DOCSLIB.ORG

Succinct Data Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Succinct Data Structures SUCCINCT DATA STRUCTURES by Ankur Gupta Department of Computer Science Duke University Date: Approved: Jeffrey Scott Vitter, Supervisor Pankaj Agarwal Roberto Grossi Xiaobai Sun Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Computer Science in the Graduate School of Duke University 2010 ABSTRACT SUCCINCT DATA STRUCTURES by Ankur Gupta Department of Computer Science Duke University Date: Approved: Jeffrey Scott Vitter, Supervisor Pankaj Agarwal Roberto Grossi Xiaobai Sun An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Computer Science in the Graduate School of Duke University 2010 Copyright c 2010 by Ankur Gupta All rights reserved Abstract The world is drowning in data. The recent explosion of web publishing, XML data, bioinformation, scientific data, image data, geographical map data, and even email communications has put a strain on our ability to manage the information contained there. The influx of massive data sets with all kinds of features presents a number of difficulties with efficient management of storage space, organization of informa- tion, and data accessibility. A primary computing challenge in these cases is how to compress the data but still allow them to be queried quickly. This thesis addresses theoretical and algorithmic issues arising from these practical concerns for the prob- lem of compressed text indexing, where we want to maintain efficient data storage and rapid response to queries on data. The premise of data compression comes from many real-life situations, where data are often highly compressible. This compressibility constitutes a major opportunity for saving space and data query latency, and is a critical bottleneck for many applica- tions. In mobile applications, for instance, space and the power to access information are at a premium. In a streaming environment, where new data are being generated constantly, compression can also aid in prediction of upcoming trends. In the case of bioinformatics, analyzing succinct representations of DNA sequences could lead to a deeper understanding of nature, perhaps even giving hints on secondary and tertiary structure, gene evolution, and other important topics. We use text data as the subject of this particular study. We introduce a num- ber of compressed data structures for compressed text indexing that enable arbitrary searching for patterns in the provably best possible time. The methodology is distinct in that the process of searching also encompasses decoding; therefore, the original document is no longer needed. Together, these data structures can be used at mul- iv tiple levels of a compression-retrieval hierarchy to arrive at an overall text indexing solution. Some structures can be used individually as well, within or beyond the scope of text indexing. For each data structure, we provide a theoretical estimate of its space usage and query performance on a suite of operations crucial to access the stored data. In each case, we relate its space usage to the compressed size of the original data and show that the supported operations function in near-optimal or optimal time. We also present a number of experimental results using our methodology. These experiments validate our theoretical findings, and we establish that our methodology is competitive with the state-of-the-art. v Acknowledgements First and foremost, I would like to thank my advisor Jeffrey Scott Vitter. I'm not sure where I would be without his continued support and guidance. Jeff's insistence on clarity and precision is a necessary foundation for any serious graduate student, and I am grateful to have benefited from such a firm vision. I would also like to thank my committee members Roberto Grossi, Xiaobai Sun, and Pankaj Agarwal for providing careful comments on my doctoral work. Special thanks go to Roberto Grossi, who served as a collaborator and co-advisor throughout my graduate career and helped shape who I have become. I would also like to thank Rahul Shah and Wing-Kai Hon, both of with whom I enjoyed working and socializing. Finally, I would like to thank Jon Sorenson, who offered a sounding board in the final stages of writing. I would like to thank my family for providing love and encouragement. My par- ents, Umesh and Manju Gupta, and my brother Parag Gupta, were always there when I most needed someone. I could not have completed this work without them. I cannot begin to express in words the impact my wife Diksha had on me during the final stages of my studies; her concern for and patience with long hours and demanding schedules is truly amazing. Finally, my grandfather Ramswaroop Gupta has always been a quiet strength in my life, with a deep calm and a focus on the simple things. I hope one day to reach that pedestal. I have a long list of friends whose companionship has broadened my life: Matt Taylor, Rex Robinson, Sharlotte Greer, Tylan Watts, Andrew Strack, Priya Mahade- van, Justin Moore, Kristina Killgrove, Patrick Reynolds, David Cherryholmes, and Aaron Miller to name just a few. I am glad to have met them. I would like to offer thanks to Michael E. Durbin, who advised me while I was vi at the University of Texas at Dallas. His mentorship played a big part in fueling my enthusiasm towards Computer Science. I would also like to thank Diane Riggs, in the Department of Computer Science at Duke University. She was always there to offer help to students, whether it be paperwork, scheduling, or just a sympathetic ear. vii Notice of Revision Owing to a mistake on the part of the author, the thesis document submitted for graduation on December 14, 2007 contained errors in the results and analysis of Section 2.4.4 regarding t-subset encoding, as pointed out by the thesis committee. As of the date of this revision, the author had not fixed these errors. This revised document removed the section in question, as well as all references to this material. No results in the revised document depend on the removed section. Some fixes were made in other parts of Section 2.4 as well. Discussion of related work was not updated and hence may not reflect new advances after December 2007. This version of the thesis supersedes all previous versions of the thesis. The author regrets the mistake and apologizes for the introduction of errors in technical results. The author also apologizes for any inconvenience to interested readers, who may have been led astray by the earlier incorrect results. The revised version of this thesis and this notice were approved on 2 July 2010. viii Contents Abstract iv Acknowledgements vi Notice of Revision viii List of Figures xiv List of Tables xv 1 Introduction 1 1.1 Text Compression and Text Indexing . 3 1.2 Dictionaries and Data-Aware Measures For Set Data . 5 1.3 Dynamizing Succinct Data Structures . 6 2 An Algorithmic Framework for Compression and Text Indexing 8 2.1 Introduction . 8 2.1.1 Text Compression . 8 2.1.2 Compressed Text Indexing . 11 2.1.3 Outline of Chapter . 16 2.2 High-Order Empirical Entropy . 16 2.2.1 Empirical Probabilistic High-Order Entropy . 16 2.2.2 Finite Set High-Order Entropy . 19 2.3 The Unified Algorithmic Framework: Tighter Analysis for the BWT . 22 2.3.1 The BWT and (Compressed) Suffix Arrays . 23 2.3.2 Context-Based Partitioning of the BWT . 25 2.4 Encoding Sublists in High-Order Entropy . 29 ix 2.4.1 Individually Encoded Sublists . 30 2.4.2 The Space Redundancy of Encoding Multiple Sublists . 33 2.4.3 The Wavelet Tree . 38 2.5 Encoding the Empirical Statistical Model . 42 2.5.1 Definitions and a Simple Bound . 43 2.5.2 Nearly Tight Upper Bound on M(T; Σ; h) . 44 2.6 Nearly Tight Lower Bounds for the BWT . 49 2.6.1 Constructing δ-resilient Texts . 51 2.6.2 Encoding a δ-resilient Text . 55 2.7 Random Access to the Compressed Representation of LF and Φ . 57 2.7.1 Wavelet Trees as Succinct Dictionaries . 58 2.7.2 Random Access to the Compressed Representation of Φ . 62 2.7.3 Random Access to the Compressed Representation of LF . 66 2.8 Using the Framework for Compressed Suffix Arrays . 67 2.8.1 Compressed Suffix Arrays (CSAs) . 67 2.8.2 High-Order Entropy-Compressed Suffix Arrays . 71 2.9 Applications to Text Indexing . 78 2.9.1 High-Order Entropy-Compressed Text Indexing . 79 2.9.2 A Pattern Matching Tool . 82 2.10 Conclusions . 85 3 When Indexing Equals Compression: Experiments with Compressing Suffix Arrays and Applications 88 3.1 Introduction . 88 3.1.1 Our Results . 89 x 3.1.2 Outline of Chapter . 91 3.2 A Simple Yet Powerful Dictionary . 91 3.2.1 Practical Dictionaries . 92 3.2.2 Empirical Distribution of RLE Values and γ Codes . 96 3.2.3 Statistical Evidence Justifying γ Codes . 99 3.2.4 Fast Access of Experimental-Analysis-Driven Dictionaries . 102 3.3 Review of Wavelet Trees . 104 3.3.1 Efficient Construction of the Wavelet Tree . 108 3.3.2 Compression with bwt2wzip . 109 3.3.3 Decompression with wzip2bwt . 112 3.3.4 Performance and Experiments for wzip . 113 3.4 Practical Suffix Arrays: Indexing Equals Compression . 115 3.4.1 Compressed Suffix Arrays (CSA) . 115 3.4.2 Practical Considerations for Compressed Suffix Arrays . 117 3.4.3 Suffix Array Compression . 120 3.4.4 Suffix Array Functionalities . 122 3.5 Space-Efficient Suffix Trees . 123 3.6 Conclusions . 126 4 Compressed Dictionaries and Data-Aware Measures 128 4.1 Introduction .
Load more

Recommended publications
  • Persistent Predecessor Search and Orthogonal Point Location on the Word
    Persistent Predecessor Search and Orthogonal Point Location on the Word RAM∗ Timothy M. Chan† August 17, 2012 Abstract We answer a basic data structuring question (for example, raised by Dietz and Raman [1991]): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in 1,...,U under an arbitrary sequence of n insertions and deletions, with O(log log U) { } expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linear-space structures and improves previous near-O((log log U)2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worst-case query time and linear space. This result is again optimal in U for linear-space structures and improves the previous O((log log U)2) method by de Berg, Snoeyink, and van Kreveld [1995]. The same result also holds for higher-dimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions 3 on the RAM. ≥ Our key technique is an interesting new van-Emde-Boas–style recursion that alternates be- tween two strategies, both quite simple. 1 Introduction Van Emde Boas trees [60, 61, 62] are fundamental data structures that support predecessor searches in O(log log U) time on the word RAM with O(n) space, when the n elements of the given set S come from a bounded integer universe 1,...,U (U n).
    [Show full text]
  • Space-Efficient Data Structures for String Searching and Retrieval
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2014 Space-efficient data structures for string searching and retrieval Sharma Valliyil Thankachan Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Computer Sciences Commons Recommended Citation Valliyil Thankachan, Sharma, "Space-efficient data structures for string searching and retrieval" (2014). LSU Doctoral Dissertations. 2848. https://digitalcommons.lsu.edu/gradschool_dissertations/2848 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. SPACE-EFFICIENT DATA STRUCTURES FOR STRING SEARCHING AND RETRIEVAL A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Computer Science by Sharma Valliyil Thankachan B.Tech., National Institute of Technology Calicut, 2006 May 2014 Dedicated to the memory of my Grandfather. ii Acknowledgments This thesis would not have been possible without the time and effort of few people, who believed in me, instilled in me the courage to move forward and lent a supportive shoulder in times of uncertainty. I would like to express here how much it all meant to me. First and foremost, I would like to extend my sincerest gratitude to my advisor Dr. Rahul Shah. I would have hardly imagined that a fortuitous encounter in an online puzzle forum with Rahul, would set the stage for a career in theoretical computer science.
    [Show full text]
  • Lecture 1 — 24 January 2017 1 Overview 2 Administrivia 3 The
    CS 224: Advanced Algorithms Spring 2017 Lecture 1 | 24 January 2017 Prof. Jelani Nelson Scribe: Jerry Ma 1 Overview We introduce the word RAM model of computation, which features fixed-size storage blocks (\words") similar to modern integer data types. We also introduce the predecessor problem and demonstrate solutions to the problem that are faster than optimal comparison-based methods. 2 Administrivia • Personnel: Jelani Nelson (instructor) and Tom Morgan (TF) • Course site: people.seas.harvard.edu/~cs224 • Course staff email: [email protected] • Prerequisites: CS 124/125 and probability. • Coursework: lecture scribing, problem sets, and a final project. Students will also assist in grading problem sets. No coding. • Topics will include those from CS 124/125 at a deeper level of exposition, as well as entirely new topics. 3 The Predecessor Problem In the predecessor, we wish to maintain a ordered set S = fX1;X2; :::; Xng subject to the prede- cessor query: pred(z) = maxfx 2 Sjx < zg Usually, we do this by representing S in a data structure. In the static predecessor problem, S does not change between pred queries. In the dynamic predecessor problem, S may change through the operations: insert(z): S S [ fzg del(z): S S n fzg For the static predecessor problem, a good solution is to store S as a sorted array. pred can then be implemented via binary search. This requires Θ(n) space, Θ(log n) runtime for pred, and Θ(n log n) preprocessing time. 1 For the dynamic predecessor problem, we can maintain S in a balanced binary search tree, or BBST (e.g.
    [Show full text]
  • On Data Structures and Memory Models
    2006:24 DOCTORAL T H E SI S On Data Structures and Memory Models Johan Karlsson Luleå University of Technology Department of Computer Science and Electrical Engineering 2006:24|: 402-544|: - -- 06 ⁄24 -- On Data Structures and Memory Models by Johan Karlsson Department of Computer Science and Electrical Engineering Lule˚a University of Technology SE-971 87 Lule˚a, Sweden May 2006 Supervisor Andrej Brodnik, Ph.D., Lule˚a University of Technology, Sweden Abstract In this thesis we study the limitations of data structures and how they can be overcome through careful consideration of the used memory models. The word RAM model represents the memory as a finite set of registers consisting of a constant number of unique bits. From a hardware point of view it is not necessary to arrange the memory as in the word RAM memory model. However, it is the arrangement used in computer hardware today. Registers may in fact share bits, or overlap their bytes, as in the RAM with Byte Overlap (RAMBO) model. This actually means that a physical bit can appear in several registers or even in several positions within one regis- ter. The RAMBO model of computation gives us a huge variety of memory topologies/models depending on the appearance sets of the bits. We show that it is feasible to implement, in hardware, other memory models than the word RAM memory model. We do this by implementing a RAMBO variant on a memory board for the PC100 memory bus. When alternative memory models are allowed, it is possible to solve a number of problems more efficiently than under the word RAM memory model.
    [Show full text]
  • Predecessor Search
    Predecessor Search GONZALO NAVARRO and JAVIEL ROJAS-LEDESMA, Millennium Institute for Foundational Research on Data (IMFD), Department of Computer Science, University of Chile, Santiago, Chile. The predecessor problem is a key component of the fundamental sorting-and-searching core of algorithmic problems. While binary search is the optimal solution in the comparison model, more realistic machine models on integer sets open the door to a rich universe of data structures, algorithms, and lower bounds. In this article we review the evolution of the solutions to the predecessor problem, focusing on the important algorithmic ideas, from the famous data structure of van Emde Boas to the optimal results of Patrascu and Thorup. We also consider lower bounds, variants and special cases, as well as the remaining open questions. CCS Concepts: • Theory of computation → Predecessor queries; Sorting and searching. Additional Key Words and Phrases: Integer data structures, integer sorting, RAM model, cell-probe model ACM Reference Format: Gonzalo Navarro and Javiel Rojas-Ledesma. 2019. Predecessor Search. ACM Comput. Surv. 0, 0, Article 0 ( 2019), 37 pages. https://doi.org/0 1 INTRODUCTION Assume we have a set - of = keys from a universe * with a total order. In the predecessor problem, one is given a query element @ 2 * , and is asked to find the maximum ? 2 - such that ? ≤ @ (the predecessor of @). This is an extension of the more basic membership problem, which only aims to find whether @ 2 -. Both are fundamental algorithmic problems that compose the “sorting and searching” core, which lies at the base of virtually every other area and application in Computer Science (e.g., see [7, 22, 37, 58, 59, 65, 75]).
    [Show full text]
  • Advanced Data Structures 1 Introduction 2 Setting the Stage 3
    3 Van Emde Boas Trees Advanced Data Structures Anubhav Baweja May 19, 2020 1 Introduction Design of data structures is important for storing and retrieving information efficiently. All data structures have a set of operations that they support in some time bounds. For instance, balanced binary search trees such as AVL trees support find, insert, delete, and other operations in O(log n) time where n is the number of elements in the tree. In this report we will talk about 2 data structures that support the same operations but in different time complexities. This problem is called the fixed-universe predecessor problem and the two data structures we will be looking at are Van Emde Boas trees and Fusion trees. 2 Setting the stage For this problem, we will make the fixed-universe assumption. That is, the only elements that we care about are w-bit integers. Additionally, we will be working in the word RAM model: so we can assume that w ≥ O(log n) where n is the size of the problem, and we can do operations such as addition, subtraction etc. on w-bit words in O(1) time. These are fair assumptions to make, since these are restrictions and advantages that real computers offer. So now given the data structure T , we want to support the following operations: 1. insert(T, a): insert integer a into T . If it already exists, do not duplicate. 2. delete(T, a): delete integer a from T , if it exists. 3. predecessor(T, a): return the largest b in T such that b ≤ a, if one exists.
    [Show full text]
  • Dynamic Integer Sets with Optimal Rank, Select, and Predecessor
    Dynamic Integer Sets with Optimal Rank, Select, and Predecessor Search ∗ Mihai Pˇatra¸scu† Mikkel Thorup‡ University of Copenhagen Abstract We present a data structure representing a dynamic set S of w-bit integers on a w-bit word RAM. With S = n and w log n and space O(n), we support the following standard operations in O(log n/ log| |w) time: ≥ insert(x) sets S = S x . • ∪{ } delete(x) sets S = S x . • \{ } predecessor(x) returns max y S y < x . • { ∈ | } successor(x) returns min y S y x . • { ∈ | ≥ } rank(x) returns # y S y < x . • { ∈ | } select(i) returns y S with rank(y)= i, if any. • ∈ Our O(log n/ log w) bound is optimal for dynamic rank and select, matching a lower bound of Fredman and Saks [STOC’89]. When the word length is large, our time bound is also optimal for dynamic predecessor, matching a static lower bound of Beame and Fich [STOC’99] whenever log n/ log w = O(log w/ log log w). Technically, the most interesting aspect of our data structure is that it supports all operations in constant time for sets of size n = wO(1). This resolves a main open problem of Ajtai, Komlos, and Fredman [FOCS’83]. Ajtai et al. presented such a data structure in Yao’s abstract cell-probe model with w-bit cells/words, but pointed out that the functions used could not be implemented. As a partial solution to the problem, Fredman and Willard [STOC’90] introduced a fusion node that could handle queries in constant time, but used polynomial time on the updates.
    [Show full text]
  • MIT 6.851 Advanced Data Structures Prof
    MIT 6.851 Advanced Data Structures Prof. Erik Demaine Spring '12 Scribe Notes Collection TA: Tom Morgan, Justin Zhang Editing: Justin Zhang Contents 1 1. Temporal data structure 1 4 Scribers: Oscar Moll (2012), Aston Motes (2007), Kevin Wang (2007) 1.1 Overview . 4 1.2 Model and definitions . 4 1.3 Partial persistence . 6 1.4 Full persistence . 9 1.5 Confluent Persistence . 12 1.6 Functional persistence . 13 2 2. Temporal data structure 2 14 Scribers: Erek Speed (2012), Victor Jakubiuk (2012), Aston Motes (2007), Kevin Wang (2007) 2.1 Overview . 14 2.2 Retroactivity . 14 3 3. Geometric data structure 1 24 Scribers: Brian Hamrick (2012), Ben Lerner (2012), Keshav Puranmalka (2012) 3.1 Overview . 24 3.2 Planar Point Location . 24 3.3 Orthogonal range searching . 27 3.4 Fractional Cascading . 33 4 4. Geometric data structure 2 35 2 Scribers: Brandon Tran (2012), Nathan Pinsker (2012), Ishaan Chugh (2012), David Stein (2010), Jacob Steinhardt (2010) 4.1 Overview- Geometry II . 35 4.2 3D Orthogonal Range Search in O(lg n) Query Time . 35 4.3 Kinetic Data Structures . 38 5 5. Dynamic optimality 1 42 Scribers: Brian Basham (2012), Travis Hance (2012), Jayson Lynch (2012) 5.1 Overview . 42 5.2 Binary Search Trees . 42 5.3 Splay Trees . 45 5.4 Geometric View . 46 6 6. Dynamic optimality 2 50 Scribers: Aakanksha Sarda (2012), David Field (2012), Leonardo Urbina (2012), Prasant Gopal (2010), Hui Tang (2007), Mike Ebersol (2005) 6.1 Overview . 50 6.2 Independent Rectangle Bounds . 50 6.3 Lower Bounds .
    [Show full text]
  • Succinct Permutation Graphs and Graph Isomorphism [5]
    Succinct Permutation Graphs Konstantinos Tsakalidis∗ Sebastian Wild† Viktor Zamaraev‡ October 9, 2020 Abstract We present a succinct, i.e., asymptotically space-optimal, data structure for permuta- tion graphs that supports distance, adjacency, neighborhood and shortest-path queries in optimal time; a variant of our data structure also supports degree queries in time indepen- dent of the neighborhood’s size at the expense of an O(log n/ log log n)-factor overhead in all running times. We show how to generalize our data structure to the class of circular permutation graphs with asymptotically no extra space, while supporting the same queries in optimal time. Furthermore, we develop a similar compact data structure for the special case of bipartite permutation graphs and conjecture that it is succinct for this class. We demonstrate how to execute algorithms directly over our succinct representations for sev- eral combinatorial problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Moreover, we initiate the study of semi-local graph representations; a concept that “in- terpolates” between local labeling schemes and standard “centralized” data structures. We show how to turn some of our data structures into semi-local representations by storing only O(n) bits of additional global information, beating the lower bound on distance labeling schemes for permutation graphs. 1 Introduction As a result of the rapid growth of electronic data sets, memory requirements become a bottle- neck in many applications, as performance usually drops dramatically as soon as data structures do no longer fit into faster levels of the memory hierarchy in computer systems.
    [Show full text]
  • Advanced Rank/Select Data Structures: Succinctness, Bounds, and Applications
    Universita` degli Studi di Pisa Dipartimento di Informatica Dottorato di Ricerca in Informatica Settore Scientifico Disciplinare: INF/01 Ph.D. Thesis Advanced rank/select data structures: succinctness, bounds, and applications Alessio Orlandi Supervisor Roberto Grossi Referee Referee Referee Veli M¨akinen Gonzalo Navarro Kunihiko Sadakane December 19, 2012 Acknowledgements This thesis is the end of a long, 4 year journey full of wonders, obstacles, aspirations and setbacks. From the person who cautiously set foot in Pisa to the guy who left his own country, none of these steps would have been possible without the right people at the right moment, supporting me through the dark moments of my life and being with me enjoying the bright ones. First of all, I need to thank Prof. Roberto Grossi, who decided that he could accept me as a Ph.D. Student. Little he knew of what was ahead :). Still, he bailed me out of trouble multiple times, provided invaluable support and insight, helping me grow both as a person and as a scientist. I also owe a lot to Prof. Rajeev Raman, who has been a valuable, patient and welcoming co-author. I also want to thank Proff. Sebastiano Vigna and Paolo Boldi, who nurtured my passion for computer science and introduced me to their branch of research. Also Prof. Degano, as head of the Ph.D. course, provided me with non-trivial help during these years. In Pisa, I met a lot of interesting people who were crucial for my happiness, was it when studying on a Sunday, on a boring the-proof-does-not-work-again day, in moments of real need, or on a Saturday night.
    [Show full text]
  • Computational Geometry Through the Information Lens Mihai Pdtracu
    Computational Geometry through the Information Lens by Mihai Pdtracu Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2007 © Massachusetts Institute of Technology 2007. All rights reserved. A uthor ........................................... L............... ........ Department of Electrical Engineering and Computer Science May 22, 2007 Certified by .......................... Erik D. Demaine Associate Professor Thesis Supervisor Accepted by.... .. .. .......... .. .......... ... ..-- - ............... Arthur C. Smith Chairman, Department Committee on Graduate Students BARKEfR MASSACHUSETTS INMTITUTE OF TECHNOLOGY AUG 16 2007 URRIEFS cl Computational Geometry through the Information Lens by Mihai Pdtra§cu Submitted to the Department of Electrical Engineering and Computer Science on May 22, 2007, in partial fulfillment of the requirements for the degree of Master of Science Abstract This thesis revisits classic problems in computational geometry from the modern algorithmic perspective of exploiting the bounded precision of the input. In one dimension, this viewpoint has taken over as the standard model of computation, and has led to a powerful suite of techniques that constitute a mature field of research. In two or more dimensions, we have seen great success in understanding orthogonal problems, which decompose naturally into one dimensional problems. However, problems of a nonorthogonal nature, the core of computational geometry, have remained uncracked for many years despite extensive effort. For example, Willard asked in SODA'92 for a o(n lg n) algorithm for Voronoi diagrams. Despite growing interest in the problem, it was not successfully solved until this thesis. Formally, let w be the number of bits in a computer word, and consider n points with O(w)-bit rational coordinates.
    [Show full text]
  • MTAT.03.238 Succinct Trees
    Algorithmics (6EAP) MTAT.03.238 Succinct Trees Jaak Vilo Thanks to S. Srinivasa Rao 2020 Jaak Vilo 1 succinct Suppose that is the information-theoretical optimal number of bits needed to store some data. A representation of this data is called • implicit if it takes bits of space, • succinct if it takes bits of space, and • compact if it takes bits of space. ! http://en.wikipedia.org/wiki/Succinct_data_structure • In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but (unlike other compressed representations) still allows for efficient query operations. The concept was originally introduced by Jacobson[1] to encode bit vectors, (unlabeled) trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first. A related notion is that of a compressed data structure, in which the size of the data structure depends upon the particular data being represented. Succinct Representations of Trees S. Srinivasa Rao Seoul National University Outline p Succinct data structures n Introduction n Examples p Tree representations n Motivation n Heap-like representation n Jacobson’s representation n Parenthesis representation n Partitioning method n Comparison and Applications p Rank and Select on bit vectors Succinct data structures p Goal: represent the data in close to optimal space, while supporting the operations efficiently. (optimal –– information-theoretic lower bound) Introduced by [Jacobson, FOCS ‘89] p An “extension” of data compression. (Data compression: n Achieve close to optimal space n Queries need not be supported efficiently ) Applications p Potential applications where n memory is limited: small memory devices like PDAs, mobile phones etc.
    [Show full text]