Succinct and Implicit Data Structures for Computational Geometry
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Advanced Data Structures
Advanced Data Structures PETER BRASS City College of New York CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521880374 © Peter Brass 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-43685-7 eBook (EBL) ISBN-13 978-0-521-88037-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page xi 1 Elementary Structures 1 1.1 Stack 1 1.2 Queue 8 1.3 Double-Ended Queue 16 1.4 Dynamical Allocation of Nodes 16 1.5 Shadow Copies of Array-Based Structures 18 2 Search Trees 23 2.1 Two Models of Search Trees 23 2.2 General Properties and Transformations 26 2.3 Height of a Search Tree 29 2.4 Basic Find, Insert, and Delete 31 2.5ReturningfromLeaftoRoot35 2.6 Dealing with Nonunique Keys 37 2.7 Queries for the Keys in an Interval 38 2.8 Building Optimal Search Trees 40 2.9 Converting Trees into Lists 47 2.10 -
A Pointer-Free Data Structure for Merging Heaps and Min-Max Heaps
Theoretical Computer Science 84 (1991) 107-126 107 Elsevier A pointer-free data structure for merging heaps and min-max heaps Giorgio Gambosi and Enrico Nardelli Istituto di Analisi dei Sistemi ed Informutica, C.N.R., Roma, Italy Maurizio Talamo Dipartimento di Matematica Pura ed Applicata, University of L’Aquila, L’Aquila, Italy, and Istituto di Analisi dei Sistemi ed Informatica, C.N.R., Roma, Italy Abstract Gambosi, G., E. Nardelli and M. Talamo, A pointer-free data structure for merging heaps and min-max heaps, Theoretical Computer Science 84 (1991) 107-126. In this paper a data structure for the representation of mergeable heaps and min-max heaps without using pointers is introduced. The supported operations are: Insert, DeleteMax, DeleteMin, FindMax, FindMin, Merge, NewHeap, DeleteHeap. The structure is analyzed in terms of amortized time complexity, resulting in a O(1) amortized time for each operation except for Insert, for which a O(lg n) bound holds. 1. Introduction The use of pointers in data structures seems to contribute quite significantly to the design of efficient algorithms for data access and management. Implicit data structures [ 131 have been introduced in order to evaluate the impact of the absence of pointers on time efficiency. Traditionally, implicit data structures have been mostly studied for what concerns the dictionary problem, both in l-dimensional [4,5,9, 10, 141, and in multi- dimensional space [l]. In such papers, the maintenance of a single dictionary has been analyzed, not considering the case in which several instances of the same structure (i.e. several dictionaries) have to be represented and maintained at the same time and within the same array-structured memory. -
Lecture III BALANCED SEARCH TREES
§1. Keyed Search Structures Lecture III Page 1 Lecture III BALANCED SEARCH TREES Anthropologists inform that there is an unusually large number of Eskimo words for snow. The Computer Science equivalent of snow is the tree word: we have (a,b)-tree, AVL tree, B-tree, binary search tree, BSP tree, conjugation tree, dynamic weighted tree, finger tree, half-balanced tree, heaps, interval tree, kd-tree, quadtree, octtree, optimal binary search tree, priority search tree, R-trees, randomized search tree, range tree, red-black tree, segment tree, splay tree, suffix tree, treaps, tries, weight-balanced tree, etc. The above list is restricted to trees used as search data structures. If we include trees arising in specific applications (e.g., Huffman tree, DFS/BFS tree, Eskimo:snow::CS:tree alpha-beta tree), we obtain an even more diverse list. The list can be enlarged to include variants + of these trees: thus there are subspecies of B-trees called B - and B∗-trees, etc. The simplest search tree is the binary search tree. It is usually the first non-trivial data structure that students encounter, after linear structures such as arrays, lists, stacks and queues. Trees are useful for implementing a variety of abstract data types. We shall see that all the common operations for search structures are easily implemented using binary search trees. Algorithms on binary search trees have a worst-case behaviour that is proportional to the height of the tree. The height of a binary tree on n nodes is at least lg n . We say that a family of binary trees is balanced if every tree in the family on n nodes has⌊ height⌋ O(log n). -
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. -
23. Metric Data Structures
This book is licensed under a Creative Commons Attribution 3.0 License 23. Metric data structures Learning objectives: • organizing the embedding space versus organizing its contents • quadtrees and octtrees. grid file. two-disk-access principle • simple geometric objects and their parameter spaces • region queries of arbitrary shape • approximation of complex objects by enclosing them in simple containers Organizing the embedding space versus organizing its contents Most of the data structures discussed so far organize the set of elements to be stored depending primarily, or even exclusively, on the relative values of these elements to each other and perhaps on their order of insertion into the data structure. Often, the only assumption made about these elements is that they are drawn from an ordered domain, and thus these structures support only comparative search techniques: the search argument is compared against stored elements. The shape of data structures based on comparative search varies dynamically with the set of elements currently stored; it does not depend on the static domain from which these elements are samples. These techniques organize the particular contents to be stored rather than the embedding space. The data structures discussed in this chapter mirror and organize the domain from which the elements are drawn—much of their structure is determined before the first element is ever inserted. This is typically done on the basis of fixed points of reference which are independent of the current contents, as inch marks on a measuring scale are independent of what is being measured. For this reason we call data structures that organize the embedding space metric data structures. -
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 -
A Dynamic Data Structure for Approximate Range Searching∗
A Dynamic Data Structure for Approximate Range Searching∗ David M. Mount Eunhui Park Department of Computer Science Department of Computer Science University of Maryland University of Maryland College Park, Maryland College Park, Maryland [email protected] [email protected] ABSTRACT 1. INTRODUCTION In this paper, we introduce a simple, randomized dynamic Recent decades have witnessed the development of many data structure for storing multidimensional point sets, called data structures for efficient geometric search and retrieval. a quadtreap. This data structure is a randomized, balanced A fundamental issue in the design of these data structures variant of a quadtree data structure. In particular, it de- is the balance between efficiency, generality, and simplicity. fines a hierarchical decomposition of space into cells, which One of the most successful domains in terms of simple, prac- are based on hyperrectangles of bounded aspect ratio, each tical approaches has been the development of linear-sized of constant combinatorial complexity. It can be viewed as a data structures for approximate geometric retrieval prob- multidimensional generalization of the treap data structure lems involving points sets in low dimensional spaces. In of Seidel and Aragon. When inserted, points are assigned this paper, our particular interest is in dynamic data struc- random priorities, and the tree is restructured through ro- tures, which allow the insertion and deletion of points for tations as if the points had been inserted in priority order. use in approximate retrieval problems, such as range search- In any fixed dimension d, we show it is possible to store a ing and nearest neighbor searching. -
Worst Case Efficient Data Structures
BRICS Basic Research in Computer Science BRICS DS-97-1 G. S. Brodal: Worst Case Efficient Data Structures Worst Case Efficient Data Structures Gerth Stølting Brodal BRICS Dissertation Series DS-97-1 ISSN 1396-7002 January 1997 Copyright c 1997, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Dissertation Series publi- cations. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory DS/97/1/ Worst Case Efficient Data Structures Gerth Stølting Brodal Ph.D. Dissertation Department of Computer Science University of Aarhus Denmark Worst Case Efficient Data Structures A Dissertation Presented to the Faculty of Science of the University of Aarhus in Partial Fulfillment of the Requirements for the Ph.D. Degree by Gerth Stølting Brodal January 31, 1997 Abstract We study the design of efficient data structures. In particular we focus on the design of data structures where each operation has a worst case efficient implementations. The concrete prob- lems we consider are partial persistence, implementation of priority queues, and implementation of dictionaries. The first problem we consider is how to make bounded in-degree and out-degree data structures partially persistent, i.e., how to remember old versions of a data structure for later access. -
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. -
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. -
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. -
MASTER of SCIENCE December, 1999 Oklaho.,..,,'" C '"I/O I It
COMPARATIVE STUDY OF PRIORITY QUEUES IMPLEMENTAnON By DONGHONG WEI Bachelor ofForeign Languages Shanxi University Taiyuan, China 1990 Submitted to the Faculty ofthe Graduate College ofthe Oklahoma State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE December, 1999 Oklaho.,..,,'" C '"I/o I It ... :• •~ ,._ .oJ., •, 'f """"':Iry COMPARATIVE STUDY OF PRIORITY QUEUES IMPLEMENTATION Thesis Approved: DeaOfthe Graduate College ii ACKNOWLEDGMENTS I wish to express my deepest appreciation to my advisor Dr. Jacques Lafrance for accepting to be my major advisor. His enthusiastic support, constructive guidance, encouragement, and friendship throughout time of acquaintance have been a constant source of inspiration and motivation that helped me gain confidence academically and professionally. My sincere appreciation also extends to Dr. John P. Chandler whose intelligent suggestion and guidance have made this thesis feasible. r would like to thank Dr. H. K. Dai for his constructive criticism, direction and wisdom. Grateful appreciation also goes to the faculty and students ofthe Department of Computer Science for their interest, guidance, and friendship. I wish to sincerely thank my husband for his unconditional love, spiritual support and endless encouragement. To my parents, Shuren Wei and Shuzhen Ma, I want to tell them how much I love them and that r could not have made it through graduate school without their total support and consistent encouragement. I dedicate this thesis to my dearest daughter, Connie Yang, who is the reason why I have worked so feverishly to complete this project. Finally, I would like to praise God for providing me with confidence, wisdom, persistence and strength that guided me through the entire process.