DOCSLIB.ORG

Amortized Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Amortized Analysis DESIGN AND ANALYSIS OF ALGORITHMS (DAA 2018) Juha Kärkkäinen Based on slides by Veli Mäkinen Master’s Programme in Computer Science 06/09/2018 1 ANALYSIS OF RECURRENCES & AMORTIZED ANALYSIS Master’s Programme in Computer Science DAA 2018 week 1 / Juha Kärkkäinen 06/09/2018 2 ANALYSIS OF RECURRENCES • Analysing recursive, divide-and-conquer algorithms • Step 1: Divide problem into subproblems • Step 2: Solve subproblems recursively • Step 3: Combine subproblem results • Three methods • Substitution method (Section 4.3 in book) • Recursion-tree method (Section 4.4) • Master method (Section 4.5) • Quicksort (Chapter 7) • We will continue on Week II with this topic with advanced recursive algorithms Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 3 QUICKSORT pivot 4 7 8 1 3 6 5 2 9 1 3 2 4 7 8 6 5 9 1 3 2 4 6 5 7 8 9 2 3 4 5 6 7 8 9 2 3 5 6 Bad pivot causes recursion tree to be skewed O(n2) worst case. We learn next week how to select a perfect pivot (median) in linear time! Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 4 QUICKSORT WITH PERFECT PIVOT … … … … … … … log n levels … … O(n) work on each level O(n log n) time This is called the recursion tree method. Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 5 QUICKSORT WITH PERFECT PIVOT • Running time can also be stated as a recurrence (recursively defined equation): recursive calls • T(n) = 2T(n/2) + O(n) divide and combine • T(1) = O(1) base case • Assumes n=2k for some integer k>0 (why is this fine to assume?). • Substitution method: 1. Guess a solution (with unknown constants) 2. Prove the solution by induction a. Assume solution holds for inputs smaller than n b. Substitute according to induction assumption c. Check that that the solution holds (with appropriate constants) d. Check (and adjust if necessary) the base case Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 6 SUBSTITUTION METHOD EXAMPLE • Observation: big-O() notation is not compatible with substitution method, as we need more exact claims for induction to work. Hence we solve T(n) = 2T(n/2)+an and T(1)=a for some constant a>0. 1. Guess: T(n) ≤ c n log n for some c>0 when n≥n0 2. Prove by induction a. Induction assumption: T(n/2) ≤ cn/2 log (n/2) = cn/2 log n – cn/2 b. Substitute: T(n) = 2T(n/2)+an ≤ cn log n-cn + an. c. Check: T(n) ≤ cn log n, for any c≥a. d. Base case: T(1) = a > c 1 log 1 = 0, but T(2) = 4a ≤ c2 log 2 = 2c, when c≥2a. Thus we can choose e.g. c=2a and n0=2. • Here induction base case (n=2) is different from recurrence base case (n=1). Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 7 MASTER METHOD • Master Theorem characterizes many cases of recurrences of type T(n) = aT(n/b)+f(n). • Depending on the relationship between a,b, and f(n), three different outcomes for T(n) follow. • Let α = logb a. The cases are • If f(n)=O(nα-ε) for some constant ε>0, then T(n)=Θ(nα). • If f(n)=Θ(nα), then T(n)=Θ(nα log n). • If f(n)=Ω(nα+ε) for some constant ε>0 and if af(n/b)≤cf(n) for some constant c<1 and all sufficiently large n, then T(n)=Θ(f(n)). • Example: T(n) = 2T(n/2)+Θ(n). α α • α = log2 2=1 and f(n)=Θ(n ), thus T(n)=Θ(n log n)=O(n log n). Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 8 AMORTIZED ANALYSIS • Consider algorithms whose running time can be expressed as (time of a step) * (number of steps) = tstep * #steps = ttotal • E.g. linked list: O(1) append * n items added = O(n) • Sometimes a single step can take long time, but the total time is much smaller than what the simple analysis gives • Work done on heavy steps can be charged on the light steps • Amortized cost of a step = ttotal / #steps • Examples: • Cartesian tree construction (separate pdf) • Dynamic array (Section 17.4.2) Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 9 CARTESIAN TREE Cartesian tree on array CT(A) • root = smallest element • left subtree = Cartesian tree of subarray to left of root • right subtree = Cartesian tree of subarray to right of root A = 7 9 1 5 8 3 4 2 3.5 Naive construction needs Θ(n2) time in the worst case. Incremental left-to-right construction runs in linear time. Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 10 CARTESIAN TREE CONSTRUCTION 7 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 11 CARTESIAN TREE CONSTRUCTION 7 9 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 12 CARTESIAN TREE CONSTRUCTION 7 9 1 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 13 CARTESIAN TREE CONSTRUCTION 7 9 1 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 14 CARTESIAN TREE CONSTRUCTION . 7 9 1 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 15 CARTESIAN TREE CONSTRUCTION . 7 9 1 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 16 CARTESIAN TREE CONSTRUCTION General step: Compare new element to elements on the right-most path starting from bottom and insert in appropriate place. 7 9 1 5 8 3 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 17 CARTESIAN TREE CONSTRUCTION General step: Compare new element to elements on the right-most path starting from bottom and insert in appropriate place. 7 9 1 5 8 3 Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 18 CARTESIAN TREE Comparing a new item to all items in the right-most path may take O(n) time. But after comparing an old item, you either insert the new item, or never compare that old item again (by-pass). 7 9 1 5 8 3 4 2 3.5 The total running time is #by-passes + #insertions, which both are O(n). Hence, the amortized cost of modifying CT(A[1..n-1]) into CT(A[1..n]) is O(1). Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 19 DYNAMIC ARRAY / TABLE Insert (to full array) Double array size Delete Bad idea Half array size Insert Double array size … Worst case: each insert and delete needs O(n) time for doubling/halfing Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 20 DYNAMIC ARRAY / TABLE Insert (to full array) Double array size Delete … More deletes Half array size after n/4 deletions Each doubling/halfing of array of size n is followed by O(n) inserts/deletes before another doubling/halfing → constant amortized time for insert/delete Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 21 STRATEGIES FOR AMORTIZED ANALYSIS • Aggregate method (Section 17.1) • Show that each step grows some quantity that is bounded. The bound on the quantity can be used to show that total time used for all steps is proportional to that same bound. • In Cartesian tree construction, each step added one to #by-passes or #insertions. Both are bounded by n, and hence the total number of steps is at most 2n. • Accounting method (Section 17.2) • Pay in advance the expensive operations by charging them from the cheap operations. Then show that any sequence of operations has more operations in bank account than the number of true operations. • In Dynamic array we pay for 2 copy operations at each insertion or deletion. Consider any sequence of operations after a halfing / doubling to size n until next: ‒ Halfing: n/4 deletions have gathered n/2 credits which is sufficient to copy n/4 elements to a new location. ‒ Doubling: n/2 insertions have gathered n credits which is sufficient to copy n elements to a new location. Master’s Programme in Computer Science DAA 2018 lecture 1 / Juha Kärkkäinen 06/09/2018 22 STRATEGIES FOR AMORTIZED ANALYSIS • Potential method (Section 17.3) • Let p(t), p(t) ≥0, be a potential of data structure after t operations with p(0)=0. • Let at(t)=c(t)+p(t)-p(t-1) be the amortized time of operation t, where c(t) is the actual cost of that operation. • By telescoping cancellation one can see that the sum of amortized times of n operations is at(1)+at(2)+...+at(n) = c(1)+c(2)+…+c(n)+p(n) and thus an upper bound for the actual running time. • To show e.g. that total running time is linear, it is sufficient to show that for each type of operation amortized time is constant! • This kind of analysis requires a good guess on p(t).
Load more

Recommended publications
  • Amortized Analysis
    Amortized Analysis Outline for Today ● Euler Tour Trees ● A quick bug fix from last time. ● Cartesian Trees Revisited ● Why could we construct them in time O(n)? ● Amortized Analysis ● Analyzing data structures over the long term. ● 2-3-4 Trees ● A better analysis of 2-3-4 tree insertions and deletions. Review from Last Time Dynamic Connectivity in Forests ● Consider the following special-case of the dynamic connectivity problem: Maintain an undirected forest G so that edges may be inserted an deleted and connectivity queries may be answered efficiently. ● Each deleted edge splits a tree in two; each added edge joins two trees and never closes a cycle. Dynamic Connectivity in Forests ● Goal: Support these three operations: ● link(u, v): Add in edge {u, v}. The assumption is that u and v are in separate trees. ● cut(u, v): Cut the edge {u, v}. The assumption is that the edge exists in the tree. ● is-connected(u, v): Return whether u and v are connected. ● The data structure we'll develop can perform these operations time O(log n) each. Euler Tours on Trees ● In general, trees do not have Euler tours. a b c d e f a c d b d f d c e c a ● Technique: replace each edge {u, v} with two edges (u, v) and (v, u). ● Resulting graph has an Euler tour. A Correction from Last Time The Bug ● The previous representation of Euler tour trees required us to store pointers to the first and last instance of each node in the tours.
    [Show full text]
  • Lecture Notes of CSCI5610 Advanced Data Structures
    Lecture Notes of CSCI5610 Advanced Data Structures Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong July 17, 2020 Contents 1 Course Overview and Computation Models 4 2 The Binary Search Tree and the 2-3 Tree 7 2.1 The binary search tree . .7 2.2 The 2-3 tree . .9 2.3 Remarks . 13 3 Structures for Intervals 15 3.1 The interval tree . 15 3.2 The segment tree . 17 3.3 Remarks . 18 4 Structures for Points 20 4.1 The kd-tree . 20 4.2 A bootstrapping lemma . 22 4.3 The priority search tree . 24 4.4 The range tree . 27 4.5 Another range tree with better query time . 29 4.6 Pointer-machine structures . 30 4.7 Remarks . 31 5 Logarithmic Method and Global Rebuilding 33 5.1 Amortized update cost . 33 5.2 Decomposable problems . 34 5.3 The logarithmic method . 34 5.4 Fully dynamic kd-trees with global rebuilding . 37 5.5 Remarks . 39 6 Weight Balancing 41 6.1 BB[α]-trees . 41 6.2 Insertion . 42 6.3 Deletion . 42 6.4 Amortized analysis . 42 6.5 Dynamization with weight balancing . 43 6.6 Remarks . 44 1 CONTENTS 2 7 Partial Persistence 47 7.1 The potential method . 47 7.2 Partially persistent BST . 48 7.3 General pointer-machine structures . 52 7.4 Remarks . 52 8 Dynamic Perfect Hashing 54 8.1 Two random graph results . 54 8.2 Cuckoo hashing . 55 8.3 Analysis . 58 8.4 Remarks . 59 9 Binomial and Fibonacci Heaps 61 9.1 The binomial heap .
    [Show full text]
  • Fundamental Data Structures Contents
    Fundamental Data Structures Contents 1 Introduction 1 1.1 Abstract data type ........................................... 1 1.1.1 Examples ........................................... 1 1.1.2 Introduction .......................................... 2 1.1.3 Defining an abstract data type ................................. 2 1.1.4 Advantages of abstract data typing .............................. 4 1.1.5 Typical operations ...................................... 4 1.1.6 Examples ........................................... 5 1.1.7 Implementation ........................................ 5 1.1.8 See also ............................................ 6 1.1.9 Notes ............................................. 6 1.1.10 References .......................................... 6 1.1.11 Further ............................................ 7 1.1.12 External links ......................................... 7 1.2 Data structure ............................................. 7 1.2.1 Overview ........................................... 7 1.2.2 Examples ........................................... 7 1.2.3 Language support ....................................... 8 1.2.4 See also ............................................ 8 1.2.5 References .......................................... 8 1.2.6 Further reading ........................................ 8 1.2.7 External links ......................................... 9 1.3 Analysis of algorithms ......................................... 9 1.3.1 Cost models ......................................... 9 1.3.2 Run-time analysis
    [Show full text]
  • Introduction to Algorithms, 3Rd
    Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein Introduction to Algorithms Third Edition The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected]. This book was set in Times Roman and Mathtime Pro 2 by the authors. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Introduction to algorithms / Thomas H. Cormen ...[etal.].—3rded. p. cm. Includes bibliographical references and index. ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper) 1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H. QA76.6.I5858 2009 005.1—dc22 2009008593 10987654321 Index This index uses the following conventions. Numbers are alphabetized as if spelled out; for example, “2-3-4 tree” is indexed as if it were “two-three-four tree.” When an entry refers to a place other than the main text, the page number is followed by a tag: ex. for exercise, pr. for problem, fig. for figure, and n. for footnote. A tagged page number often indicates the first page of an exercise or problem, which is not necessarily the page on which the reference actually appears. ˛.n/, 574 (set difference), 1159 (golden ratio), 59, 108 pr. jj y (conjugate of the golden ratio), 59 (flow value), 710 .n/ (Euler’s phi function), 943 (length of a string), 986 .n/-approximation algorithm, 1106, 1123 (set cardinality), 1161 o-notation, 50–51, 64 O-notation, 45 fig., 47–48, 64 (Cartesian product), 1162 O0-notation, 62 pr.
    [Show full text]
  • Draft Version 2020-10-11
    Basic Data Structures for an Advanced Audience Patrick Lin Very Incomplete Draft Version 2020-10-11 Abstract I starting writing these notes for my fiancée when I was giving her an improvised course on Data Structures. The intended audience here has somewhat non-standard background for people learning this material: a number of advanced mathematics courses, courses in Automata Theory and Computational Complexity, but comparatively little exposure to Algorithms and Data Structures. Because of the heavy math background that included some advanced theoretical computer science, I felt comfortable with using more advanced mathematics and abstraction than one might find in a standard undergraduate course on Data Structures; on the other hand there are also some exercises which are purely exercises in implementational details. As such, these notes would require heavy revision in order to be used for more general audiences. I am heavily indebted to Pat Morin’s Open Data Structures and Jeff Erickson’s Algorithms, in particular the parts of the “Director’s Cut” of the latter that deal with randomization. Anyone familiar with these two sources will undoubtedly recognize their influences. I apologize for any strange notation, butchered analyses, bad analogies, misleading and/or factually incorrect statements, lack of references, etc. Due to the specific nature of these notes it can be hard to find time/reason to do large rewrites. Contents 1 Models of Computation for Data Structures 2 2 APIs (ADTs) vs Data Structures 4 3 Array-Based Data Structures 6 3.1 ArrayLists and Amortized Analysis . 6 3.2 Array{Stack,Queue,Deque}s ..................................... 9 3.3 Graphs and AdjacencyMatrices .
    [Show full text]
  • Amortizedanalysis-2X
    Data structures Static problems. Given an input, produce an output. DATA STRUCTURES I, II, III, AND IV Ex. Sorting, FFT, edit distance, shortest paths, MST, max-flow, ... I. Amortized Analysis Dynamic problems. Given a sequence of operations (given one at a time), II. Binary and Binomial Heaps produce a sequence of outputs. Ex. Stack, queue, priority queue, symbol table, union–find, …. III. Fibonacci Heaps IV. Union–Find Algorithm. Step-by-step procedure to solve a problem. Data structure. Way to store and organize data. Ex. Array, linked list, binary heap, binary search tree, hash table, … 1 2 3 4 5 6 7 8 7 Lecture slides by Kevin Wayne 33 22 55 23 16 63 86 9 http://www.cs.princeton.edu/~wayne/kleinberg-tardos 33 10 44 86 33 99 1 ● 4 ● 1 ● 3 ● 46 83 90 93 47 60 Last updated on 11/13/19 5:37 AM 2 Appetizer Appetizer Goal. Design a data structure to support all operations in O(1) time. Data structure. Three arrays A[1.. n], B[1.. n], and C[1.. n], and an integer k. ・INIT(n): create and return an initialized array (all zero) of length n. ・A[i] stores the current value for READ (if initialized). ・READ(A, i): return element i in array. ・k = number of initialized entries. ・WRITE(A, i, value): set element i in array to value. ・C[j] = index of j th initialized element for j = 1, …, k. ・If C[j] = i, then B[i] = j for j = 1, …, k. true in C or C++, but not Java Assumptions.
    [Show full text]
  • Amortized Analysis & Splay Trees
    Lecture 2: Amortized Analysis & Splay Trees Rafael Oliveira University of Waterloo Cheriton School of Computer Science [email protected] September 14, 2020 1 / 66 Overview Introduction Meet your TAs! Types of amortized analyses Splay Trees Implementing Splay-Trees Setup Splay Rotations Analysis Acknowledgements 2 / 66 Meet your TAs Thi Xuan Vu Office hours: Wed morning (put exact time here) Anubhav Srivastava Office hours: Thursday afternoon (put exact time here) 3 / 66 With regards to the final project: I highly encourage you all to explore an open problem (but survey is also completely fine! :) ). The reason I wanted to mention is that this may be a unique opportunity for many of you to explore! So be bold! :) If you solve an open problem, you also automatically get 100 in this course and get to publish a paper! (and also get to experience the exhilarating feeling of solving a cool problem!) Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover. - Mark Twain Admin notes Late homework policy: I updated the late homework policy to be more flexible. Now each student has 10 late days without penalty for the entire term. 4 / 66 Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor.
    [Show full text]
  • Purely Functional Data Structures
    Purely Functional Data Structures Kristjan Vedel November 18, 2012 Abstract This paper gives an introduction and short overview of various topics related to purely functional data structures. First the concept of persistent and purely functional data structures is introduced, followed by some examples of basic data structures, relevant in purely functional setting, such as lists, queues and trees. Various techniques for designing more efficient purely functional data structures based on lazy evaluation are then described. Two notable general-purpose functional data structures, zippers and finger trees, are presented next. Finally, a brief overview is given of verifying the correctness of purely functional data structures and the time and space complexity calculations. 1 Data Structures and Persistence A data structure is a particular way of organizing data, usually stored in memory and designed for better algorithm efficiency[27]. The term data structure covers different distinct but related meanings, like: • An abstract data type, that is a set of values and associated operations, specified independent of any particular implementation. • A concrete realization or implementation of an abstract data type. • An instance of a data type, also referred to as an object or version. Initial instance of a data structure can be thought of as the version zero and every update operation then generates a new version of the data structure. Data structure is called persistent if it supports access to all versions. [7] It's called partially persistent if all versions can be accessed, but only last version can be modified and fully persistent if every version can be both accessed and mod- ified.
    [Show full text]
  • Mechanized Verification of the Correctness and Asymptotic Complexity of Programs
    Universit´ede Paris Ecole´ doctorale 386 | Sciences Math´ematiquesde Paris Centre Doctorat d'Informatique Mechanized Verification of the Correctness and Asymptotic Complexity of Programs The Right Answer at the Right Time Arma¨elGu´eneau Th`ese dirig´eepar Fran¸coisPottier et Arthur Chargu´eraud et soutenue le 16 d´ecembre 2019 devant le jury compos´ede : Jan Hoffmann Assistant Professor, Carnegie Mellon University Rapporteur Yves Bertot Directeur de Recherche, Inria Rapporteur Georges Gonthier Directeur de Recherche, Inria Examinateur Sylvie Boldo Directrice de Recherche, Inria Examinatrice Mihaela Sighireanu Ma^ıtrede Conf´erences, Universit´ede Paris Examinatrice Fran¸coisPottier Directeur de Recherche, Inria Directeur Arthur Chargu´eraud Charg´ede Recherche, Inria Co-directeur Abstract This dissertation is concerned with the question of formally verifying that the imple- mentation of an algorithm is not only functionally correct (it always returns the right result), but also has the right asymptotic complexity (it reliably computes the result in the expected amount of time). In the algorithms literature, it is standard practice to characterize the performance of an algorithm by indicating its asymptotic time complexity, typically using Landau's \big-O" notation. We first argue informally that asymptotic complexity bounds are equally useful as formal specifications, because they enable modular reasoning: a O bound abstracts over the concrete cost expression of a program, and therefore abstracts over the specifics of its implementation. We illustrate|with the help of small illustrative examples|a number of challenges with the use of the O notation, in particular in the multivariate case, that might be overlooked when reasoning informally. We put these considerations into practice by formalizing the O notation in the Coq proof assistant, and by extending an existing program verification framework, CFML, with support for a methodology enabling robust and modular proofs of asymptotic complexity bounds.
    [Show full text]
  • Analysis of Algorithms
    Analysis of Algorithms CSci 653, TTh 9:30-10:50, ISC 0248 Professor Mao, [email protected], 1-3472, McGl 114 General Information I Office Hours: TTh 11:00 – 12:00 and W 3:00 – 4:00 I Textbook: Intro to Algorithms (any edition), CLRS, McGraw Hill or MIT Press. I Prerequisites/background: Linear algebra, Data structures and algorithms, and Discrete math Use of Blackboard I Announcements I Problem sets (aka assignments or homework) I Lecture notes I Grades I Check at least weekly Lecture Basics I Lectures: Slides + board (mostly) I Lecture slides ⊂ lecture notes posted on BB I Not all taught in class are in the lecture notes/slides, e.g., some example problems. So take your own notes in class Course Organization I Mathematical foundation I Methods of analyzing algorithms I Methods of designing algorithms I Additional topics chosen from the lower bound theory, amortization, probabilistic analysis, competitive analysis, NP-completeness, and approximation algorithms Grading I About 12 problem sets: 60% I Final: 40% (in-class) Grading Policy (may be curved) I [90;100]: A or A- I [80;90): B+, B, or B- I [70;80): C+, C, or C- I [60;70): D+, D, or D- I [0;60):F Homework Submission Policy I Hardcopy (stapled and typeset in LaTex) to be submitted at the beginning of class on the due date I Extensions may be permitted for illness, family emergency, and travel to interviews/conferences. Requests must be made prior to the due date Homework Policy I Homework must be typeset in LaTex.
    [Show full text]
  • 8 Priority Queues 8 Priority Queues Dijkstra's Shortest Path Algorithm
    8 Priority Queues 8 Priority Queues A Priority Queue S is a dynamic set data structure that supports the following operations: ñ S: build(x1; : : : ; xn): Creates a data-structure that contains An addressable Priority Queue also supports: just the elements x1; : : : ; xn. ñ handle S: insert(x): Adds element x to the data-structure, ñ S: insert(x): Adds element x to the data-structure. and returns a handle to the object for future reference. ñ element S: minimum(): Returns an element x S with 2 ñ S: delete(h): Deletes element specified through handle h. minimum key-value key[x]. ñ S: decrease-key(h; k): Decreases the key of the element ñ element delete-min : Deletes the element with S: () specified by handle h to k. Assumes that the key is at least minimum key-value from and returns it. S k before the operation. ñ boolean S: is-empty(): Returns true if the data-structure is empty and false otherwise. Sometimes we also have ñ S: merge(S ): S : S S ; S : . 0 = [ 0 0 = 8 Priority Queues © Harald Räcke 303 © Harald Räcke 304 Dijkstra’s Shortest Path Algorithm Prim’s Minimum Spanning Tree Algorithm Algorithm 19 Prim-MST(G (V , E, d), s V) Algorithm 18 Shortest-Path(G (V , E, d), s V) = ∈ = ∈ 1: Input: weighted graph G (V , E, d); start vertex s; 1: Input: weighted graph G (V , E, d); start vertex s; = = 2: Output: pred-fields encode MST; 2: Output: key-field of every node contains distance from s; 3: S.build(); // build empty priority queue 3: S.build(); // build empty priority queue 4: for all v V s do 4: for all v V s do ∈ \{ } ∈ \{ } 5: v.
    [Show full text]
  • Scribe Notes
    15-750: Graduate Algorithms January 20, 2016 Lecture 2: Amortized Analysis, Heaps and Binomial Heaps Lecturer: Gary Miller Scribe: Zhong Zhou, Daniel Keenan 1 Amortized Analysis This section is written with reference to CLRS Chapter 17, and Avrim Blum's Amortized Analysis notes. 1.1 Introduction There are various time analysis methods: worst-case analysis, e.g., Strassen Algorithm for matrix multiplication, best-case analysis, and average case, e.g., Quick Sort pivoting on the first element. Amortized analysis is focused on worse-case input and average run time. For example, the random- ized algorithm is focusing on worst-case input and average over coin flips like randomized quicksort. The problems we can apply amortized analysis is often the time where we have a series of oper- ations, and the goal is to analyze the time taken per operation. For example, rather than being given a set of n items up front, we might have a series of n insert, lookup, and remove requests to some database, and we want these operations to be efficient. Definition 1.1. Amortized Cost. The amortized cost per operation for a sequence of n operations is the total cost of the operations divided by n There are generally three methods for performing amortized analysis: the aggregate method, the accounting method, and the potential method. All of these give the same answers, and their usage difference is primarily circumstantial and due to individual preference. • Aggregate analysis determines the upper bound T(n) on the total cost of a sequence of n operations, then calculates the amortized cost to be T(n) / n.
    [Show full text]