ec i i y y es so e o a ime ee e is O (
3 e e a y o e o - ea egio e Q ι Q Q e a a i a y
co ec i i y es ec i e y e y es ese o co ec i i y e c i egio s 1 y es( seo o- o es s a e com i e oug e o e a i g c ossi gs a e e im- i a e ese e ices om e esu i g g a a a si g e o ou e e ges c ossi g e commo ace s a ese e ices I a c ossi g o a commo ace is a e e a c ossi g i o e c i egio u o i e o e e e com i a io is a i I a a i g a is o ai e om e com i a io e i s co ec i i y y e is com u - e y e o mi g e ecessa y co ac io y e ume a i g a com i a io s e 11
mi imum-cos o eac co ec i i y y e a i s co es o i g mu ig a i ca e ou
e o a u i g ime o e com i a io co ac io is o ious y i ea i e o- a si e o ese co ec i i y types, a is ( Si ce e e a e a mos ( ( °ίτ
i a y a e e mi imum-cos mu ig a o eac co ec i i y y e o e oo egio is com u e e o ima mu ig a s o e o igi a o em ca e easi y ou y us aki g e o e wi mi imum cos amo g ose co ec i i y y es w ic co sis o a ee wi only o e e e a i g co ec i i y equi eme o
a some iso a e e ices e o gi g o Ρ
Correctness of the dynamic programming. ow a ske c wi e gi e o o e a e mu ig a s o ai e sa is ies e co ec i i y equi eme s a as mi imum cos i s e mu ig a wi e s ow o mee e co ec i i y equi eme o eac i u oi is is o ious y ue i o e ooks a co ec i i y y e o is mu ig a s ou I is a
ee oge e wi some oi s o Ρ so a oi s o Ρ1 U a e co ec e u e mo e o y o e e e a i g co ec i i y equi eme o is mea s a oi s e o g o Ρ a e co ai e i a wo-e ge-co ec e com o e s e ese e y is e e e i wi e o e a its cos is mi imum amo g a suc mu ig a s I is e oug o s ow a e so u io is o ima i eac egio i e i eac egio e mu ig a s o e so u io wi i it has mi imum cos amo g a mu ig a wi i is egio a i g e same co ec i i y y e is ca e o e y co a ic io Su ose e so u io is o o ima i some egio s ake a y suc a egio a e owes e e e ace e mu ig a s i is egio o e so u io y e o ima so u io a i g 115
e same co ec i i y y e i is egio I is easy o see a a ew mu ig a s sa is yi g e co ec i i y equi eme s u wi owe cos a e so u io is o ai e u is is co a ic o e ac a e so u io as mi imum cos amo g a mu ig a s a i g e e same co ec i i y y e
5 9 E e sio s 1 A e a gume s om Sec io 5 e ce e y amic og ammi g a ca e eas- i y ge e a i e o i c u e 1 - d - nn t v t f r lt r ph ega - i g e y amic og ammi g o e ca com i e e y amic og ammi g o e e ge-co ec i i y om [ 5] wi e me o o ea i g wi o u i o m co ec- i i y equi eme s us e a ia o e geome ic su i a i i y o em w e e e e ge-co ec i i y equi eme s sa is y „ E 1 k} a k = (1 a mi s a so a ASs e u i g ime is owe e sig i ka y g ea e a ( og oug i is s i o y omia o co s a a
Usi g e same a gume s as i [3] o e ca e e e ASs o i c u e o e me ics as we
3 e As cou e a so e e e o a i i i e omai o S ei e ca i a e oi s
(e g i Ρσ = Χ 1 o e as e Euc i ea com e e S ei e o em o i e o e ca e e mi e e se Y (o i s goo a o ima io as as as c aime i emma 5 1 Figure 5.3 I us a io o e oo o emma 5 1 1 (a A e am e o i u co igu a io (
ays Αχ Αν a Α (c Co s uc io o υ ( o ice a e same co s uc io is a i e e w e ΑΖ oes o ie e wee A a Α ν
5.10 AuDiliary Claims
Lemma 5.10.1 Let and be any positive real numbers and let and be two concentric dndimensional balls of radius r and R, respectively. If and u are points contained in Β and is not contained within of B R then
Proof: i s o ice a i is su icie o o e e emma o d = u e mo e i is easy o see a i is e oug o co si e e case w e a u a e o e ou a y o (o e wise o e cou mo e e ia g e Δ ( u o a e a u o e ou a y o w i e s i a i g o co ai e within a is o e ou a y o (o e wise o e cou mo e o e ou a y o wi ou c a gi g I Ί a wi ec easi g I I + u i ow see igu e 5 3 o a i us a io o e emai i g o e oo aw ee
ays ΑΧ Au a ΑΑ om e ce e o oug oi s u a es ec i e y (see
igu e 5 3 (b)). e i e a y o e e oi s o e i e sec io o e ou a y o a ΑΧΕ Au es ec i e y e e the oi o e ay ΑΙ a ies o e ou a y o (See igu e 5 3 (c o a i us a io y e S a es eo em 117 ow co si e e qua i a e a o (see igu e 5 3 (c Si ce e segme s a C a e a a e o is a a e oi u e mo e ( = ( CHAPTER 6
APPROXIMATION SCHEMES FOR MINIMUM 2-EDGE-CONNECTED AND RECONNECTED GRAPHS IN PLANAR GRAPHS
6.1 Introduction is c a e co si e s a o ima io a go i ms o e mos asic case o e su i a e e wo k esig o em i w ic e esu i g su g a s ou e esis a o e emo a o a single e ge o e e S e wo c assica o ems co si e e e e a e o i a mi - imum -e ge-co ec i i y ( - C s a i g su g a (a -ECSS o a - e e -co ec e ( - C o eco ec e s a i g su g a (a -ECSS o a a g a s As me io e e o e ese o ems a e ma -S - a [ 5] e e o u weig e g a s o w e u ica e e ges a e a owe ; e e o e o e ca o e ec a SASs u is oes o ec u e a AS o es ic e c asses o g a s i a icu a a ESAS e is s o o o ems i geometric graphs of co s a ime sio [ 5] S e goa i is c a e is o esig ASs o e wo o ems i a a g a s Ma y o y omia ime a o ima io a go i ms a e a ea y k ow o ese o - ems see e su ey [7 ] a mo e ece a a ces [ 19 39 7 ] I u weig e g a s e es cu e y k ow a o ima io a ios a e / o -ECSS o em [ ] a a /3 o e -ECSS o em [ 11 ] o o o ems i weig e a a g a s e es k ow su e o e ia - ime a o ima io gua a ee is s i [73 9 ] A ese a o ima io s a e ac ie e y o y omia - ime a go i ms wo ki g o ge e a weig e g a s S is c a e describes a AS o e mi imum -e ge-co ec i i y o em a a AS o e mi imum eco ec i i y o em o o u weig e a a g a s Mo e ge e a y w e e a a g a as e ge weig s e a go i ms a o ima e y so e e mi imum-cos o ems i ime 1 w e e Ύ is e a io o e o a e ge cos o e
118 119 o imum so u io cos S is is a AS w e γ is ou e ; o e a γ is ou e ( y 3 w e e e ge weig s a e u i o m S e ew ge e a a oac esem es e a o ima io sc emes o me ic- S i a a g a s [ 7 ] I uses a se a a o eo em ie a c ica ecom osi io a y amic og ammi g S e ew se a a o i s ow-cos cyc es i a a a g a so a a e co ac i g ose cyc es (a commi i g ei e ges o e a o ima e so u io e emai i g g a as a oga i mic si e e e se a a o Usi g is e i u g a G is ecu si e y i i e i o ieces o mi g a ecom osi io ee Τ o oga i mic e Eac iece as a oga i mic um e o " o a " e ices co ec i g i o e es o G o eac iece o e ca e ume a e a e i e e ways a some su g a o G (ou si e is iece may i ue ce e co ec i i y co s ai s wi i is iece S ese a e ca e e
xt rn l t p o e iece a o e ca s ow a e um e o suc y es is a sim e e o e ia i e um e o o a s o eac iece i Τ a o eac e e a y e o e mus i a ea mi imum cos su g a o e iece so a oge e wi e e e a y e ca mee e g o a co ec i i y co s ai s S ese o ems a e so e y y amic og ammi g wo ki g u Τ om e ea es o e oo G I e o owi g - C e o es " -e ge-cο ec e " -EC e o es " - e e -co ec e " (o i ico ec e -EC e o es " -e ge-cο ec e s a i g su g a " - C e- o es " - e e -co ec e s a i g su g a " a c ( e o es e o a e ge cos o a su g a S e mai esu s a e summa i e i e o owi g wo eo ems 1
As ema ke a o e eac c aime a go i m is a E AS w e e a io y = c(G /O is ou e I a icu a S eo ems 1 1 a 1 im y a AS o e nl ht d mi - imum -e ge-co ec i i y a e mi imum ico ec i i y o ems
I e o owi g S eo em 1 1 wi e iscusse i s e e ew i eas equi e o S eo em 1 wi e ese e S oug ou e c a e i is assume a g a s a e u i ec e wi ou se - oo s u ossi y wi a a e e ges Eac e ge e as a
o ega i e cos ce ; a su g a o mi o i e i s e ge cos s om i s a e g a a c( e o es e o a e ge cos o Η 121
A cycle as o e ea e e e u i may co sis o wo e ices oi e y wo
e e ges o A A e co ac io se - oo s a e isca e u a a e e ges a e e ai e A minor o g a G is a g a o ai e om G oug a se ies o suc e ge co ac io s a e ge/ e e e e io s
Definition 6.2.1 A e a i io of a set is a pair of nonemply subsets S1 S } such that Si US =Α and S e Π S = Q.
Suppose ti and are (k—EC, P) -types of two graphs sharing the vertex subset P;
I ui i e y e (k-EC - y e esc i es ow Α is c osse y e ge cu s usi g ess a k e ges Usua y o y e y e w e G is (k-EC, P)-safe is i e es i g mea i g a a e ge cu s o G o c ossi g P use a eas k e ges S e e e a ce o suc y es o e k-EC o em o ows om is sim e c aim
S e i co i io a o e ca e a e ia e y sayi g a H 1 a H2 (o 1 a e y e o Hl, o ice e sa a e compatible. 1
1 o eac ossi e y e oe o a su g a o G1 i a mm-cos (k-EC -sa e s a -
i g su g a s o G com a i e wi i
o eac ossi e y e o a su g a s o G i a mm-cos (k-EC -sa e s a -
i g su g a 1 o G1 com a i e wi l
3 Co si e a ai s o 1 om S e 1 a om S e k e u e mm-cos com a i e ai
A simi a a oac is use i e E AS o S eo em 1 1 I is ecessa y o a e i a - icu a a o y omia ou o e um e o is i c su g a y es co si e e is ee e O e may succi c y e ese e (k-EC - y e o G y a sma e g a (G w ic co - ai s Ρ a as e same y e I a icu a a mi o o G co ai s as o g as o e e o P is e e e o wo e ices o P a e co ac e oge e I e s ecia case o k = o e ca co s uc (G om G y a yi g e o - owi g u es u i o e a y
1 I a cyc e C as a c o (a e ge e ¢ E (C co ec i g wo e ices o C e e e e c o
I a cyc e C as a mos o e e e i co ac C o a oi
3 I a e e ¢ as eg ee co ac i wi a eig o
S e co ec ess o e a o e o ows om e o se a io a a -e ge cu s a 1-e ge cu s o a e i a ia u e e a o e u es o e a i G is a a e so is (G A so -sa e e so is ( G o e a ica io co si e e i is c a e o e ee s o co si e e si ua io w e e G is em e e i a isk wi e e ices o Α o
e ou a y I e e wo emmas e si e o (G a e o a um e o ossi e (k-EC - y es i uce y su g a o G a e ou e
emma 3 Suppose G ( —ΣC - f pl n r r ph b dd d n th d th the vertices of P on the disk boundary. Then ( G is a planar graph embedded in the same way, with 0(113 1) vertices. 1 3
oo y co si e i g e ee u es use o o m ( i is a so a a a (k—EC - sa e g a em e e i e isk wi o e ou a y E e y i e a ace o ( as a eas wo o a s I as e ac y wo o a s o e aws a a c e i si e e wee ose wo o a s I as 3 o a s o e aws a cyc e o a cs wi i co ec i g e o a s S ese a cs o m a ou e a a g a Α o e o a s O e ca c aim a Α as o a a e e ges Su ose i s ea a wo o a s q Ε Ρ a e co ec e y wo a a e a cs c1 a coi om aces 1 a Si ce a aces mus i o e a eas wo o a s o e ca c oose ii a a co secu i e a so a 1 a s a e a eas o e e ge ow co si e e a o (G aw e wee Ι a a i as o cyc es ( y u e k a is co ec e so i is a ee ecause ( is (k—EC -sa e i is a a om o q y u e 3 i mus e a e ge i ec y e wee a q u e i is a c o e wee 1 a so i s ou a e ee e e e y u e 1 S e e o e Α is a sim e ou e a a g a o e e se so i as ess a 1 a cs u e i o e a a cs om e ou e aces o ( ( ose aces ou e y a segme o e ou a y e e a e a mos ee a a e a cs e ai o o a s e e o e a mos 1 1 a cs o eac Ε i s eg ee i (G is a mos e um e o a ace a cs; e e o e e sum o e eg ee o i ( o e a Ε is a mos = 1 1 1 ow i o e e ases eac o a a a i i i esima eig o oo a ou i e g a ( is a s o me i o a o es ( y u e k wi ea es a a i e a e ices o eg ee a eas 3 ( y u e 3 S e ( as ess a e ices o i o i o e wo s ( as ess a 131 1 e ices o e a ❑
emma With and embedded as in the previous lemma, the number of distinct
oo e e a su g a o co ai i g y immi g o e ca make i (k—EC -sa e wi ou c a gi g i s ( —EC - y e I e e ious oo i is s ow a (G ca e esc i e y a a a o es wi a mos 1 ea es i e a e ices 124 o eg ee a eas ee a eac ea a e e y some Ε w e e e a e s o a gi e a e o co secu i e ea es y s a a ee cou i g ec iques e e a e (1 1 suc
6.3 Planar Separators Suppose o e ee s o a o ima e y so e e -ECSS o em i a a a g a G em- e e on a s e e I a ow-cos sim e cyc e C i G ca e ou e o e may i i e the problem into su o ems y co ac i g C S is o ows om wo o se a io s
Fact 6.3.2 When C is contracted, the sphere pinches into two spheres kissing at the new contracted vertex (a cut-point). Therefore the 2-ECSS problem in G/C is equivalent to two disjoint 2-ECSS problems, one on h sphere.
Sherefore o a o ima e y so e e -ECSS o em i G o e may co ac C a a - proximately so e e wo i e e e -ECSS su o ems S e i e e ges o ose two solutions ack o G a a e e ges o C S e o ai e g a is a -ECSS i G. The additive e o o is so u io ( e i e e ce e wee i s cos a e o ima cos is at most the sum o e e o s o e wo su o ems us c (C O e ca o a ways ucki y i a ig cyc e w ic oes a goo e oug o o se a a i g G e e o e a mo e ge e a ki o se a a o com i i g cyc es wi a Jordan cut is co si e e a o a cu o G is a c ose o a cu e i e em e i g of G that does not c oss (i e sec e i e io o a y e ge Gi e a o a cu e e y e ge is either in the interior o e e e io o u ose e ices a aces i e sec e y J a e o cou e i ei e e i e io o e e e io o S e o owi g eo em is om [ 1 ] w ic is a mo i ka io o Mi e s a a se - a a o eo em as a ea y use o e a a AS [ 7 9 1 ] 125
Theorem 6.3.3 Let G be a connected planar graph on n > 3 vertices embedded in the plane. Suppose G has non-negative weights on its vertices, edges and faces, and non- negative costs on its edges. Let W be the total light of the graph and let M be its total cost and assume that no edge has weight more than (3/ W Then, for any positive integer k one can find a subgraph of G and a closed Jordan curve J in O( time such that:
1. F is the union of at most two vertex-disjoint simple cycles (maybe none). The total cost of the edges on each cycle is at most M/k If contains two cycles and then i ( C i (
2. h nt r r f Β nd th xt r r f Α ( f th x t b th h v ht t t W/
3. Denote by G th b dd d r ph th t r lt ft r d l t n th nt r r f Β nd th exterior of Α ( f they exist) and contracting each cycle in t v rt x f ht Ο Then is a Jordan curve through the new contracted vertices, which intersects edges of G only at their endpoints. The interior and exterior of J both have weight at most (3/ W
4. The set of vertices of G on has size (k
S e ee ossi e y es o e se a a o (acco i g o e um e o cyc es i a e i us a e i igu e 1 o e a i is ecessa y o assume a eac e ge weig s a mos (3/ W I a g a as a e ge e a as weig a ge a (3/ W a as cos e cee i g M/k e o suc se a a o wou e is ecause o i s ig cos e ca o e o a y o e cyc es i a ue o i s ig weig i ca o e i e co ac e i e io o Β o i e co ac e e e io o e ce e is a so a e ge i G a a y o a cu e i e co ac e g a G as e ei e i i s i e io o e e io a so wou o sa is y e i o e y 11 -
igu e 1 S e ee i e e y es o e se a a o I e i s wo cases e o a cu e is c ose a e e co ac io
W e e a o e eo em is a ie Q is sai o e e se o ew 1 portal e ices i o uce y e se a a o S e o igi a g a G as ee i i e i o a mos ou a s o weig a mos (3/ W e i e io o e cyc e e o e y G (wi co ac e
e e e io o e cyc e A e o e y G A (wi A co ac e e i e io o a e e e io o e G1 e o e Q oge e wi e su g a o G i e io o a e G
e o e Q oge e wi e su g a o G e e io o I is way G1 U G = G Ε(G1 Ε(G = Ο, a ν(G1 Π ν(G = Q I e i e i e em e i g o G1 (o G a e ices o Q a ea o a si g e ew ace w ic is ca e a portal face; e o aces a i e sec e a e go e W e so i g -EC i G o e wi see a e su o ems i GA a G ecome i e e e -ΕCSS o ems u e su o ems i G1 a G a e e e e ecause ey s a e Q
e -ECSS A go i m
e G e e i u w ic is a em e e a a -ECSS g a wi e ices o - ega i e e ge cos s a a a ame e S e a go i m assig s weig 1 o eac e e a weig Ο o eac ace y e is i g a o ima io a go i ms o e es ima es ΟΡ (G e i "Α" is ese e 1 e o e a o a s i a g a ew o o A oo e ecom osi io ee is ui om G as o ows Eac o e o s o es a em e e a a g a G w ic as e ge cos s e e / ace weig s a some is i - guis e su se Α o " o a " e ices S e oo o s o es G i se wi o o a s Eac o e o as a mos ou c i e e i e i uc i e y as o ows e G e e g a s o e a a o e o a e W e i s o a e e / ace weig
I W is ( e is o e is a ea o O e wise a y S eo em 3 3 o a i io G i o a mos ou ieces (i e io o a e e io o A a G1 G eac o o a weig a mos (3/ W co ac io o a e ices om G emai as o a s i eac iece w e e
ey a ea ; e g a s G1 a Gl eac ge a mos k ew o a s e se Q. I G1 a Gl, assig a weig o W/(1 o eac ew o a a weig W/1 ό o e ew o a ace y e Se a a o S eo em e um e o ew o a s o ai e i is ase is a mos k
us e weig o eac o G1 a G2 is a mos W -I k- ( 1 ΙίW + i W = s W I e g a GA a G assig e ew ( o - o a e e weig 1 a a e e emai i g e ices a e e same weig as i G Si ce eac c i as weig a mos co s a ac io o e weig o e a e e ee as e O ( og a si e O ( og u e mo e i G is k-e ge co ec e y e se a a o o e ies eac G i is (k-EC -sa e y e co s uc io Α is e se o o a s i G w ic a e ee i o uce y some o a cu u o ye cu o y a cyc e co ac io o a o e o a cu ow co si e e um e o o a s a aces o e sma e g a s Si ce G A a G o o a e ew
o a s a o a aces o e ee s co si e G1 a G2 o y S e co s uc io e su es a
e o a s a G1 a Gl i e i e om G a e a ways ea ie a e ew o a s i e e ices o a e i e i e o a aces a e a ways ea ie a e ew o a ace
S is im ies a i G1 a Gl, e e y o a as weig a eas W a e e y o a ace
as weig a eas i6 W Si ce G1 a G2 eac as weig a mos s W o e ca co c u e a G1 a G2 eac co ai s a mos 1 o a s a a mos 13 o a aces Eac o a 128
ace co ai s a o e ma e y a o a cu a some a ces o o G i Τ o e a a o a cu mig cu (sim y ac oss a e is i g o a ace i w ic case some o o a s may a ea o e ew o a ace u is s i cou s as a si g e o a ace O i e ms o a em e i g o a s e e wi o es a o o es c osse y e o a cu isa ea wi segme s o ei ou a ies i co o a e i o e o e ew o e ou a y G is co ec e ia o e es o Go ( ea y a i c e a co ac e e sio o G w ic ca e em e e as is oi ieces o e i eac o a ace o G S e e o e e (k—EC - y es im ose o y e es o Go ecom oses i o i e e e y es o e i eac o a ace o G y a yi g emma k o eac o a ace o e may ou a e ume a e e °e1 1 = °(Y/ε ) i e e ( —EC - y es a may e im ose o y e es o Go Ca is is e is o external types o G I is mo e e kie a oug o esse ia i eac e e a y e o G is e ese e as a a a g a o si e 1 (see emma k 3 em e e i e o a aces o G I a icu a a e oo o Τ e i u g a G as o o a s a e e o e i as e "em y" e e a y e o a i g com u e Τ a ese e e a y e is s o e may ow e i e a se o su o ems a ee o e a o ima e y so e
Definition 6.4.1 For G n Τ ( th p rt l t nd n xt rn l t p t f r G the su o - em (G is this: find a mm-cost (2—EC, P)-safe spanning subgraph H of G which is compatible with t, or else declare that (G, t) is infeasible (no such
C ecki g easi e i y is sim e us c eck w e e is com a i e wi G i se S e o- a um e o su o ems (o e a c oices o G a is °1 " a e su o em (G is e o igi a -ECSS o em S e su o ems wi e a o ima e y so e s a i g a e ea es o Τ a i is i g a e oo Usi g y amic og ammi g e so u io s a e s o e o a oi ecom i a io I e ase case G is a ea o Τ a as si e = ( S e e ume a i e me o ca e a ie ase o i o - a a se a a o s [ 5] o e ac y so e suc su - 1 9 o ems i ( " = °(Ύ/ε ime ( is may e ega e as a co i ua io o ou me o usi g o a cu s wi ou cyc e co ac io s O e wise G is o a ea a as u o ou c i e i Τ as ou y S eo- em 3 3 S e e e a y e is ecom ose i o i e e e e e a y es o e o eac o a ace o G
e G c e o e ei e G A o G a e Mc e e su y e o i uce y e o a
aces o G ooku e so u io e o e su o em (G c c a i e e ges o He a e e ges o C o e a o e a o ima e so u io Η o e (G su o em
ow co si e e wo emai i g c i e G1 a Gl. As i S eo em 3 3 e G e w a is e o G a e e (u o wo cyc es a e co ac e a ei i e io o e e io is i c e ; so G1 U Gl = G e e o e e e e a y e i uce y i e o a
aces o G o k owi g e o ima c oice o e e a y es i a o G1 a Gl, o e ee s o y em a S a is o e e y ai ( 1 w e e su o ems (G1 a
(G we e ou easi e o e ooku s ei so u io s Η 1 a Η a c eck w e e = Η1 U Η is com a i e wi S e c ea es com a i e ou a e ake a i s e ges a e i e ack o G S ese e ges o oge e wi e C a Η e ges me io e ea ie com ise e a o ima e so u io Η o e (G su o em A oug G is o ac ua y associa e wi a o e o Τ o e a o e ca s i s eak se si y o e (G su o em as e i e a o e I ac i wou e a sim e ma e o e o mu a e as a e a y ee i c u i g G a eac i e a o e o Τ o e wou ei e i c o e cyc e o a y a o a cu
A a ysis S e a o e a go i m so es °ίΥ/ε su o ems eac i iO(y/C ime so e
o a u i g ime is °(Y/ε Co si e a easi e su o em (G y a a i y ecom oses i o i e e - e y es i eac o a ace a ese aces ca o c oss a cyc e; e e o e eac cyc e- i c e su o em (G c c a e emai i g su o em (G a e a easi e ak- 13 i g e e e a y e o G1 i uce y G U as oe we see a (G1 o is easi e Su - osi g ( y i uc io a e a go i m ou some so u io Η 1 o (G1 1 e Η1 U i uces a e e a y e o G suc a (G is a so easi e S e e o e y i uc io u e a go i m i s some so u io o eac easi e (G su o em ow su ose Η is e so u io a e a go i m i s o a easi e su o em (G e i e e rr r o (G as e i e e ce e wee e cos A ( a e mi imum ossi e cos o eac i c e cyc e C (u o wo y ac s 3 1 a 3 su o em (G wi i e i e e o o ( e us a a i io a a i i e e o o a mos c (C A e i c i g cyc es e emai i g e o o (G is a om (G eca _ _ _ _ _ Cie e u k ow o ima so u io o (G e e o e e e e a y e o G1 i uce y ( Γ G U a simi a y e e o e e e e a y e S e (Gi i as e o ima so u io Gi ( o i = 1 a ( is a com a i e y e ai co si e e y e a go i m; i ese wo su o ems a e so e o ima y e so u io cos wou e A ( S e e o e e e o o (G is a mos e sum o e e o s o (G1 a (G e e oug o e mig o ac ua y i e es so u io usi g is ai S e e o e e o e ms sim y a a a o a cu S e e o e e o a e o o e oo o em (G is a mos e sum o c(C o e a cyc es co ac e i Ι is easy o see a o a y e e o e o a e ge cos o a e e is a mos A( G S e e o e e o a e ge cos o a cyc es co ac e o a e e is O (A (Go /k Summi g o e a O ( og e e s o e o a e o om a e e s o Τ is O ((A (Go /k og y a a o ia e c oice o e ea i g co a 1e i i σ i is is a mos • ΟΡΤ(G S e e o e e i a so u io as cos a mos o i g S eo em 1 1 131
5 e - CSS A go i m S e mai i ea o e k-ECSS a go i m is simi a o e k-ECSS a go i m Gi e e i u a e g a G e se a a o eo em is a ie o ecom ose G ie a c ica y i o sma ieces o eac iece y es a e use o e ume a e e i e e ways a e " es " o e g a may i ue ce e co ec i i y co s ai s wi i is iece S e y amic og ammi g is use o a o ima e y i e mi imum su g a com a i e o eac y e o eac iece Simi a y o e ca o e a e um e o e e a y es o eac iece is a sim e e o e ia i e um e o o a s ( emma 5 yie i g e same u i g ime a a ysis Agai e o y sou ce o e o is i e weig o e se a a i g cyc e e ges yie i g e same e o a a ysis owe e e e a e some i icu ies e e i g o e om usi g e same ec ique o so e e -ECSS o em S e i ci e i ku y is cyc e co ac io I e -ECSS a go i m e ecom osi io y e e i i io a y amic og ammi g a e a e o me o e mi o s o G a i is s ow a is is su kie u i is o o ge easo a e o co ac e cyc es i e k-ECSS o em ecause is c a ges e o em (i a icu a i may i o uce a a se e e S e e o e o eac o e i Τ a ai o g a s a e ke e com esse su g a G as e i e i e k-ECSS a go i m a i s co es o i g
n pr d su g a G a is e su g a o G i uce y a e ices w ic a ea (a e cyc e co ac io s as some e e i G As e o e e se a a o eo em is s i a ie o G a e ecom osi io o G is o ai e a u a y u e y e a y amic og ammi g wi e e i e o G S e u com esse g a G a so co ai s o a e ices a o a aces w ic a e e esages o e o a s a o a aces i G S eci ka y e e e o a cu w e e se a a o eo em is a ie o G e e a e e o I is ma e o a si g e e e o G e is a ew o a a wi e co ai e i e su g a o G O e wise is ma e o ma y e ices w ic a e o a se ies o cyc es i G e a mos k o ese e ices wi e s eci ie as ew o a s o G S ese wo e ices a e w e e e o a cu 13 wou i e sec G i e cyc es e ese e y a e u co ac e S us e e a e s i (k o a s i G Eac o a a ea s o some o a ace i G co es o i g o e o a aces o G a e o a aces i e i y w e e " e es o G " wou a ea i e em e i g
S e e ges o eac se a a i g cyc e wi a ea i G as h rd d w ic a e commi e o e i a so u io as i e -ECSS a go i m S e o io o y es mus a so e e e e o e i e ico ec e co e so a o e may agai use y es o c a ac e i e e ossi e cou e a s o a u com esse g a ; is is e oi o emma 5 k e ow o co e ie ce y es a e sim y e ese e as g a s ( a e a a s ac cu a es e ese e y g a s as i Sec io k o owi g is a e i i io a a ogous o "(k-EC -sa e" g a s
5 1 y es o (k- C -Sa e a a G a s
e i i io o e y e S e e i i io o y e is o e a io a a i is o ie e owa s
emmas 5 k a 5 e ow e = ( £ H e a y ( - C -sa e g a ( oes o a e o e a a wi a is i guis e o a se Α a a is i guis e se o a e ges Ε τ S e y e ( o is e i e y e o mi g a se ies o o e a io s o See igu e k o a i us a io 133
1 e e e g a co sis i g a o a s o a a e e ges o E om co s uc a sim i ie g a w ic is a o es o me y a co ec io o ( ossi y co ec e s a s
(a A e ices o a e i c u e i
( E e y o a o is ma ke as a p r-p rt l
(c o eac e e q ΦΑ i i is a e e o a e e e o a iso a e e ge
i ma k i as a su e - o a s i
( o eac ock o ( a ock wi a eas wo e ices c ea e a ew e e
i as a p r-p rt l a co ec y a e ge e su e - o a s o e e y e e i e ock
(e Assig is i c I s o a su e - o a s
e ace e su g a o y e g a a emo e ose e ices a a e o Ι a a e a ace o y o a su e - o a s ( o ice a e emo e e ices a e a co ec e o y o e ices o a a ock o Η e e o ai e g a e 11
3 o eac ock o a is o a i ge i i as e ac y wo e ices a oes o co ai a su e - o a s co ac i i o a si g e e e
e ea e o owi g wo o e a io s u i o e a y
(a emo e a c o s i a y cyc e o ii
( o eac a π o Γ wi o su e - o a s as i e a e ices a a i e a
e ices wi eg ee e ac y ntr t π o a e ge wi e same e e ices as Π
No ice a simi a i y o is co s uc io o e s a a co s uc io o cu e e - ees see e g [3 ] 134
5 e H' e e g a o ai e a e S e s 1- S e y e ( is a g a wi some e ices labeled ( a i g I s ese a e e ices co es o i g o su e - o a s a wi e ca e o a o es I is co s uc e om as o ows
(a A eac cu e e i o ( a e e o C as a vertex i ( I i is a su e - o a s i e i is a o a - o es i ( wi e I o
( o eac ock i a oes o co ai a y su e - o a co ac i o a e e i ( S is e e i ( is e e e as a block-vertex.
(c o eac ock i H' co ai i g e ac y o e su e - o a s co ac is ock o a ock- e e I e ock is o a i ge a is o a cu e e i e is ock- e e is a o a o e i ( wi e I o
( Co ec a ock- e e y a e ge i ( o eac cu e e a ace o i o o e e ices o e i ge i e ock- e e is o ai e om e i ge i H'.
(e I a ock as wo o mo e su e - o a e kee e ock i ( as i is i
( i a y o eac a π o ( wi o o a - o es as i e a e ices a a i e a e ices wi eg ee e ac y contract π o a si g e e ge wi e same e e ices as π
o e e co ce o super-portals ca e a ie o a y g a a co ai s o a s a a e ges a i is com e e y e e mi e y e o a se a e a e ge se o H.
Properties of the types. Some asic o e ies o e y es a e is e e e i s o a e um e o a e s o ( is i e ica o e um e o su e - o a o Seco y
o ice a e y e ( is u ique y e i e e 1 a e wo ( - C -sa e g a s o e same e e se a wi e same se o o a s a a e ges S e 1 a 135 a e sai o a e th t p i ( 1 a ( a e i e ica wi es ec o ei I s S i y a e ices a a e o o a - o es a e eg ee a eas 3 e o ice a i H is eco ec e a e o a se a a e ges a e em y e ( co ai s a si g e e e ( ock- e ices i a y i is easy o see a e co ec io s amo g o a s i H a e ese e a ese e y e co ec io s o o a o es i (G S ese o e ies ca e summa i e i e o owi g emma
Lemma 6.5.2 t G nd H b t ( - C - f r ph th t h v th t f p rt l
P nd th t f h rd d Ε t t ( b th t p f H. h n f r n p nn n
b r ph G f th t nt n ll h rd d n Ε ( nd t ( h v th t f p rt l-n d th r p t t I nd G U H b nn t d f nd nl f ( U (G
- d nn t d nd th tv rt f (G U ( r tv rt f t ( r (G
r th p r p rt l r lt n fr ntr t n f h rd bl d f n d n th t p
e H e a cou e a o G wi e same se o o a s a a e ges i U H is ico ec e e is sai o e com a i e wi (G o c eck w e e is com a i e wi ( y e a o e emma i is e oug o c eck w e e ( U ( is -e ge co ec e a e e ices o ( G U ( a e cu e ices o (G o ( o su e - mo a s e ese i g a ocks o G a H.
e e um e o y es s eci ka y e um e o xt rn l t p o ( - C )- sa e g a spl n is co si e e e G e a ( - C -sa e a e g a wi a a e ge se Ε a a o a ace se e e su e - o a s se o G e e mi e y a E. e Ps .
I H e a cou e a o G a ca e em e e i 3 e e y e (G o H is sai o e a e e a y e o wi es ec o S e ocus wi e o e cou e a H o G
a a so is ( - C -sa e a s a es e same se o su e - o a s S (i e H a G a e e same se o o a s a a e ges S e e y e ( is ca e a e e a y e o wi es ec o S a S e goa is o s ow a e um e o e e a y es o G wi
a is Η is aw o e a e suc a a e ices a e ges o Η a e co ai e i e o a aces (i c u i g e ou a ies o a aces i 136
es ec o S a is sma a is i is o y e o e ia i SI S e o owi g emma s ows a y e o a a e g a Η ca a ways mai ai some o o ogy o Η
Lemma 6.5.3 t Η = ( E be a planar graph with a set of portals P C that is embedded on the plane. Then, one can draw t(H) on the plane such that t(H) is planar and each portal node of t(H) is drawn at the same location as one of the corresponding portals in Η
Proof: E e y si g e o e a io e o me o Η w i e co s uc i g ( sa is ies is
o e y a e ce a y seque ce o suc o e a io s sa is ies i oo ❑
o owi g is e e mai emma ou i g e um e o y es
Lemma 6.5.4 Let G = ( G EG be a (2-VC, P)-safe plane graph with a hard edge set
a super-portal set P S and a portal face set G is embedded in a way such that all super portals of P S will be on the border of the portal faces if the graph consisting of all portals and hard edges is replaced with a as in the definition of type. Then the number of external types of G with respect to P S and F induced by (2-VC, P)-safe graphs is at most ίιρ 1
Proof: e Η e a y ( - C -sa e a e g a a as su e - o a se a is em e e i o eac o a ace i e e su g a o Η em e e i i e Ai .
ecause Η is a e Ai is is oi wi Η 5 o i Co es o i g y e su g a o ( i i e e aces a e a so is oi e ce e y e o e su g a i eac ace ca e co si e e i e e e y
i a y ace i Ε a e su g a Η 1 o Η i i e i s y e e i ow co si e
e s uc u e o i S e e ices i i w ose eg ee is a mos o e a e goi g o e ca e as leaves. Α ea mus e a o a o e o ecause a e ices a a e o o a o es a e eg ee a eas 3 y e co s uc io o ( Seco y e e may e some a a e e ges
4Α p rt l n d n t( rr p nd t n p rt l b th h rd d r pr d 137 i i w ic mus e e wee wo o a o es (su e - o a s ecause o e wise i wi e co ac e acco i g o S e a- o e e i i io I i co ai s a ock e e ock mus e a cyc e S is is ecause y assum io a emma 5 3 a o a o es a e o
e ou a y o i us ca o e i si e a ock u a e S e 1-3 o e e wi emai i si e e cyc e u e mo e y co s uc io o i eac suc cyc e i i co ai s a eas wo o a o es a a o - o a e ices o e cyc e mus a e eg ee a eas 3
o e u ose o cou i g o e ca a s o m i i o a o es wi o y 1 (3i 1 ea es I o e e ases eac o a o e a a i i i esima eig o oo a ou i e a
o es wi e o ai e wi a i e a o es a e eg ee a eas 3 wi o I assig e
S e ea es co es o o o a s- o es ey a e e same I as e co es o i g o a -
o es I ca e c aime a e um e o e ea es is (Α 1 5 S us e e a e a mos
°e1311) suc o es s
e ce e um e o y es i i ace i is o e o e 0(1P10 ecause i a a e i e e e o i Ο e o a um e o ossi e e e a y es wi es ec o Α a
e ( e a ai s o e i a o e o Τ w e e Μ is e u com esse ( - C Α -
sa e a eΑ a a g a a e e a e ge Su ose y e se Ε Ε as e a o a se o Ε is sai o e com a i e wi i (I U is -e ge co ec e a e cu e ices o (I U a e cu e ices o (I o o su e - o a s e ese i g a ocks y emma 5 e um e o ossi e e e a y es is e e mi e y e um e o su e - o a s o Μ ow i wi e s ow a e um e o su e - o a s i eac su g a o ai e om se a a o eo em is (k y e i i io e um e o su e - mo a s o Ε is a mos e um e o o a s i i us e um e o se a a i g cyc es ( a cyc es a e cu e ices e wee ese cyc es ecause e e o e ecom osi io
e o e a io ca e see as wo s e s i s eak e cyc es a e o a s o e cyc e y e asi g o y e eig o oo o e cyc e e ge us o e ge s a mos 1Ρ1Ι ew ea es a e o ai e g a is a o es wi i e a e ices eg ee a eas ee e ce ose o a o es e e o e e um e o e ges o e o es is ( 11 o make a o a o es e ea es o e ca o e same o e a io o e i e a o a o es a ge s e i a o es 13
ee is a mos ( og1 a a eac o e a mos "cyc es" a e i o uce 6 the number of hard "cyc es" is a mos ( ogii e ce e um e o su e - o a s o ΕΙ is (k +
( og1 = (k y emma 5 Ε as a mos (1 = a(Υ/ε e e a y es
5 Recursive Decomposition In is sec io a oce u e is ese e o ui a ecom osi io ee T i a way sui a e o so e e k-ECSS o em i a a g a s S e oce u e is simi a as o e k-ECSS o em owe e a eac o e o T, a ai o g a s a e ke e com esse su g a G as e i e i e -ECSS a go i m a i s co es o i g uncompressed su g a G I
e oo e i u g a G a G = C a e s o e e e a osi i e ea i ce ai k = Θ((γ/ og o e es o is sec io e G a C e e ai o g a s s o e a some o e o T. G as a se o o a s a se o o a aces F a a se o a e ges S e Se a a o S eo em is a ie o e com esse g a G o o ai u o wo cyc es Τ a a o a cu a i g a mos k e ices S is ecom oses C (e ici y a G (im ici y i o a mos 4 smaller graphs:
• For each cyc e C in Υ ake e su g a o C co ai e i si e (ou si e i e cyc e is A C oge e wi e cyc e C O e ca co ac C o a si g e e e (a e ges i ci e o a y e e o C a e ow i ci e o e ew e e wi a se - oo s emo e a ca e esu i g g a Cc Assig weig 1 o e ew e e
e e a e no new o a s i G u Gc may i e i some o o a s a o a s aces om C S e weig o Cc is a mos W, w e e W is e weig o G
• e G e e i e as i e Se a a o S eo em O e ee s o e i e wo o e com- esse g a s G1 a C a a e e i e io a e e e io o e o a cu i G es ec i e y A e ices i a e a e o e se o o a s o G1 a G A - i io a y e a go i m assig s e weig o W o eac ew o a a weig
i W o e ew o a ace us e weig o eac o C1 a G is a mos W
ey may o e sim e cyc es i I { u a se o cyc es g ue oge e 139
W i e G is ecom ose e ici y e ecom osi io o C is o ai e im ici y Co es o i g o C Cc w ic is C c u o cyc es a e co ac e esi es e e ges o C wi e ew a e ges i C c
e o e e i i g C1 a C w ic co es o o C1 a C es ec i e y o e ee s o i s e i e e ew o a s Ι ui i e y ese a e e e ices w e e e o a cu wou i e sec e u com essio g a C e e a o a o e o a cu Ι is ma e o a si g e e e o C e i se is a ew o a o C O e wise is ma e o ma y e ices w ic a e o a se ies o cyc es i C Co si e ose e ges E a a e i ci e o some e e o some o ese cyc es a a e i e i e io o i G Assume i is o em y; o e wise co si e ose e ges i e e e io o i C e e e e e mos e ge i E o e i e em e i g Su ose e is i ci e o w ic is o o e o e cyc es S e wi e esig a e as a ew o a i C Ι e e a e e ges o E o i ci e o ick e ig mos e ge e i e em e i g Su ose e is i ci e o w ic is o o e o e cyc es S e is a so esig a e as e ew o a Ι is way o eac ew o a ma e o a cyc e i C a mos ew o a s a e e i e
ow C1 a G wou e e g a s i si e a ou si e es ec i e y wi o cyc e co ac e u e ey wi o o y i e i some o o a s a a e ges om G u a so a e a e ew a e ges a o a s e i e as a o e A i io a y esi es
e i e i e o a aces o C1 a C a e a ew o a ace y e Se a a o S eo em a e e i i io o ( - C -sa e i is easy o see
a i C is ( - C 13 -sa e e Cc a C1 a C a e a so ( - C -sa e wi es ec o ei o a s ow co si e e um e o o a s a aces As o e -ECSS y e way o se i g e weig o ew o a s a ew o a aces o e ca s ow a G1 a G eac co ai s a mos 1 k o a s a a mos 13 o a aces S e um e o o a aces i C1 a G a e e same as a o G1 a G u e um e o o a s i G1 a G may
e a ge ecause wo o a s i C1 a C may e ma e o a si g e o a i G1 a 1
Gl. owe e ese o a s mus e o e cyc es i G1 a Gl ou y e Se a a o S eo em Eac ime e Se a a o S eo em is a ie a mos "cyc es" (a so ca e a cyc es a e i o uce i e u com essio g a C 7 O e o e a eac ime e ecu si e ca e uces e weig o e g a y a co s a a io us e e o e ecu sio is a mos ( og e ce e um e o a "cyc es" is a mos ( og
S e e o e e um e o o a s i C 1 a C is a mos e um e o o a s o G1 a
Gl us ( og i e (k e um e o su e - o a s o C1 o Cl is a mos e um e o o a s i em us e um e o a cyc es a e cu e ices e wee ese cyc es w ic is (k + ( og = (k Si ce C1 a Cl a e co s a um e o o a aces y emma 5 ey a e a mos k = ° ε e e a y es eac Simi a y i ca e s ow a C c as co s a um e o o a aces (k su e - o a s a °ί É e e a y es wi es ec o Ας a i s o a aces
5 3 y amic og ammi g A e e ecom osi io e e s e o e a go i m is usi g y amic og ammi g o so e e o em e C e a a e g a wi a is i guis se o a e ges A su g a Η o G is sai o e consistent i is a s a i g su g a o G a co ai s a a e ges o C S e su o ems a e e i e as o ows a e goa is o so e e su o ems a o ima e y
As e o e e o a um e o su o ems o e a c oices o C a is (k =
°(Y/ε S e su o em (C is e o igi a k-ECSS o em w e e o is em y a
7Τ ey may o e sim e cyc es i G u e g ue wi a se o o e cyc es om e ious ecu si e ca s 1 1
G is e i u g a wi o o a s a o a e ges Eac e e i G as weig 1, eac ace a e ge as weig Ο As o e -ECSS o em e o ems associa e wi e ea es w e e e com esse C as si e Ν = (k a e co si e e i s A a e ges o G a e i c u e i e so u io o (G S e su o em i s a ce a e so e e ac y i ° ( ? Ν = °(Ύ/ε ime y a yi g i o - a o S eo em [ 5] o C (e ici y a G (im ici y e (G e a su o em associa e wi a i e a o e o A ase o e so u io s o e su o ems associa e wi e c i e o es Su ose e o em i s a ce o e c i e o C CA C C1 a Gl, a e ee so e
e G c e ese ei e GA o GB. e (C e e su g a o a is i uce y e su e - o a s o Gc S e mi imum su g a E c o Gc com a i e wi y e (C ca e ou y ooki g u e so u io o e o em i s a ce o Gc e = — (C S e so u io ] I o (C ca e ou y com i i g e so u io s o (C1 o a (C2 w e e oe U = S e E cisU a o ima e y ΕΙ e mi imum co sis e su g a o C o e e a y e
Analysis of the algorithm As e a go i m o -ECSS e o y sou ce o e o comes om e cyc es i e e se a a o i e e a e ges Usi g simi a a a ysis o e ca s ow a y se ec i g a a o ia e co s a i k = ((γ/ og e e o is a mos imes e o ima o e ime com e i y e ime s e i e ecom osi io ase is mai y o e u o e Se a a o S eo em w ic is a ie a mos Ο ( og imes so e o a
ime is (11 og 1t). I e y amic og ammi g e e a e 1Y/ε su o ems eac ca e so e i (Y/ε ime So e u i g ime o e w o e a go i m is υ οίγ/ε S is o es S eo em 1 1 CHAPTER 7
APPROXIMATION SCHEMES FOR MINIMUM RECONNECTED SPANNING
GRAPHS IN WEIGHTED PLANAR GRAPHS
7.1 Introduction I is c a e e a o ima io a go i ms o e k-ECSS a k-ECSS o em i a - e a y weig e a a g a s a e s u ie A s a a e a a io o e k-ECSS o -
em is a so co si e e w ic is o i a mi imum weig -ECSS s a i g b- lt r ph ( -ECSS o C mea i g a a e ge o G ca e use mu i e imes i (co se- que y i s weig is a so cou e mu i e imes i H). A o e c assica e e sio is e 1- - nn t v t pr bl eac e e is assig e a co ec i i y y e „ Ε f1, } e o em is o i a mi imum weig s a i g su g a suc a o a y ai o e ices
u E e e a e a eas u = mi u „} e ge- is oi o e e - is oi a s e wee a S e 1- -e ge-co ec i i y is e o e y { 1 }-EC a 1- - e e -co ec i i y y { 1 }-EC S e e a e 1- -e ge-co ec i i y o em w e e eac e ge may e use mo e a o ce is a so co si e e
7.1.1 Related Results A e o ems me io e a o e a e ee e e si e y s u ie i e i e a u e Si ce a ese o ems a e - a e mai esea c as ee e o e o esig i g e kie a - o ima io a go i ms see e su ey [7 ] a mo e ece a a ces [19 39 7 ] I ge e a o e wou e e o esig a AS owe e a e o ems co si e e i is c a e a e ma -S E- a [ 5] S e e o e ey o o a e a AS u ess Ρ = u is oes o ec u e a E AS o es ic e c asses o g a s i ee i e e ious c a e i as ee s ow a e e a e As o o k-ECSS a -ECSS o ems i ht d pl n r r ph as I ac e a o ima io sc emes o i C a e a -
143 7.1.2 New Contributions and Techniques
Eflkient a o ima io sc emes o a e a o e me io e o ems i weig e a a graphs will e ese e S ese a o ima io a go i ms e e i a c ucia way o e new co s uc io o l ht p nn r o a a g a s e C e a weig e g a e ( e o e e weig e s o es a is a ce e wee e e ices a i C A p nn r o G is a s a i g su g a Η o G suc a A ( ν s • AGE ( o a A s a e o i es a a o ima e e ese a io o e s o es a me ic (1-co ec i i y i C u i may e muc ig e a G A δ e e a 111 esig e a sim e g ee y a go i m a o a a i a y g a G com u es a s a e Η o G o a y s 1 I e case f pl n r g a s i is s ow i [1]
is ou s e a io w( /O i e ms o us s I a weig e g a s i a g a ami y a e s a e s wi suc a ou o ω( /O ( e e i g o y o s), e e ami y is sai o a e l ht p nn r o is o em ig s a e s a e k ow o e e y 1 5 use u o so i g a ious o imi a io o ems o g a s o e am e a a g a s a e ig s a e s o me ic- S e i s s e i e me ic- S AS o weig e a a g a s [ ] is o e ace e i u g a wi a accu a e e oug s-s a e s (usi g [1] us e ec i e y ou i g w(G /O o e emai e o e a go i m S a e s a e a so use i com e e geome ic g a s o esig e icie A s o geome ic S a e a e o ems [1 1] a o esig A s o e -e ge a - e e -co ec i i y o ems [ 7] y com e i g e s a e co s uc e i [1] wi e a a se a a o ecom osi- io a oac u e o a a y e eco ec e g a s i C a e i wi e s ow a o e ca esig a E AS o e eECSS o em a a AS o e { 1 }-ECSSM o em owe e is a oac o e aci g e i u g a wi a s-s a e s ai s o e eECSS a - CSS o ems S e easo is a a s a e oes o a e o e eco ec e us e s a e may o co ai e o ima o a ea o ima so u io i mos cases a u a y o e may i k o use ig f lt-t l r nt s a e s (see e g i [ ] w ic a e su g a a e sis as s-s a e e e a e e e i g a co s a um e o e ices o e ges U - o u a e y is co ce is o use u o weig e a a g a s si ce sim e e am es s ow a ig au - o e a s a e s o o e is i weig e a a g a s o e e o a si g e e ge e e io o so e e o em me io e a o e e mai co i u io is esc i e a ew g ee y s a e co s uc io w ic o uces a ig a a s a e wi ce ai esi a e o e ies S eci ka y gi e a weig e a a g a G a co ec e s a i g su g a
A o G a 1 i com u es a s-s a e s o G co ai s A as a su g a a S us i e a go i m is e wi α- a o ima e so u io s o e a ious co ec i i y o ems i a weig e a a g a G e a Ο (α/(s — 1 -a o ima io o a o em is o ai e w e e is a s- sca e o G a e same ime u e mo e o e ca s ow a w i e ee o co ai 1 a (1 + -a o ima e so u io S e um e o e ges o S "c ossi g" eac ace o ( emma 7 is ou e
O ga i a io i s a E AS o e eECSSM o em is ese e i Sec io 7 S is sec io co ai s a so a esc i io o e mai a go i mic a oac use i ou a o i- ma io sc emes w ic is a com i a io o e use o s a e s a ecu si e a oac i e y a a ia o e a a se a a o eo em a y amic og ammi g e i Sec io s 7 3 a 7 e ew co s uc io o s a e s is esc i e a e s ecia o e ies o e s a e s a e iscusse I Sec io 7 5 quasi- o y omia a o ima io sc emes o
7 E AS o e EECSSM o em e G e a co ec e weig e g a A eECSSM o G is a s a i g su -mu ig a o G i w ic e ges ca a e some mu i ici y a i w ic e e y ai o e ices is co ec e y a eas wo e ge- is oi a s o e a G may o a e a y mu i e e ges a a I a e ge is use mu i e imes i i s weig a so co i u es mu i e imes o e weig o Si ce i e e e s o use a e ge mo e a wice o e may ca
a e ge mu i ici ies a wo o owi g is a AS o is o em w ic u s i ° ( Ι/ε ime uses o y e ges om H. Su ose S is a o ima - ECSSM i G wi w(e = O ow S is mo i ie suc a i uses o y e ges om H. o eac e ge e o S o i H, emo e e a a a s o es a om H o o a weig a mos s • wee W e e a is a e e e ges a e a e wi mu i ici y u ca e a wo S e esu o a ese mo i ka io s is a o e -ECSSM S usi g o y e ges om H, eac e ge use a mos wice wi w(S s • O e e a go i m a ies e eECSS a o ima io a go i m om C a e o e g a H', w ic is H wi eac e ge u ica e I summa y o e ca ge e o owi g eo em
Theorem 7.2.1 t ε nd l t G be a connected lighted planar graph with vertices.
There is an algorithm running in t 1 Ο(1 /ε 2) th t tp t -ECSSΜ f G whose weight is at most (1 + t) times the minimum.
Why this technique fails for other problems: S e a o e ec ique oes o wo k o o e o ems co si e e i is c a e ecause i ese o ems we a e o a owe o u ica e e ges om G i e ou u g a I s ea ou a o ima io sc emes mus co si e e ossi i i y a e ea -o ima S ee s some "e a" e ges om ou si e e s a e I Sec io s 7 3 a 7 a ew y e o ig a a s a e s a e e e o e a e um e a a a geme o ose e a e ges ou si e e s a e a e imi e
7.3 Augmented Planar Spanners I is sec io a ew g ee y a go i m is ese e o co s uc s-s a e i weig e a a g a s esem i g e s a a g ee y a go i m [1] o ge e a g a s us as i e s a a a go i m e a go i m akes a co ec e weig e g a G a a a ame e s > 1, a o uces a s a e U ike e ge e a a go i m ou G mus e a a a o eac e ge e o G o i i is gua a ee a s • wee is a eas e e g o 148
Figure 7.1 Α o -sim e ace i a c o e a wa ks ι a Α some a i e ace o co ai i g e S e ew a go i m is a so o i e wi a i a gume a "see " s a i g su g a A co ai i g e ges a mus a ea i H. I Sec io 7 A wi e use o e o ce some eco ec i i y o e ies i e s a e
Su ose G is a weig e pl n r ph ( a is a em e e a a g a a H is a su g a A h rd e o H is a e ge o G o i H. o e a H a e i e i em e i gs
om G o eac c o e Y (e is e i e o e e e g o e s o es wa k co ec i g e e oi s o e a o g e ou a y o e ace o H co ai i g e Mo e ecise y i e e oi s o e a e isco ec e i e e i e w (e = +oo O e wise e co ec s wo e ices i a com o e o H, a e is em e e i some ace o is com o e S e ou a y o is a cyc ic wa k o (o ie e e ges wi o a weig w( ; o e a a cu -e ge may a ea wice i e ou a y (o ce e o ie a io a i s weig wou e cou wice i w( Simi a y a cu - e e may a ea mu i e
imes S e e ge e s i s e ou a y seque ce i o wo wa ks Α a Α o co ec i g e e oi s o e wi w( +w(Ρ = w( ow e i e we (e = mi (w( ω(Α (see igu e 7 1 Gi e G s a A as a o e = υgme (G A is com u e as o ows 149
add e to Η return H
like the general greedy spanner algorithm [1], except that WHO replaces Ad.
Theorem 7.3.1 Let G be a weighted plane graph, s > 1, and B a spanning subgraph of
G. Then H = HZgment( G, s, A) is an spanner of G. If H is connected, then w(H) < (1 +2/(s - 1)) - w(H).
Proof : To show that H is an spanner it suflkes to show that each edge of G is s- approximated in For e not in H, at the moment it was rejected it must be the case that we (e) < s - weed). Note that W H (e) may only decrease after that, so Ad (e) < Ed (e) < s - wee) at the end of the algorithm. For the second part one needs to show that the weight of all edges in H u not A is at most (2/(s - 1)) - w(A). Suppose e is such an edge; then e is not a cut edge in H since H is a connected spanning subgraph. Sherefore e is bounded by two distinct faces. Let H be either face bounding e. First one can claim that w(H) > (1 + s) - w(e). To see this, consider the last edge e' added to H whose boundary consists of a path P plus e Si ce e 150
7.4 Spanners and 2-EC subgraph Su ose a weig e a e eECSS g a G is gi e w e e e goa is o i a (1 + - a o ima e - eECSS i s o e ca co s uc a au i ia y su g a as o ows
Gi e a ace i H*, e c o s o a e e e ges o G em e e i si e is ace acco i g o G s em e i g A Aface-edge e o is a a s ac e ge co ec i g wo e ices o ; u ike a c o a ace-e ge is o ecessa i y a e ge o G (I e ices a ea mo e than once o o e mus s eci y w ic a ea a ces is e e oi s o e S e ace e ge e is said o crosses a c o c i c is a c o o e same ace ei e oi s a e is i c vertex a ea a ces o a ey a ea i cyc ic "ecec" o e a ou e ou a y o Note that e may be embedded i si e so e i e sec s o y e c osse c o s Su ose S is a eEC i G a a e ge c o S is o i H*. S e c is a c o o some ace o H*. e Α e e a i co ec i g e e oi s o c suc a w (Ρε • w (c S e e chord move at c is e o owi g mo i ka io o 5 a
o S a e e ges o a a we e o a ea y i S a emo e om S a y c o s i si e the cycle c U Ας (see igu e 7 (a Si ce H* is eECSS e cyc e as o e ea e e ges a e e o e S is s i a - eECSS a e e c o mo e S e c o mo e is improving i w (S ec eases; is a e s w e e e w (Α a (o ‚/ • w (c is ess a e weig o e isca e c o s A y o i ia c o mo e i gs S c ose o H* (i ammi g is a ce us a mos Ο ( im o i g c o mo es a y o a y gi e eEC S Proof: First one can suppose that S has no improving chord move at a chord crossing e, since such a move could only emo e some chords crossing e. Let C be the set of chords in S crossing e. Arrange C in "left to right" order, according to how they intersect e. Let e Ε C e e c o wi ma imum weig Say a a c o e Ε C is short if w (c) < • w(c )o(2\/). Now if there are short chords to the left of co, perform a chord move at the rightmost one, e . Similarly if there are short chords to the right of c^, perform 152 a c o mo e a e e mos o e C S is e esu o ese (a mos wo c o mo es;
o e a C co ai s o s o c o s e ce ossi y c 1 a A
7.5 ApproDimation Sc emes for the 2-reECSS and 2-VACS Problems Ι is sec io i wi e s ow ow o use ou ew s a e co s uc io o i quasi- o y omia ime a o ima io sc emes o e - eECSS a e - ACS o ems i weig e a a g a s o owi g is e AS o eECSS o em A simi a amewo k as a i e AS o -ECSSΜ o em i Sec io 7 is use u i s ea o usi g e s a e co s uc e as i [1] ow o e may use e augme e s a e H* as co s uc e i Sec io 7 i s S eo em 3 3 is a ie o H* wi k = O( og1 / o ecom ose H*. ow- e e i e e om e AS o e eECSS o em H* may o co ai a ea - o ima so u io o e eECSS o em S us o e ca o wo k o e ieces o H* i- ec y o u a e y emma 7 gua a ees a e e e is s a ea -o ima so u io wi 153 a mos (k og (1 / e ges c ossi g e o a cu e S ese c ossi g e ges ca e guesse y yi g a °(k1°g(1/ε ossi i i ies S e guesse e ges a e a e o e co e- s o i g ieces a e e ices o a o g oge e wi e e oi s o e guesse e ges e e mi e e se o o a s o e ew ieces o eac ossi e ew iece wi e guesse e ges weig s a e assig e o e ew o a s suc a eac ew iece as cos a mos co s a ac io o a (k o a s S e e ew ieces a e ecu si e y ecom ose As i e E AS o e e CSS o em o eac iece e ge-co ec i- i y y es a e e i e o esc i e ow ese o a s may e co ec e ou si e o e o e ieces i a (1 + a o ima io so u io S e um e o y es o eac iece is °( 1°g( / e o e - ia i e um e o o a s S e y amic og ammi g is use o so e e su o ems as e o e a e cyc e e ges a e commi e o e so u io S e a o ima io sc eme o e e CSS o em is simi a a o y e i e - e ces a e me io e e e i s is e e i e as ema ke a e e o Sec io 7 a e e e -co ec i i y y es a e e i e usi g e same ec iques as i C a e S e e o o e i a so u io comes om wo sou ces i s e e ges o e cyc es a a ose om e a ica io o e se a a o eo em o e so u io a e a e Si ce eac iece i e ecom osi io as weig a mos co s a ac io o i s a e weig e e o e ecu si e ca s is Ο ( og ii As e o e e o a e o e ecu si e e e is O((w( /k og w e e k = O( og / a w( = Ο (ΟΡΡ/ y a a o ia e c oice o e ea i g co s a e i i g is is a mos (ε/ • O Mo eo e eac ime a ace o (o i s ieces is cu y a o a cu e Ο ( og(1 /ε c ossi g e ges a e guesse I ese e ges a e guesse o ima y ( ey we e e ges i some o igi a o ima e e y emma 7 a a i i e e o o a mos ε/ imes e weig o ese guesse e ges is ai Summi g o e e e i e assem y o a ossi e so u io e o a o ese e o s is a mos ( / - O 154
S e omi a i g ac o i e u i g ime comes om yi g a 1° 1°g / os- si i i ies o e guesse e ges S e weig s o e su o ems a e o y a co s a imes e weig o ei es ec i e a e s a e e o e a u e ecu si e a oac (wi ou y- amic og ammi g ea s o a ime ou o A(ii 1 οϊ"°g(Ι/ε (c • ( c 1), wi so u io ((1/ 1°g(Ι/ε 1°g S is ou may e im o e y a oga i mic ac o i e e o e y usi g y amic og ammi g a y a mo e ca e u cou o su o - ems S e o owi g emma o e i [ 1 ] ou s e um e o ways ow a g a ca e ecom ose ega ess o i s weig sc eme
Lemma 7.5.1 Let G be a planar graph on n vertices with non-negative edge costs, em- bedded in the plane, and a parameter k 1 Then one can find a list of (τ separations of G such that for any valid weight scheme of the vertices, edges and faces of G some separation in this list satisfies the properties of Theorem 6.3.3.
emma 7 5 1 s ows a a iece is a i io e i o y ( i e e ways o ma e ow ma y a a geme s o e e ices a o g e o a cu e a i ci e wi e "guesse " e ges S is im ies e o owi g emma
Lemma 7.5.2 The total number of distinct pieces (contracted graphs) of the
originalH*thatTherefore1ο(1° n occur theduring number our recursive decomposition is of distinct subproblems (a piece, P = O ( og (1 /t) portals selected in the piece, and an external connectivity type on those portals) is 1 ο(1°g Ο(Ι Ι Ο(Ι Ι = ο(κι°g(1/ε
7.6 EDtensions to the f1, connectivity Problem In is sec io o e ca e e e e ious esu s o e 1 co ec i i y o ems i weig e a a g a s O e ee s o ocus o e a go i m o e 1 }-ECSS o em 155 o y S e a go i m o e {1 }-ECS o em ca e o ai e simi a y S e a go i ms ese e a e mo i ka io s o e es ec i e a go i ms i Sec io s 7 a 7 5 i s co si e e 1 }-ECSS o em w ic is a e a e e sio o e 1 }-ECS o em w e e u ica e e ges a e a owe As i Sec io 7 i ca e s ow a e e is a (1 + -a o ima e 1 }-ECSS AM a uses o y e ges om a ig (1 + - s a e So i s ea o G o e ca wo k o wi u ica e e ges S e mai i e e ce om Sec io 7 is e y amic og ammi g a S e co - ec i i y y es ee o e e e i e o e ec e o u i o m co ec i i y equi eme o is o e ca use e co ec i i y y e co s uc io i C a e 5 I o ma y e mai i e e ce is a eac ime a eco ec e com o e o a is co ac e e ig es co ec i i y equi eme amo g a co ac e e ices is assig e o e ew e e S is i c eases e um e o y es om °( 1 1 o °( 1 1 w e e is e se o o a s i e gi e g a Agai a E AS is o ai e wi u i g ime ° ( /ε • ow co si e e 1 }-ECS o em i s a -a o ima e so u io Α ca e ou usi g a go i ms om [ ] (o [3 ] o 1 }-ECS S e Α is augme e i o a ig s a e as i Sec io 7 Usi g simi a a gume s as i e oo o emma 7 o e ca s ow a e e is a (1 + a o ima e 1 }-ECSS e so a o eac icke ace-e ge e o y ( og(1 / e ges o e c oss e ow e e i e e co ec i i y y es as a o e i a y y amic og ammi g is use o so e e o em S e u i g ime is s i omi a e y e um e o su o ems °(Ιogυ·Ι g(Ι/ε /ε e ce a Q AS is o ai e i is case S e esu s i is sec io a e summa i e as o ows 156
Theorem 7.6.2 t t nd l t G b ht d pl n r r ph th n v rt h r
n l r th r nn n n t l Αι 1ο (1/ε /ε th t tp t 1 }-ECSSGΗ f h th t
w(H) < (1 + t • OPT.
Theorem 7.6.3 t t nd l t G b l ht d pl n r r ph th n v rt h r
n l r th r nn n n t 1 gτι οg(1/ε /ε th t tp t 1 }- CSS Η f G h th t
w(H) < (1 + t - OPT. C A E
AU - O E A GEOME IC S A E S
1 I o uc io S is c a e co si e s geome ic au - o e a s a e s i Euc i ea s aces w ic a e o ma y e i e as o ows
1 1 e ious esu s S is o em o e kie y co s uc i g goo au o e a s a e s as ee o ose ece y y e co ou os e a [ 3] S ey ese e i [ 3] ee a go i ms co s uc i g e e au - o e a s a e s S ei i s a go i m is ase o e o se a io a e (k+ s- owe o a s a e is a (k - S (a em owe o a g a G is a g a wi e same e e se as G a i co ai s a e ge e wee a y ai o e ices a a e co ec e y a a i G wi a mos s e ges S e e o e i o e s a s wi e s a e co s uc io om
157 15
[55] e i ime ( og + (k o e ca o ai a (k - S o ma imum eg ee
° 1 a e o a cos o (k imes e cos o a MS o a co s a k is co s uc io ea s o a au - o e a s a e wi asym o ica y o ima a ame e s S e o e wo a go i ms esc i e y e co ou os e a [ 3] use e ll- p r t d p r d p t n [ 13] I is s ow a a e e au - o e a s a e ca e co s uc e (i i ( og +
ime wi ( e ges o (ii i ( k og ime wi ( k og e ges ei e e ma imum eg ee o e o a cos o e au - o e a s a e is ou e i ese wo a go i ms I a o ow-u a e uko s ki [ ] ga e a co s uc io o (k - S wi e o ima um e o e ges ( ; e u i g ime o is a go i m is ( og —Ι + k og ogi uko s ki ese e a so a co s uc io o (k - S wi e ma i- mum eg ee o ( a i es iga e au - o e a s a e s a a ow e use o S ei e oi s S e mai i ea o a co s uc io i [ ] is o ake a ig -qua i y ( a is wi co s a ma imum eg ee a o a cos o o io a o e cos o MS see [5 55] - sca e a e u a ou eac oi k S ei e oi s so a e is a ce e wee e oi a a y S ei e oi associa e wi i is i i i esima y sma Eac oi is co ec e wi a S ei e oi s associa e wi i u e mo e w e e e e e is a e ge i e s a e e wee a ai o i u oi s a u e is e ge is e i ou au - o e a s a e a k ew e ges a e c ea e y co ec i g i a ma c i g e S ei e oi s associa e wi a u I is o i ku o see a e o ai e g a as e ac y k a i io a S ei e oi s e eg ee o eac e e (ei e i u oi o a S ei e o e is u e ou e y (k a a e o a cos o e g a is (k imes e cos o mi imum s a i g ee u e mo e o e ca s ow a i o e emo es a y se o a mos k e ices om a g a e o a y ai o e ices co es o i g o e emai i g oi s e e is a a co ec i g ese oi s w ose e g is u e ou e y + imes e Euc i ea is a ce e wee e oi s 159
8.1.2 New Contributions She mai o e o em e i [ 3] a [ ] is w e e e e e is au - o e a s a e s a i g goo ou s o e ma imum eg ee a e o a cos S e es ou s o ai e i e io co s uc io s we e e au - o e a s a e s a i g e ma imum eg ee o (k y uko s ki [ ] a e o e a i g e o a cos o (1 imes e cos o a MS y e co ou os e a [ 3]
S e i s esu i is a e eso es a o em a gi es a co s uc io o pt -
l ( t - S
Theorem 8.1.3 Let V be a set of points in 11 Let t > 1 and let k be a positive integer.
Then, one can construct a (k t) -EFTS for V that has maximum degree (k nd whose total cost is 07 (k WMS h r MST denotes the cost an MST of V. The constants implicit
n th -n t t n depend on t and
o ice a y e a gume s a o e is esu im ies e i e ica esu o k-e ge au - o e a s a e s I is o i ku o see a e s a e omise i S eo em 1 3 as asym o - ically optimal bounds both for the maximum degree and the total cost. Indeed, si ce a (k, t)-EFTS ((k -E S as o e k + 1-e ge co ec e e e y e e mus a e eg ee
(k o see a e o a cos mus e Ω(k MS i e wo s -case co si e the o - owi g co s uc io (see a so [ 3] see igu e 1 Su ose a k is e e e e a c e wo oi s wi Ί e i e = 1 a e « 1 / Co si e oi s suc a k/ o them are contained in a a Β ι o a ius ce e e a e ι a e emai i g - k/ oi s a e co - tained in another ball Β o a ius wi e ce e a cl. Si ce a y MS o ese oi s has only a si g e e ge e wee Β 1 a Β MS = (s owe e si ce e minimum degree of any e e (o k-e ge au - o e a s a e is k + 1 e e y e e i Β 1 as o be connected to a eas k/ + e ices co ai e i a Β S e e o e e e a e (k e ges e wee Β Ι a a e cos o a y e e (o k-e ge au - o e a s a e is
7(k MS 160
Figure 8.1 Any EFTS for points in Ai and mus a e weig a eas Ω(k ), while the MST h weight 0(1).
S e co s uc io i S eo em 1 3 is a ge e a i a io o e g ee y a go i m a as ee use e o e o co s uc s a e s [1 5 15 3 55] S e mai co i u io i is co e is e i s ecise a a ysis o e au - o e a s a e s o ai e i a co s uc- io S e e co i u io gi es a e icie co s uc io o goo au - o e a s a - e s S e co s uc io om S eo em 1 3 gi es au - o e a s a e s a i g o ima a ame e s u i oes o ea o a e icie a go i m o co s uc i g e s a e s S e o owi g eo em s ows a o e ca co s uc e y e kie y a au - o e a s a - e w ose o a cos is us s ig y a ge a o ima (a e ma imum eg ee emai s o ima
Ou e kie a go i m om S eo em 1 is based on a new, interesting property of Euc i ea g a s a gi es a su icie co i io (characterization) for graphs to be (k, t)-EFTSs. 1 1 e imi a ies I is c a e ma y a go i ms i es iga e wi co si e ai s o oi s (e ges i some seque ia o e o co e ie ce e o a o e o e cos s o e e ges i E is i o-
As me io e e o e both, directed and undirected Euclidean graphs wi e co - si e e S oug ou e c a e ( u wi e o e both, a u i ec e a a i ec e e ge om o u I e i s a o e c a e i Sec io 3 undirected g a s a u i ec - e au - o e a s a e s a e a a y e w i e i e seco a o e c a e i Sec io o co e ie ce directed g a s a ei s a e s wi e co si e e o ice owe - e a is is i c io is ea y o co e ie ce o y si ce a y u i ec e s a e ca e co e e o i ec e o e y e aci g eac e ge wi wo i ec e e ges; simi a y a y i- ec e s a e ca e a s o me o e u i ec e y maki g e e y e ge u i ec e a e emo i g e a a e e ges e wee a ai s o e ices
1 Me e s eo em a I s Co seque ces S e o owi g emma a o ows easi y om Ma ge s eo em (see [11 C a e III eo em 5] wi e use a e
• either ( Ε Ε or there are s internally vertex-disjoint vu-paths in G and
• either ( u E E or there are s internally vertex-disjoint vu-paths in G
Then, there are s internally vertex-disjoint vu-paths in G In particular, if one removes any s - 1 vertices in u} om G then the obtained graph still contains a vu-path. ❑
O e ca easi y e e e a o e emma o o ai e o owi g Then, there are s internally vertex-disjoint vu-paths in G. In particular, f one removes any s — 1 vertices in V \ fv, u} from G, then the obtained graph still contains a v path. ❑
3 k- e e au - o e a S a e s o ow eg ee a ow C e I this section, the following algorithm will be analyzed to show it constructs (k -E S of both low degree and low cost.
C aim 3 1 The k-Greedy Algorithm constructs a (k, t)-EFTS. 1 3 oo e e a y su se o a i g si e a mos k i s o e a G is a s a e s o
ema k 3 Notice that Claim 8.3.1 holds even f the edges are taken in an arbitrary order (that is, not necessarily nondecreasing).
Furthermore, this claim holds not only for Euclidean graphs and the assumption
3 1 A a y i e Ma imum e ee S is sec io is e o e o o e a e (k -E S co s uc e y e k-G ee y A go- i m as ma imum eg ee O (k S e a a ysis is i a simi a s i i as e a a ysis o e g ee y a go i m o o ma s a e s see e g [1 15] o owi g is a au i ia y c aim a wi e use i a e a a ysis 1
oo Let p be any t-spanner ux-path in G'. Let q be the ux-path obtained by taking the edge (u, u) followed by p. Shen
u e mo e i Ζ e o es e oi o e segme u suc a (u u = (u u = π/ then
e o se e a ΙνΖΙ = Ιuν • si ( (Ζuu Ιόν • si a u i = ΙυνΙ • cos( ( uu
ΙυνΙ • cos Θ S e e o e com i i g ese wo i e i ies wi ( 1 a ( o e o ai s 1 5 e ges i G a a e s o e a (u Co si e i s a y e ge (u Ε Ε υ suc a ( Ε y C aim 3 3 e u - a co sis i g o e e ges (u ( is o e g u e ou e y • Iii k
e co si e a y e ge (u Ε υ suc a ( S e si ce e k- G ee y A go i m as o ake e ge (u o E e e mus e is k + 1 i e a y e e - is oi -s a i e uu- a s a eac e ge o ese a s is s o e a ( w ic is ess a (u S e e o e y C aim 3 3 e e e is k + 1 -s a i e u - a s e wee u a u suc a a a s egi wi e ge (ό a e a e i e a y e e - is oi
Summa i i g e e a e k + 1 e ices 1 k+ suc a o eac i 1 i k + 1 (i (u i Ε Ε a (ii ei e ( i u Ε E a (υτ + Ι iν • Ιόν o G co ai s k + 1 -s a i e u - a s e wee ό a suc a a a s egi wi e ge (u i a e a e i e a y e e - is oi a eac e ge o eac a is s o e a (u e e o e o e ca a y emma 1 o co c u e a G co ai s a eas k+ 1 i e a y e e - is oi -s a i e u - a s eac a usi g o y e ges s o e a (u S is owe e co a ic s o e ac a (ό u is a e ge o G a e ce is com e es e oo o e c aim ❑
I [ 1 ] i was s ow a e e is a co s a c suc a o a y oi Ε a a y a g e θ π e e is a ways a co ec io C o ((c/ -1 co es i IE
a i g e a e a oi suc a (i UcEC C = a (ii eac co e C Ε C as e a gu a iame e a mos θ O e ca i co o a e is u e ou o e um e o co es i C wi C aims 3 1 a 3 o o ai e o owi g emma
emma 3 5 Let be any point set in a Euclidean space 1 Let k be any non-negative integer and let be any real number, 1 Then, the Greedy Algorithm returns a
(k t) -EETS for having maximum degree of 0 ( (e/ -1 k where cos θ - si θ In particular, if the dimension and are constant, then the maximum degree is Oak). ❑ 166
8.3.2 Upper Bound for the Cre of Spanners Generated by the k-Greedy Aldorithm S e mos i ku a c a e gi g a o e oo o S eo em 1 3 is e a a ysis o e o a cos o e s a e ge e a e y e k-G ee y A go i m Ι o e o ou e cos o a s a e o e ee s o i s i o uce e co ce o "l pfr pr p rt " [ 9] w ic yie s a ou o e o a cos o a se o e ges i e ms o e e a i e osi io o ese e ges i Euc i ea s ace o co e ie ce e e = (ΙΙ o ν e G e e e g a o ai e om G a e
emo i g a e ices i e oge e wi ei i ci e e ges a e y emo i g a e ges o s o e a (Ι1 ν y e iscussio a o e a y Ma ge s eo em a y
υ ν - a i C e mus a e o a e g g ea e a • Ιυ ν Ι S e goa is o s ow a 8.4 Efficient Construction of Fault Tolerant Spanners In e e ious sec io i as ee o e a e k-G ee y A go i m ge e a es (k - Ss wi ow ma imum eg ee a ow o a cos S e isa a age o is a go i m owe e is a i is o k ow ow o im eme i e kie y I is sec io a a e a i e 1 9 a oac o co s uc au o e a s a e s wi e iscusse o co e ie ce i is sec io directed g a s a ei s a e s wi e co si e e i s e o io o gap property a near parallel edges is i o uce a e a ew su kie o e y o g a s o e (k -E Ss is a e e S e o e ca use is c a ac e i a io o esig a sim e a go i m a ge e a es goo (k -E Ss i o y o- mia ime i a y a go i m is u e o decrease the running time to U ((n k log s n +
k log k)) .
1 asic Au i ia y o e ies She following are wo im o a o io s o i ec e e ges a wi e ea i y use i e c a e
e i i io 1 (Near parallel edges) Two directed edges (p, q) and ( u are called α- near parallel if after translating vector xi such that coincides with that is, to the vector p± with z = u — (x - the angle between vectors p^ and 51 is upper bounded by
e i i io (Gap property) t ω Ο t G = ( be a directed Euclidean graph. A set of edges Ρ C satisfies w-ga o e y if for any two edges (νι iii (" υ Ε Ρ the distance betlen the heads and the tails of ( i iii and (ν u is greater than w times the length of the shorter of the two edges, that is,
mi {Ινι ν 1 Ιuι ιι } ω • mi 1νιυι1 1ν υ 1}
S e o owi g esu is s ow i [15] ( e co s a im ici i e - o a io oes o e e o A Ou e c aim s a es i o ma y a i wo e ges i a i ec e Euc i ea g a a e ea a a e a c ose o eac o e e o e ca o m a s a e a e wee e e oi s o e " o ge " e ge y co ca e a i g e "s o e " e ge a e s a e a s e wee e oi s o e wo e ges S is c aim is esse ia y o e i [ emma ]
8.4.2 Sufficient Conditions for Beind a k-VerteD Fault-Tolerant Spanner In is sec io a ew su icie co i io o a Euc i ea g a o e a (k -E S wi e ese e a e i wi e s ow ow is co i io ca e use o o ai a e kie a go i m o co s uc i g (k -E S S e a oac is mo i a e y a simi a c a ac- e i a io o s a e s e e o e y A ya a Smi i [ ] Notice that in Lemma 8.4.6, it is allowed that some of the ui and vi are equal to ii o u o e wise a o e e oi s o e e ges i ( ui ν1 (1 +1 + } mus e
ai wise is i c
oo In order to prove that G is a (k, t) -HEFTS, one need to show that for any two e ices u Ε ei e (u u Ε E o G co ai s k + 1 disjoint t-spanner uv-path, each ii - a a i g a e ges s o e a uνΙo cos α S e oo is y i uc io o e a k o
e is a ces e wee e ai s o oi s i I ΙυνΙ as e mi imum is a ce amo g a ai s o e ices e (u mus e a e ge o E a e ce e c aim o s o u e o e ca ocee y i uc io Co si e a ai o e ices ii Ε y i uc io o a o e e ai s o e ices y E wi I y Ι ΊυνΙ ei e ( y Ε E o G co ai s k + 1 disjoint t-spanner by-paths, each having a e ges s o e a Ι υ Ιo cos α S e goa is o o e a ei e (ii Ε E o G co ai s k + 1 disjoint t-spanner uv-paths in G, each having all edges shorter than Iu Io cos α One only has to consider the case when (u, v) ςΙ E. Shen, by the lemma's assump- 17
(i o eac 1 i k + 1 e ge ( ui νι is s o e a (υν1/ cos α
(ii a ui a νι a a e ei e u o a e ai wise is i c
(iii o eac 1 i k + 1, e ges (u i a (u a e - ea a a e a
(i o eac 1 i k + 1, mi u k } ω•ιι1ν
ick a y e ge (u i Assume wi ou oss o ge e a i y a ιuu i = mi Iuu 11 }
Si ce (υ1 Ai a (u u a e α- ea a a e y (iii iii ίΙ ίυν1/ cos a y (i a Ιυυι ω • ΊυίνίΙ y (i C aim (i im ies a νiν IIu e ce y i uc-
io ei e ( i Ε E o e e a e k + 1 is oi -s a e A iν- a s i G a usi g o y e ges s o e a cos i Simi a y si ce ουί ΙυνΙ either (u Ai) Ε E o
G co ai s k+ 1 is oi -s a e uuk- a s a use o y e ges s o e a Iu / cos α u e mo e y C aim (ii eac u - a co sis i g o a -s a e uuk- a s (o e ge
(u u1 e ge (1k i a a -s a e A iν- a (o e ge ( i is a -s a e u - a So a i as ee o e a e e a e +1 e ges (u1 ν1 (υ (k+1 ±i suc a o eac i 1 i k + 1 (i ei e (u ui Ε E o G co ai s k + 1 is- oi -s a e i - a s a use o y e ges s o e a Ιυ1kΙ/ cos a a (ii ei e
( i Ε E o e e a e k + 1 is oi -s a e uuk- a s i G a use o y e ges s o - e a Iu I / cos a ow e c aim o ows i ec y om e Me e s eo em ( o mo e e ai s see emma ❑
Esse ia y i e ica a gume s ca e use o o e e o owi g c a ac e i a io o e ge au - o e a s a e s
emma 7 t α ω b r l n b r h th t α π/ ω (cos α - si α — t G = ( E b E l d n r ph S pp th t f r n t v rt u and
f th r (u Ε E or th r r k + 1 d f( l A (k+1 k+ } C E such that
• for 1 < +1 Iuiνi ΙυνΙ/ cos α 173
igu e S e k-Ga -G ee y A go i m
S e c a ac e i a io o (k -E Ss i emma a mos imme ia e y im ies a sim e o y omia - ime a go i m o co s uc i g suc s a e s w ic is ca e k-Ga - G ee y A go i m a esc i e i a o m o a me a-a go i m i igu e S e o owi g ce a emma esc i es mai o e ies o e k-Ga -G ee y Ago- 17 oo i s i wi e o e a G is a (k -E S y s owi g a o a y o e e ai o e ices u o e o e wo co i io s o emma is sa is ie I (u Ε e e i s co i io is o ious y ue O e wise (u E a co si e e i e a io o e a go i m w e (u is c ose S e o y easo a (u is o a e o is a I k + 1 u is im ies a e seco co i io o emma is sa is ie o (u u S e e o e G is a (k -E S y emma e i wi e o e a e ma imum ou - eg ee a e ma imum i - eg ee o eac e e is (k e u e a y e e i s o e e ou - eg ee o u is (k e
Cu e a y co e i wi e a e a u a e a gu a iame e a mos α e Εcυ e
e se o e ges i G egi i g a u a a e co ai e i e co e C u I wi e o e a Ι Ε Ι k + 1 w ic imme ia e y im ies a e ou - eg ee o u is (k ow a a y e e e a io o e a go i m a e mome w e ΙΕ Ι = k+ 1 a e a go i m co si e s a ew e ge (u wi Ε C Σ O se e a i a case o e wi a e Ecυ C Ε a e ce E wi e o si e a eas k + 1 S e e o e e a go i m wi o a e e ge
(u o e s a e is im ies a E cu k + 1 a e ce e ou - eg ee o u is (k O e ca use esse ia y i e ica a gume s o o e e i - eg ee o u is (k (Simi a a gume s s ow a u is e ea / ai o a mos k + 1 ai wise - ea a a e e ges i a y o e ee s o o e a G as sma cos O e ca ocee simi a y as i e oo o emma 3 a i s a i io E i o is oi se s Ε Ε suc a eac E is a ma ima se o e ges w ic a e a - ea a a e y e iscussio i e oo o
emma 3 e e a e ( (e/c -1 suc is oi se s Ε Ε e us i o e se Ε a i i e e e ges i Ε i o a mi ima um e o g ou s suc a e e ges i e same g ou sa is y e w-ga o e y I wi e o e a ( g ou s a e su kie I is e oug o s ow a o a y e ge E Ε e e a e a mos ( e ges e E i s o e a e suc a e e } oes o sa is y e w-ga o e y 175
i a e ge (u Ε Ε e (u = {A Ε ](A q Ε Ε ΙAgΙ uνΙ
Ιu Ι w • q} a e Ε e e se o e ges i Ε a (i egi a e ices i ίu a (ii a e s o e a i o e a si ce Ε (u „ ς Ε a e ges i Ε ν a e ai wise
α- ea a a e O e ca s ow a ΙE (u ν Ι (k + s y co a ic io Su ose a ( + s a co si e e i e a io i w ic e e ge (u is icke y e a go i m e E e e se ake y e k-Ga -G ee y A go i m w e e e ge (ii » is co si e e I wi e o e a Ε k + 1 w ic co a ic s o e assum io
a (u Ε e Ε e e se o a e ges i s o e a (u u suc a o e e y
e ge e Ε E (i e is a- ea a a e o (u a (ii e (u } oes o sa is y e w- ga o e y o e a ecause e a go i m icks e ges i o ec easi g o e e co i io 3 o e a go i m im ies a o e o e e ges i E sa is ies w-ga o e y wi (u S us E ca e e i e as e sum o ce ai x l t h n i Σ a e se o a e ges i Σ a ei e egi a u o e a S e e o e o o e a Ι k + 1 i is su kie o s ow a e e y ma ima ma c i g i Σ co ai s a eas k + 1 e ges u e mo e ecause o e we k ow e a io e wee e ca i a i y o e ma imum ma c i g a e mi imum ca i a i y o a ma ima ma c i g i is e oug o s ow a e ma imum ma c i g i Σ co ai s a eas (k + s e ges O e ca o e is o e y y co si e i g e e ges i (u i s o se e a y e e i i io (u „ ς Σ S e e o e o e o y mus s ow a Ε (u ν as a ma c i g o si e a eas (k + s o e a e e a e a mos k + 1 e ges i 1Ξ s a i g a eac e e as o e a o e Si ce i is assume
a ΙEίu ν ( + s o e ca co c u e a se au ν mus co ai a eas ( + s is oi e ges S e e o e e e is a ma c i g o si e a eas (k + s i Σ u is ea s
o a co a ic io a e ce o e o e a (u (k + s
Symme ica y o e ca e i e (u ν a o e a E(u ν (k+ s S e e o e
ΙE(u ν U E(u„ ( + 1 e ce o e ca co c u e a o e ee s a mos ( + s g ou s o e ges om i suc a e e ges i eac g ou sa is y w-ga o e y us i is o e a o e ca a i io e e ges i E i o O((e/α —1 g ou s suc a e 17 e ges i eac g ou sa is y w-ga o e y o co c u e e oo o e ca a y C aim
3 o o ai a u e ou o e o a cos o E o e ( (e/ — ώ og imes e cos o mi imum s a i g ee o ❑
3 E icie Co s uc io o k au - o e a S a e I is easy o im eme e k-Ga -G ee y A go i m i o y omia ime owe e i ec im eme a io s ea o e u i g ime o Ω ( S is sec io iscusses ow a a go-
i m ca e mo i ie o ac ie e e u i g ime o ( k og s + k og w i e s i e u i g a s a e a i g e a ame e s omise i emma S e ew a oac is simi a o e o e e e o e y A ya a Smi [ ] o co s uc s a e s wi ou e eg ee a ow cos
e mai i ea e i e ew a oac is o o co si e a e O ( e ges u o a e a e kie oce u e a wi " o i " ce ai e ges wi ou co si e i g em e ici y u i g e u o e a go i m Si ce e e y e e is o eg ee (k e goa is o e su e a o y (k e ges i ci e o a y e e a e co si e e y e a go i m S e u es o i se i g a e ge i e k-Ga -G ee y A go i m wi e e a e i o e o o ai a mo e e kie im eme a io O o e a e goa is o mai ai e o e ies o e ou u s a e equi e y emma a o e o e a e aim is o ou u a s a e a i g o e ies esc i e i emma e ow e mai i ea is esc i e e i a e a a io a e a go i m i se is esc i e i Sec io 3
Mai i ea e i e mo i ie a o i m A co ec io o co es o a gu a iame e α (see Sec io 3 1 wi e use o es w e e wo e ges a e α- ea a a e o eac o e S a is i is assume a e e is a co ec io C o co es wi a e es a e o igi a e a gu a iame e α so a e co es i C co e S e a e ge (u is sai o e i a co e C i e ec o u - u Ε C Usi g is o io eac ime a ai o oi s u wi
(u Ε C a e co si e o e o y ee s o ook a ose e ges ( y Ε E a a e i C 177
Si ce i (u a ( u a e i e same co e e ey a e α- ea a a e o eac o e is o io a ows o e o e a es i g o α- ea a a e e ges O ce e co ec io C o co es is i e e co es om C wi e co si e e se a - i a e y S e s a e is ui y e i i g e e ge se E w ic wi e e u io o e e ge 1, se s co s uc e o eac co e se a a e y e Ε e o e a ose e ges o E i si e e co e C Ε C
a co e is a i ca e e icie y mai ai e y a y amic a go i m (see a so [ 1 3] u e mo e i is easy o s ow a i isc ( y isc (u e ii υνί / cos S e e o e y C aim a emma e a go i m emai s co ec (i e se se a i sa is ies e o e ies om emma e e i e ge ( y is co si e e e o e e ge (u u Summa i i g e a go i m is goi g o co si e e e ges i si e eac co e i o e- eg easi g o e o ei isc e g 17
e ( y e a e ge i a co e C Ε C that is to be taken by the algorithm. Observe that if there are already at least k + 1 edges in C with the head at x then the edge (x, y) will not be taken by the algorithm. Sherefore, whenever a vertex x already has k + 1 outgoing edges in C, then no further edge starting at x in C will be considered. Symmetrically, one only needs to consider at most k + 1 incoming edges for any point u.
She other reason for rejecting edge (x, u) is that there are many disjoint edges in
C which are shorter than (x, u) with respect to disc length and are very close to either x o u e Ε e a y i u oi wo s ecia a a s uc u es wi e use o mai ai a maximal set of disjoint edges in C that have starting or ending points close to x, respec- tively. Shese two data structures will be modified not when an edge in C incident to x is considered, but instead, they will be updated each time an edge close to x is inserted in- to Ε To be more precise, at each moment of the algorithm, a set Cc (x) is maintained for eac Ε w ic co ai s a se o is oi e ges i £ suc a o a y (υ ν Ε c ( 179 edges in Ε and V6 containing points that still can be the tails of new edges in Ε Each time the size of certain Fc (x) is greater than or equal to k + 1 is deleted from V; no fur- ther edges in C that begin at x will be considered, c (x) will not be updated anymore, nor will x be considered in any further sets N (u . Similarly, if e ( k + 1 is deleted from V6, no further edges in C ending at x will be considered, e ( will not be updated and x will not be considered in any further sets u • I this way, since both c (x) and Cc ( have a size between 0 and k + 1 the total number of operations for updating the sets c (x) and e ( in the entire algorithm (for a given cone C) will be 0 (k n) .
Observe that once a vertex x has been reported 2 (k + 1 imes i e se s (u ν head of at most k + 1 edges in C, there must be 2 (k + s disjoint edges among all these edges. Sherefore, in this case, the size of c (x) must be greater than or equal to k + 1 (see
Lemma 8.4.8 for a more detailed discussion on similar arguments). Hence, at this moment
wi e e e e a wi o e e o e i Ν u ν o a y u Ε a y mo e o e same 180
Improved k-Gap-Greedy Aldorithm In the previous subsection the main ideas behind the modifications of the k-Gap-Greedy Algorithm are presented. Shese ideas allow one to design a new algorithm that finds good fault-tolerant spanners eflkiently. A more formal description of the algorithm is presented below, on page 185. A formal proof of its cor- rectness will be given first and then an eflkient implementation of the algorithm will be discussed.
Properties of the Improved k-Gap-Greedy Aldorithm. Shis section is devoted to a formal proof of the following lemma.
Proof: One needs to prove that the output of Improved k-Gap-Greedy Algorithm has
the same properties as the output of k-Gap-Greedy Algorithm after some conditions are
relaxed. She proof is similar to that of Lemma 8.4.8.
i. (u, v) is considered explicitly in the "while" loop but is not inserted to Ε
In this case, disc (u, v) is minimum at the beginning of certain iteration. Since (u, v)
is not added to Ε there must exist either k + 1 edges with heads at u or k + 1 edges
with tails at u which are already inserted to Ε Shese edges have length at most 181
uνΙ/ cos a a e α- ea a a e o (u a ey a e o e commo e oi wi (u S e e o e e seco co i io o emma is sa is ie
ii (u is e e e w e some o e ai ( u is icke a a e o Ε i e "w i e"
o summa i e i as ee o e o eac ai ( ‚ ei e ( y is a e ge i G
1 ννi v ^^v vu is (k S e oo is asica y e same as e oo o emma e u e a y e e Acco i g o e Im o e k-Ga -G ee y A go i m i is easy o see a mos k + 1 e ges egi i g a u ca e a e o e Ε o a y gi e co e C Ε C S e e o e e ou - eg ee o u i G is (k Simi a y o e ca s ow a e i - eg ee is O(k i a y o e ee s o o e a G as sma cos i a co e C a i i e e
σ C i F i a mi ima ι m e o στ c c a e i e came g υη sa is y 182 are "shorter" than e in terms of the disc length and such that fe, e'} does not satisfy the
- gap property.
i a e ge (u Ε Ε e Ε = ( u Ε Ε disc (x, u) < disc (u, v) and
Ιu Ι • Ι u } a Ε u = ( y Ε Ε isc ( y isc (u a ay Ι y }
O se e a i a e ge ( u is "s o e " a (u wi es ec o e isc e g a i
ai s o sa is y e ga property with (u, v), then (x, y Ε Ε u U Είν Sherefore, to 5-
o e e c aim i is su icie o s ow a Ι u U ίu Ι = (k ).
First it will be proved that ΙΕΙu = O(k e (Σ = {( y Ε Ε distc (x, u) < ευ is c (υ ν and ΙυχΙ φ % • Cy }. Note that since lux φ υ Ε C ν Αη e oa -
(k ) which directly implies that Ε = O(k owi g i wi e s ow a είΣ = (u
S e e a e wo o se a io s i s ecause a e ges i ε(u are "shorter" than (u, u) with respect to disc, all these edges are picked and inserted to Ε by the algorithm before
(u, v) . Secondly, each time after one of these edges (x, y) is inserted to Ε the algorithm inserts (x, y) to c (u) if (x, u) is disjoint with edges already in c (Di). In other words, at the end of each iteration of the Improved k-Gap-Greedy Algorithm c (u) keeps a maximal ma c i g o e e ges i είν that have been inserted to Ε so far.
ow o e ca o e y co a ic io a είú <2 (k+sl Su ose a είu
(k+ s . It has already been shown that each vertex has out-degree (in-degree) in Ε of at mos k+ 1 S e e o e e ma imum ma c i g o ε ίν must contain at least 2 (k+ s edges a us a y ma ima ma c i g o Ε must have at least (k + s edges. Since Cm (u) is a ma ima ma c i g o είν c (u Ι k+1 S e e e mus e is a e ge ( y Ε ε(υ such that during the iteration when (x, y) is inserted to Ε ( y is also inserted to c (u), a c (u Ι ecomes k + 1 owe e e Im o e k-Ga -G ee y A go i m is esig e such that in that moment u must have been deleted from V and all edges with head at u including (u, v) are also deleted from EC and will not be inserted to Ε Shis contradicts
o e ac a (u E Ε S us i is o e a ευ Ι Ιε υ ν Ι <2(k±s - Details of the implementation of the Improved k-Gap-Greedy Aldorithm. She de- scription of the Improved k-Gap-Greedy Algorithm is on a high level and now it will be disscussed how one can implement that algorithm efficiently. In order to obtain an eflkient implementations one must provide eflkient data structures that allow one to query for (i) an edge (u, v) Ε Cc wi mi imum isc (ii e um e o e ges i Ε beginning or ending at u, (iii) reporting all points in N Du and N ^u ν ^, and (iv) for verifying if an edge (u, v) is disjoint to all edges in CAC or He (x) .
It is easy to see that one can maintain the data structure (ii) reporting the number of edges in Ε beginning or ending at a given vertex with constant query time and update time. Similarly, one can easily maintain the data structure (iv) verifying if an edge (u, v) is disjoint to all edges in Fc(x) or Hc(x) with 0(logd) query time and update time. (The bound for the query time follows immediately from the fact that c (x) or HC (x) contains
0(k) edges and one can use a balanced binary search tree to store the endpoints of these edges.)
Aryan and Smid [6] gave an eflkient data structure supporting queries (i) for an edge in Ec with minimum disc. As it is demonstrated (and discussed in details) in [6], the total running time needed to perform all these operations is in our case 0 (n k log s 1t) . (This bound follows from the fact that one can use query (i) 0(k) times, and as shown in [6], efficient data structure can be built in time 0(ii log s 11) that has constant query time and
0(log 11) amortized update time per edge.) 1
o e kie y e o a oi s i (u a Ν Ευ ν o e ca use y amic a a s uc- u e o o ogo a a ge que ies (see [ ] S e a go i m o y e e es oi s a e e o e
e amo i e e e io ime is ( og - a e que y ime is ( og _ ime us e um e o e o e oi s [ ] Si ce eac oi is e e e a mos o ce e o a e e-
io ime is ( og -Ι Α que y o e a io s is e o me o e y amic a a s uc u e eac ime a e a e ge is i se e so e o a ime is (1 og s= o ( k e ges
Eac e e is e o e a mos ( imes w e e o e o e ee s o e i y w e e c ( o e ( ca e e a ce y a i g a e ge e e i ica io a u a e akes ime
( og us e o a ime o e o i g e i ica io a u a e is ( og k I summa y o e ca co c u e e iscussio i is sec io wi e o owi g emma
emma 8.4.10 Let be a set of n points in Rd. Let α ω be real numbers such that 0 < α π/ and 0 < w (cos a — si a Let = 1/(cos - si a — ω There is a constant c such that the Improved k-Gap-Greedy Algorithm can be implemented to run in
time ((e/α - ( k og s + og k ❑
O e ca co c u e e iscussio i is sec io wi e o owi g mai eo em a o ows imme ia e y om emmas 9 a 1
Theorem 8.4.11 Let α w be real numbers such that 0 < a π/ ω (cos cc - si Let be a set of n points in Rd. Let t = 1 /(cos a - si a — w There is a constant c such that in ((e/a - ( k ogs + og time the Improved k- Gap-Greedy Algorithm computes a directed (k t)-EFTS having the maximum degree of
❑ ((e/α - k and the total cost of 0 ((c/a a /w og MS
o co c u e is sec io o ice a S eo em 11 im ies i ec y S eo em 1 1 5 CHAPTER 9
CONCLUSIONS
Shis dissertation studied combinatorial optimizations problems arising from two areas: scheduling and network design. She main focus was to design eflkient approximation algorithms for NP-hard problems.
9.1 Schedulind Problems
In the scheduling area, problems in master-slave model have been considered. She master- slave model finds many applications in parallel computer scheduling and industrial settings such as semiconductor testing, machine scheduling, transportation maintenance and paral- lel computing.
She complexity issues are considered first. It is shown that many makespan and total completion time problems with constraints such as canonical, order preserving and no-wait-in, are NP-hard in the strong sense.
Motivated by the computational complexity, some special cases of the problem are considered. Several eflkient algorithms are designed to minimize the total completion time and makespan. Shese algorithms are proven to have very good approximation ratio. Next more general cases are discussed. A job can have an arbitrary release time and arbitrary processing time. Shere can be a single master, multi-masters, or distinct preprocessor and preprocessor. Constant approximation algorithms are designed for the total completion time problem. It is shown that these algorithms give good approximation for makespan at the same time.
Shere are several questions that have not been answered. In this dissertation, it is assumed that there are no precedence constraints among the jobs. Shese assumption may not hold in some applications. It is worthwhile to develop approximation algorithms which
186 187 work well even with precedence constraints. Only the makespan and total completion time are considered so far. Other objectives such as maximum lateness, number of tardy jobs and total tardiness, have not been studied. In view of the NP-hardness of makespan and total completion time, minimizing these objectives must also be NP-hard. In that case, it will be desirable to have good approximation algorithms.
9.2 Network Desidn Problems
She survivable network design problem is the problem of designing graphs that resist edge and/or vertex removal. Shis is a fundamental problem in algorithmic graph theory with numerous applications in computer science and operations research. It is well-known that all nontrivial variants of the survivable network design problem are NP-hard and therefore the main research interest lies in the design of eflkient approximation algorithms.
First the geometric version of the survivable network design problem is considered.
A PTASs is developed for the f0, k-connectivity problem in which each vertex has a connectivity requirement at most 2. Shen it is shown that the techniques can be generalized to other Sp metrics and to the f0, 1,... ,B}-edge-connectivity problems for multigraphs.
PTAs or Quasi-PTASs are designed for the minimum 2-connectivity problem and some of its variations in weighted or weighted planar graphs.
Shen the problem of eflkient construction of fault-tolerant spanners are considered.
A greedy algorithm is presented which finds the fault-tolerant spanners with both maximum degree and total cost asymptotically optimal. An eflkient algorithm is then developed to find a fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.
Two important components used in all the approximation schemes are hierarchical decomposition and dynamic programming. Shrough the decomposition, a large graph is decomposed into smaller ones which have small interface with the rest of the graph. In the Euclidean space, random shifted dissection is used to decompose a set of points into 188
egio s S e i e ace e wee wo egio s co ai s o y co s a um e o o a s I a a g a s a mo i ie se a a o eo em is e e o e a a ie o ecom ose e g a s i o sma ieces Eac sma iece as a mos oga i mic um e o o a s S e sma i e ace e wee su a a ows o e o so e e o em y y amic og am- mi g i o y omia ime I wou e g ea i o e ca e e e a o e ec iques o o e e a e o ems o owi g a e some ossi e o ems k-connectivity problem in doublind metrics. S e ou i g ime sio o a me ic is e sma es k suc a a y a o a ius ca e co e e usi g k a s o a ius S is co ce o a s ac me ics as ee o ose as a a u a a a og o e ime sio o a Euc i ea s ace ou i g me ic a ea s i se e a ac ica a ica io s suc as ee - o- ee e wo ks a a a a a ysis I [ 11 ] aiwa s owe a o ow ime sio a me ics e e e is quasi- o y omia ime a o ima io sc emes o AS a o e o imi a io o ems S e geome ic e sio o e k-co ec i i y o em as ee co si e e a o y- omia a o ima io sc emes a e ee esig e k-co ec i i y o em i geome ic g a s y C uma a i ge s i [ ] S ei a go i m is ase o e amewo k o Au o a [3] ike e a oac use y ao a Smi [1 1] o e AS o em ei a go i m e e s o s a e s a some o e o i ia ec iques o e o em i e ow ime sio ou i g me ic e e is o k ow esu s a ou co s uc io o ig a s a se s a e s S e e o e o e ca o use e a oac as i [ ] owe e i is s i ossi e o ac ie e a Quasi- AS y usi g some o e i eas a ose om [11 ] [ 7] [ ]
Minimum strondly connected subgraph in planar draphs. o e ge e a u ig e g a s co s a a o ima io a go i ms a e ee gi e y Sc u e e a i [71 7 ] 189
She existing separator theorems only work for undirected graphs. A ETAS for the problem
in planar graphs will heavily depend on light, directed cycle separators.
Forbidden minor 2-ECSS (or 2-ECSS). Shere are two diflkulties here: first, the usual
separator theorems do not produce cycles (just trees). Also the appropriate generalization
of Lemma 6.2.4 may require a careful application of the Robertson-Seymour theory, which
appears difficult. For the special case of bounded-genus graphs, both of these issues look
more tractable.
For the fault tolerant spanners, an obvious open question that is left is whether one can design an efficient algorithm that outputs fault-tolerant spanners having properties from
Sheorem 8.1.3. It is believed that it should be possible to design an 0 (Lk pollywog L) -time
algorithm for that problem.
In all existing analysis of fault-tolerant spanners, the cost of fault-tolerant spanners is compared with that of MST. Since a fault-tolerant spanner must be (k+ s connected, a natural question is how to compare its cost with the cost of the minimum (2 + s connected
spanning subgraphs. One can extend the example in Figure 9.1 to show that the cost ratio between the optimal (1,1 + t) fault-tolerant spanner and the minimum (2 + s connected
spanning subgraphs can be arbitrarily large in planar graphs. It is believed that the cost ratio for the geometric graphs should be bounded by constant. It is desirable if one could design eflkient algorithm to construct such light spanners.
Shere are many algorithmic applications of spanners, perhaps the most appealing being the recent application in an 0(L log 1t )-time approximation algorithm for the Eu- clidean ASAP [101].[10 i On the other hand, not many applications of fault-tolerant spanners are known. Actually, even in the most natural application to the connectivity problem, the fastest approximation algorithms do not use fault-tolerant spanners [25, 26, 27]. Sherefore, investigating the relationship between the connectivity problems in Euclidean graphs and the notion of fault-tolerant spanners would be an interesting topic. I is em i g (a e y i e es i g o y o e e e esu s o geome ic au o e a s a e s o o -Euc i ea g a s O e cou e e C aim 3 1 o o o a - i a y g a s a is o s ow a e g a o ai e i e k-G ee y A go i m is a (k -E S o a i a y ( a is a so o o -Euc i ea g a s u e mo e si ce e k-G ee y A go i m is a e e sio o e c assica g ee y a as ee use e e si e y i e co s uc io -s a e see e g [1] i is ausi e o ask w e e i o uces goo qua i y s a e s o a i a y g a s o o a a g a s o e am e i o ows om [ii a i e k-G ee y A go i m wi k = is u o a a a g a e i o uces a s a e s w ose o a cos is u e ou e y (1 /( — s imes e mi imum s a i g ee cos owe e is esu ca o e ge e a i e o a ge i s e e a e g a s a oes o a e ig s a e s a a see igu e 9 1 o a e am e u e mo e e e i e e is o e e g ee y a go i m ca o a ways o uce goo s a e s see igu e 9 191
Due to its potential applications in various situations, ad hoc wireless network has received signilkant attention over the last few years for its various applications. In an ad hoc network the links between neighboring nodes get up and down from time to time be- cause of mobility of the nodes and the communication capabilities of each node is usually constrained by its limited battery power. This makes fault tolerance of the network one of the fundamental and critical issues. Because the network is distributed, the global algo- rithms for survivable network design and fault-tolerant spanner in this dissertation do not work any more. Li et al. [84] studied how to set the transmission radius to achieve the k-connectivity with certain probability for a network of n devices; they also proposed a construction of fault-tolerant spanners. Although the number of links is bounded, the to- tal cost (transmission polr) may be unbounded. A natural question is how to assign the transmission power for each node to minimize the total transmission polr assignment. REFERENCES
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