Worst Case Efficient Data Structures
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
1 Suffix Trees
This material takes about 1.5 hours. 1 Suffix Trees Gusfield: Algorithms on Strings, Trees, and Sequences. Weiner 73 “Linear Pattern-matching algorithms” IEEE conference on automata and switching theory McCreight 76 “A space-economical suffix tree construction algorithm” JACM 23(2) 1976 Chen and Seifras 85 “Efficient and Elegegant Suffix tree construction” in Apos- tolico/Galil Combninatorial Algorithms on Words Another “search” structure, dedicated to strings. Basic problem: match a “pattern” (of length m) to “text” (of length n) • goal: decide if a given string (“pattern”) is a substring of the text • possibly created by concatenating short ones, eg newspaper • application in IR, also computational bio (DNA seqs) • if pattern avilable first, can build DFA, run in time linear in text • if text available first, can build suffix tree, run in time linear in pattern. • applications in computational bio. First idea: binary tree on strings. Inefficient because run over pattern many times. • fractional cascading? • realize only need one character at each node! Tries: • used to store dictionary of strings • trees with children indexed by “alphabet” • time to search equal length of query string • insertion ditto. • optimal, since even hashing requires this time to hash. • but better, because no “hash function” computed. • space an issue: – using array increases stroage cost by |Σ| – using binary tree on alphabet increases search time by log |Σ| 1 – ok for “const alphabet” – if really fussy, could use hash-table at each node. • size in worst case: sum of word lengths (so pretty much solves “dictionary” problem. But what about substrings? • Relevance to DNA searches • idea: trie of all n2 substrings • equivalent to trie of all n suffixes. -
Managing Unbounded-Length Keys in Comparison-Driven Data
Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing∗ Amihood Amir Gianni Franceschini Department of Computer Science Dipartimento di Informatica Bar-Ilan University, Israel Universita` di Pisa, Italy and Department of Computer Science John Hopkins University, Baltimore MD Roberto Grossi Tsvi Kopelowitz Dipartimento di Informatica Department of Computer Science Universita` di Pisa, Italy Bar-Ilan University, Israel Moshe Lewenstein Noa Lewenstein Department of Computer Science Department of Computer Science Bar-Ilan University, Israel Netanya College, Israel arXiv:1306.0406v1 [cs.DS] 3 Jun 2013 ∗Parts of this paper appeared as extended abstracts in [3, 20]. 1 Abstract This paper presents a general technique for optimally transforming any dynamic data struc- ture that operates on atomic and indivisible keys by constant-time comparisons, into a data structure that handles unbounded-length keys whose comparison cost is not a constant. Exam- ples of these keys are strings, multi-dimensional points, multiple-precision numbers, multi-key data (e.g. records), XML paths, URL addresses, etc. The technique is more general than what has been done in previous work as no particular exploitation of the underlying structure of is required. The only requirement is that the insertion of a key must identify its predecessor or its successor. Using the proposed technique, online suffix tree construction can be done in worst case time O(log n) per input symbol (as opposed to amortized O(log n) time per symbol, achieved by previously known algorithms). To our knowledge, our algorithm is the first that achieves O(log n) worst case time per input symbol. -
Augmentation: Range Trees (PDF)
Lecture 9 Augmentation 6.046J Spring 2015 Lecture 9: Augmentation This lecture covers augmentation of data structures, including • easy tree augmentation • order-statistics trees • finger search trees, and • range trees The main idea is to modify “off-the-shelf” common data structures to store (and update) additional information. Easy Tree Augmentation The goal here is to store x.f at each node x, which is a function of the node, namely f(subtree rooted at x). Suppose x.f can be computed (updated) in O(1) time from x, children and children.f. Then, modification a set S of nodes costs O(# of ancestors of S)toupdate x.f, because we need to walk up the tree to the root. Two examples of O(lg n) updates are • AVL trees: after rotating two nodes, first update the new bottom node and then update the new top node • 2-3 trees: after splitting a node, update the two new nodes. • In both cases, then update up the tree. Order-Statistics Trees (from 6.006) The goal of order-statistics trees is to design an Abstract Data Type (ADT) interface that supports the following operations • insert(x), delete(x), successor(x), • rank(x): find x’s index in the sorted order, i.e., # of elements <x, • select(i): find the element with rank i. 1 Lecture 9 Augmentation 6.046J Spring 2015 We can implement the above ADT using easy tree augmentation on AVL trees (or 2-3 trees) to store subtree size: f(subtree) = # of nodes in it. Then we also have x.size =1+ c.size for c in x.children. -
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. -
Optimal Finger Search Trees in the Pointer Machine
Alcom-FT Technical Report Series ALCOMFT-TR-02-77 Optimal Finger Search Trees in the Pointer Machine Gerth Stølting Brodal∗ George Lagogiannis† Christos Makris† Athanasios Tsakalidis† Kostas Tsichlas† Abstract We develop a new finger search tree with worst-case constant update time in the Pointer Machine (PM) model of computation. This was a major problem in the field of Data Structures and was tantalizingly open for over twenty years while many attempts by researchers were made to solve it. The result comes as a consequence of the innovative mechanism that guides the rebalancing operations combined with incremental multiple splitting and fusion techniques over nodes. Keywords balanced trees, update operations, finger search trees, data structures, complexity 1 Introduction The balanced search tree is one of the most common data structures used in algorithms. Assuming that the update position is known, balanced search trees with O(1) amortized update time have been presented long ago ([6, 14]). It has also been known ([6, 16]) that updates can be performed in O(1) structural changes, but the nodes to be changed have to be searched in Ω(log n) time. Levcopoulos and Overmars ([13]) presented an algorithm achieving O(1) worst case update time by using a global splitting lemma that is based on a pebble game combined with the bucketing technique of Overmars ([14]). Instead of storing single keys in the leaves of the search tree, each leaf can store a list of several keys. Unfortunately, the buckets in [13] have size O(log2 n), so they need a two level hierarchy of lists in order to guarantee O(log n) query time within the buckets. -
Bulk Updates and Cache Sensitivity in Search Trees
BULK UPDATES AND CACHE SENSITIVITY INSEARCHTREES TKK Research Reports in Computer Science and Engineering A Espoo 2009 TKK-CSE-A2/09 BULK UPDATES AND CACHE SENSITIVITY INSEARCHTREES Doctoral Dissertation Riku Saikkonen Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Information and Natural Sciences, Helsinki University of Technology for public examination and debate in Auditorium T2 at Helsinki University of Technology (Espoo, Finland) on the 4th of September, 2009, at 12 noon. Helsinki University of Technology Faculty of Information and Natural Sciences Department of Computer Science and Engineering Teknillinen korkeakoulu Informaatio- ja luonnontieteiden tiedekunta Tietotekniikan laitos Distribution: Helsinki University of Technology Faculty of Information and Natural Sciences Department of Computer Science and Engineering P.O. Box 5400 FI-02015 TKK FINLAND URL: http://www.cse.tkk.fi/ Tel. +358 9 451 3228 Fax. +358 9 451 3293 E-mail: [email protected].fi °c 2009 Riku Saikkonen Layout: Riku Saikkonen (except cover) Cover image: Riku Saikkonen Set in the Computer Modern typeface family designed by Donald E. Knuth. ISBN 978-952-248-037-8 ISBN 978-952-248-038-5 (PDF) ISSN 1797-6928 ISSN 1797-6936 (PDF) URL: http://lib.tkk.fi/Diss/2009/isbn9789522480385/ Multiprint Oy Espoo 2009 AB ABSTRACT OF DOCTORAL DISSERTATION HELSINKI UNIVERSITY OF TECHNOLOGY P.O. Box 1000, FI-02015 TKK http://www.tkk.fi/ Author Riku Saikkonen Name of the dissertation Bulk Updates and Cache Sensitivity in Search Trees Manuscript submitted 09.04.2009 Manuscript revised 15.08.2009 Date of the defence 04.09.2009 £ Monograph ¤ Article dissertation (summary + original articles) Faculty Faculty of Information and Natural Sciences Department Department of Computer Science and Engineering Field of research Software Systems Opponent(s) Prof. -
Space-Efficient Data Structures, Streams, and Algorithms
Andrej Brodnik Alejandro López-Ortiz Venkatesh Raman Alfredo Viola (Eds.) Festschrift Space-Efficient Data Structures, Streams, LNCS 8066 and Algorithms Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday 123 Lecture Notes in Computer Science 8066 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany Andrej Brodnik Alejandro López-Ortiz Venkatesh Raman Alfredo Viola (Eds.) Space-Efficient Data Structures, Streams, and Algorithms Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday 13 Volume Editors Andrej Brodnik University of Ljubljana, Faculty of Computer and Information Science Ljubljana, Slovenia and University of Primorska, -
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). -
Finger Search Trees
11 Finger Search Trees 11.1 Finger Searching....................................... 11-1 11.2 Dynamic Finger Search Trees ....................... 11-2 11.3 Level Linked (2,4)-Trees .............................. 11-3 11.4 Randomized Finger Search Trees ................... 11-4 Treaps • Skip Lists 11.5 Applications............................................ 11-6 Optimal Merging and Set Operations • Arbitrary Gerth Stølting Brodal Merging Order • List Splitting • Adaptive Merging and University of Aarhus Sorting 11.1 Finger Searching One of the most studied problems in computer science is the problem of maintaining a sorted sequence of elements to facilitate efficient searches. The prominent solution to the problem is to organize the sorted sequence as a balanced search tree, enabling insertions, deletions and searches in logarithmic time. Many different search trees have been developed and studied intensively in the literature. A discussion of balanced binary search trees can e.g. be found in [4]. This chapter is devoted to finger search trees which are search trees supporting fingers, i.e. pointers, to elements in the search trees and supporting efficient updates and searches in the vicinity of the fingers. If the sorted sequence is a static set of n elements then a simple and space efficient representation is a sorted array. Searches can be performed by binary search using 1+⌊log n⌋ comparisons (we throughout this chapter let log x denote log2 max{2, x}). A finger search starting at a particular element of the array can be performed by an exponential search by inspecting elements at distance 2i − 1 from the finger for increasing i followed by a binary search in a range of 2⌊log d⌋ − 1 elements, where d is the rank difference in the sequence between the finger and the search element. -
An..Time Algorithm for Triangulating
SIAM J. COMPUT. (C) 1988 Society tbr Industrial and Applied Mathematics Voi. 17, No. I, February 1988 010 AN O(n log log n)-TIME ALGORITHM FOR TRIANGULATING A SIMPLE POLYGON* ROBERT E. TARJANf* AND CHRISTOPHER J. VAN WYK" Abstract. Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into n-2 triangles by adding n-3 nonintersecting diagonals. We propose an O(n log logn)-time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangu- lation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result. Key words, amortized time, balanced divide and conquer, heterogeneous finger search tree, homogene- ous finger search tree, horizontal visibility information, Jordan sorting with error-correction, simplicity test- ing AMS(MOS) subject classifications. 51M15, 68P05, 68Q25 1. Introduction. Let P be an n-vertex simple polygon, defined by a list Vo,V vn- of its vertices in clockwise order around the boundary. (The interior of the polygon is to the right as one walks clockwise around the boundary.) We denote the boundary of P by 0P. We assume throughout this paper (without loss of generality) that the vertices of P have distinct y-coordinates. For convenience we define vn V o. The edges of P are the open line segments whose endpoints are vi,vi+l for 0 < < n. The diagonals of P are the open line segments whose endpoints are vertices and that lie entirely in the interior of P. -
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.