Segment Tree Lecture Notes
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Interval Trees Storing and Searching Intervals
Interval Trees Storing and Searching Intervals • Instead of points, suppose you want to keep track of axis-aligned segments: • Range queries: return all segments that have any part of them inside the rectangle. • Motivation: wiring diagrams, genes on genomes Simpler Problem: 1-d intervals • Segments with at least one endpoint in the rectangle can be found by building a 2d range tree on the 2n endpoints. - Keep pointer from each endpoint stored in tree to the segments - Mark segments as you output them, so that you don’t output contained segments twice. • Segments with no endpoints in range are the harder part. - Consider just horizontal segments - They must cross a vertical side of the region - Leads to subproblem: Given a vertical line, find segments that it crosses. - (y-coords become irrelevant for this subproblem) Interval Trees query line interval Recursively build tree on interval set S as follows: Sort the 2n endpoints Let xmid be the median point Store intervals that cross xmid in node N intervals that are intervals that are completely to the completely to the left of xmid in Nleft right of xmid in Nright Another view of interval trees x Interval Trees, continued • Will be approximately balanced because by choosing the median, we split the set of end points up in half each time - Depth is O(log n) • Have to store xmid with each node • Uses O(n) storage - each interval stored once, plus - fewer than n nodes (each node contains at least one interval) • Can be built in O(n log n) time. • Can be searched in O(log n + k) time [k = # -
Through the Concept of Data Structures, the Self- Balancing Binary Search Tree
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 01 | Jan 2019 www.irjet.net p-ISSN: 2395-0072 THROUGH THE CONCEPT OF DATA STRUCTURES, THE SELF- BALANCING BINARY SEARCH TREE Asniya Sadaf Syed Atiqur Rehman1, Avantika kishor Bakale2, Priyanka Patil3 1,2,3Department of Computer Science and Engineering, Prof Ram Meghe College of Engineering and Management, Badnera -------------------------------------------------------------------------***------------------------------------------------------------------------ ABSTRACT - In the word of computing tree is a known way of implementing balancing tree as binary hierarchical data structure which stores information tree they are originally discovered by Bayer and are naturally in the form of hierarchy style. Tree is a most nowadays extensively (diseased) in the standard powerful and advanced data structure. This paper is literature or algorithm Maintaining the invariant of red mainly focused on self -balancing binary search tree(BST) black tree through Haskell types system using the nested also known as height balanced BST. A BST is a type of data data types this give small list noticeable overhead. Many structure that adjust itself to provide the consistent level gives small list overhead can be removed by the use of of node access. This paper covers the different types of BST existential types [4]. their analysis, complexity and application. 3.1. AVL TREE Key words: AVL Tree, Splay Tree, Skip List, Red Black Tree, BST. AVL Tree is best example of self-balancing search tree; this means that AVL Tree is also binary search tree 1. INTRODUCTION which is also balancing tree. A binary tree is said to be Tree data structures are the similar like Maps and Sets, balanced if different between the height of left and right. -
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 -
Search Trees
Lecture III Page 1 “Trees are the earth’s endless effort to speak to the listening heaven.” – Rabindranath Tagore, Fireflies, 1928 Alice was walking beside the White Knight in Looking Glass Land. ”You are sad.” the Knight said in an anxious tone: ”let me sing you a song to comfort you.” ”Is it very long?” Alice asked, for she had heard a good deal of poetry that day. ”It’s long.” said the Knight, ”but it’s very, very beautiful. Everybody that hears me sing it - either it brings tears to their eyes, or else -” ”Or else what?” said Alice, for the Knight had made a sudden pause. ”Or else it doesn’t, you know. The name of the song is called ’Haddocks’ Eyes.’” ”Oh, that’s the name of the song, is it?” Alice said, trying to feel interested. ”No, you don’t understand,” the Knight said, looking a little vexed. ”That’s what the name is called. The name really is ’The Aged, Aged Man.’” ”Then I ought to have said ’That’s what the song is called’?” Alice corrected herself. ”No you oughtn’t: that’s another thing. The song is called ’Ways and Means’ but that’s only what it’s called, you know!” ”Well, what is the song then?” said Alice, who was by this time completely bewildered. ”I was coming to that,” the Knight said. ”The song really is ’A-sitting On a Gate’: and the tune’s my own invention.” So saying, he stopped his horse and let the reins fall on its neck: then slowly beating time with one hand, and with a faint smile lighting up his gentle, foolish face, he began.. -
2 Dynamization
6.851: Advanced Data Structures Spring 2010 Lecture 4 — February 11, 2010 Prof. Andr´eSchulz Scribe: Peter Caday 1 Overview In the previous lecture, we studied range trees and kd-trees, two structures which support efficient orthogonal range queries on a set of points. In this second of four lectures on geometric data structures, we will tie up a loose end from the last lecture — handling insertions and deletions. The main topic, however, will be two new structures addressing the vertical line stabbing problem: interval trees and segment trees. We will conclude with an application of vertical line stabbing to a windowing problem. 2 Dynamization In our previous discussion of range trees, we assumed that the point set was static. On the other hand, we may be interested in supporting insertions and deletions as time progresses. Unfortunately, it is not trivial to modify our data structures to handle this dynamic scenario. We can, however, dynamize them using general techniques; the techniques and their application here are due to Overmars [1]. Idea. Overmars’ dynamization is based on the idea of decomposable searches. A search (X,q) for the value q among the keys X is decomposable if the search result can be obtained in O(1) time from the results of searching (X1,q) and (X2,q), where X1 ∪ X2 = X. For instance, in the 2D kd-tree, the root node splits R2 into two halves. If we want to search for all points lying in a rectangle q, we may do this by combining the results of searching for q in each half. -
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 . -
Balanced Binary Search Trees – AVL Trees, 2-3 Trees, B-Trees
Balanced binary search trees – AVL trees, 2‐3 trees, B‐trees Professor Clark F. Olson (with edits by Carol Zander) AVL trees One potential problem with an ordinary binary search tree is that it can have a height that is O(n), where n is the number of items stored in the tree. This occurs when the items are inserted in (nearly) sorted order. We can fix this problem if we can enforce that the tree remains balanced while still inserting and deleting items in O(log n) time. The first (and simplest) data structure to be discovered for which this could be achieved is the AVL tree, which is names after the two Russians who discovered them, Georgy Adelson‐Velskii and Yevgeniy Landis. It takes longer (on average) to insert and delete in an AVL tree, since the tree must remain balanced, but it is faster (on average) to retrieve. An AVL tree must have the following properties: • It is a binary search tree. • For each node in the tree, the height of the left subtree and the height of the right subtree differ by at most one (the balance property). The height of each node is stored in the node to facilitate determining whether this is the case. The height of an AVL tree is logarithmic in the number of nodes. This allows insert/delete/retrieve to all be performed in O(log n) time. Here is an example of an AVL tree: 18 3 37 2 11 25 40 1 8 13 42 6 10 15 Inserting 0 or 5 or 16 or 43 would result in an unbalanced tree. -
Buffer Trees
Buffer Trees Lars Arge. The Buffer Tree: A New Technique for Optimal I/O Algorithms. In Proceedings of Fourth Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science Vol. 955, Springer-Verlag, 1995, 334-345. 1 Computational Geometry 2 Pairwise Rectangle Intersection A Input N rectangles B D Output all R pairwise intersections F E Example (A; B) (B; C) (B; F ) (D; E) (D; F ) C Intersection Types Intersection Identified by: : : A B Orthogonal Line Segment Intersection on 4N rectangle sides D E Batched Range Searching on N rectangles and N upper-left corners Algorithm Orthogonal Line Segment Intersection + Batched Range Searching + Duplicate removal 3 Orthogonal Line Segment Intersection Input N segments, vertical and horizontal Output all R intersections Sweepline Algorithm Sort all endpoints w.r.t. x-coordinate • Sweep left-to-right with a range tree T y4 • storing the y-coordinates of horizontal segments intersecting the sweepline y3 Left endpoint insertion into T y • ) 2 Right endpoint deletion from T y • ) 1 Vertical segment [y ; y ] • 1 2 ) sweepline report T [y ; y ] \ 1 2 Total (internal) time O(N log N + R) · 2 4 Range Trees Create Create empty structure Insert(x) Insert element x Delete(x) Delete the inserted element x Report(x1; x2) Report all x [x1; x2] 2 x1 x2 Binary search trees B-trees (internal) (# I/Os) Updates O(log2 N) O(logB N) R Report O(log2 N + R) O(logB N + B ) Orthogonal Line Segment Intersection using B-trees O(Sort(N) + N log N + R ) I/Os : : : · B B 5 Batched Range Searching Input N rectangles and points Output all R (r; p) where point p is within rectangle r Sweepline Algorithm Sort all points and left/right rectangle • sides w.r.t. -
Efficient Data Structures for Range Searching on a Grid MARK H
JOURNAL OF ALGORITHMS 9,254-275 (1988) Efficient Data Structures for Range Searching on a Grid MARK H. OVERMARS Department of Computer Science, University of Utrecht, The Netherlands Received February 17,1987; accepted May 15.1987 We consider the 2-dimensional range searching problem in the case where all points lie on an integer grid. A new data structure is presented that solves range queries on a U * U grid in O( k + log log U) time using O( n log n) storage, where n is the number of points and k the number of reported answers. Although the query time is very good the preprocessing time of this method is very high. A second data structure is presented that can be built in time O( n log n) at the cost of an increase in query time to O(k + m). Similar techniques are used for solving the line segment intersection searching problem in a set of axis-parallel line segments on a grid. The methods presented also yield efficient structures for dominance searching and searching with half-infinite ranges that use only O(n) storage. A generalization to multi-dimensional space, using a normalization approach, yields a static solution to the general range searching problem that is better than any known solution when the dimension is at least 3. Q 1988 Academic Press, Inc. 1. INTRODUCTION One of the problems in computational geometry that has received a large amount of attention is the range searching problem. Given a set of n points in a d-dimensional space, the range searching problem asks to store these points such that for a given range ([A, * . -
A Speculation-Friendly Binary Search Tree Tyler Crain, Vincent Gramoli, Michel Raynal
A Speculation-Friendly Binary Search Tree Tyler Crain, Vincent Gramoli, Michel Raynal To cite this version: Tyler Crain, Vincent Gramoli, Michel Raynal. A Speculation-Friendly Binary Search Tree. [Research Report] PI-1984, 2011, pp.21. inria-00618995v2 HAL Id: inria-00618995 https://hal.inria.fr/inria-00618995v2 Submitted on 5 Mar 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Publications Internes de l’IRISA ISSN : 2102-6327 PI 1984 – septembre 2011 A Speculation-Friendly Binary Search Tree* Tyler Crain** , Vincent Gramoli*** Michel Raynal**** [email protected], vincent.gramoli@epfl.ch, [email protected] Abstract: We introduce the first binary search tree algorithm designed for speculative executions. Prior to this work, tree structures were mainly designed for their pessimistic (non-speculative) accesses to have a bounded complexity. Researchers tried to evaluate transactional memory using such tree structures whose prominent example is the red-black tree library developed by Oracle Labs that is part of multiple benchmark distributions. Although well-engineered, such structures remain badly suited for speculative accesses, whose step complexity might raise dramatically with contention. We show that our speculation-friendly tree outperforms the existing transaction-based version of the AVL and the red-black trees. -
Algorithms for Packet Classification Pankaj Gupta and Nick Mckeown, Stanford University
Algorithms for Packet Classification Pankaj Gupta and Nick McKeown, Stanford University Abstract The process of categorizing packets into “flows” in an Internet router is called packet classification. All packets belonging to the same flow obey a predefined rule and are processed in a similar manner by the router. For example, all packets with the same source and destination IP addresses may be defined to form a flow. Packet classification is needed for non-best-effort services, such as firewalls and quality of service; services that require the capability to distinguish and isolate traffic in differ- ent flows for suitable processing. In general, packet classification on multiple fields is a difficult problem. Hence, researchers have proposed a variety of algorithms which, broadly speaking, can be categorized as basic search algorithms, geometric algorithms, heuristic algorithms, or hardware-specific search algorithms. In this tutorial we describe algorithms that are representative of each category, and discuss which type of algorithm might be suitable for different applications. ntil recently, Internet routers provided only best- Table 2 shows the flows into which an incoming packet effort service, servicing packets in a first-come- must be classified by the router at interface X. Note that the first-served manner. Routers are now called on to flows specified may or may not be mutually exclusive. For provide different qualities of service to different example, the first and second flow in Table 2 overlap. This is Uapplications, which means routers need new mechanisms common in practice, and when no explicit priorities are speci- such as admission control, resource reservation, per-flow fied, we follow the convention that rules closer to the top of queuing, and fair scheduling. -
Lecture Range Searching II: Windowing Queries
Computational Geometry · Lecture Range Searching II: Windowing Queries INSTITUT FUR¨ THEORETISCHE INFORMATIK · FAKULTAT¨ FUR¨ INFORMATIK Tamara Mchedlidze · Darren Strash 23.11.2015 1 Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Range Searching II Object types in range queries y0 y x x0 Setting so far: Input: set of points P (here P ⊂ R2) Output: all points in P \ [x; x0] × [y; y0] Data structures: kd-trees or range trees 2 Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Range Searching II Object types in range queries y0 y0 y x x0 y x x0 Setting so far: Further variant Input: set of points P Input: set of line segments S (here P ⊂ R2) (here in R2) Output: all points in Output: all segments in P \ [x; x0] × [y; y0] S \ [x; x0] × [y; y0] Data structures: kd-trees Data structures: ? or range trees 2 Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Range Searching II Axis-parallel line segments special case (e.g., in VLSI design): all line segments are axis-parallel 3 Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Range Searching II Axis-parallel line segments special case (e.g., in VLSI design): all line segments are axis-parallel Problem: Given n vertical and horizontal line segments and an axis-parallel rectangle R = [x; x0] × [y; y0], find all line segments that intersect R. 3 Dr. Tamara Mchedlidze · Dr. Darren Strash · Computational Geometry Lecture Range Searching II Axis-parallel line segments special case (e.g., in VLSI design): all line segments are axis-parallel Problem: Given n vertical and horizontal line segments and an axis-parallel rectangle R = [x; x0] × [y; y0], find all line segments that intersect R.