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KP-Trie Algorithm for Update and Search Operations
The International Arab Journal of Information Technology, Vol. 13, No. 6, November 2016 722 KP-Trie Algorithm for Update and Search Operations Feras Hanandeh1, Izzat Alsmadi2, Mohammed Akour3, and Essam Al Daoud4 1Department of Computer Information Systems, Hashemite University, Jordan 2, 3Department of Computer Information Systems, Yarmouk University, Jordan 4Computer Science Department, Zarqa University, Jordan Abstract: Radix-Tree is a space optimized data structure that performs data compression by means of cluster nodes that share the same branch. Each node with only one child is merged with its child and is considered as space optimized. Nevertheless, it can’t be considered as speed optimized because the root is associated with the empty string. Moreover, values are not normally associated with every node; they are associated only with leaves and some inner nodes that correspond to keys of interest. Therefore, it takes time in moving bit by bit to reach the desired word. In this paper we propose the KP-Trie which is consider as speed and space optimized data structure that is resulted from both horizontal and vertical compression. Keywords: Trie, radix tree, data structure, branch factor, indexing, tree structure, information retrieval. Received January 14, 2015; accepted March 23, 2015; Published online December 23, 2015 1. Introduction the exception of leaf nodes, nodes in the trie work merely as pointers to words. Data structures are a specialized format for efficient A trie, also called digital tree, is an ordered multi- organizing, retrieving, saving and storing data. It’s way tree data structure that is useful to store an efficient with large amount of data such as: Large data associative array where the keys are usually strings, bases. -
14 Augmenting Data Structures
14 Augmenting Data Structures Some engineering situations require no more than a “textbook” data struc- ture—such as a doubly linked list, a hash table, or a binary search tree—but many others require a dash of creativity. Only in rare situations will you need to cre- ate an entirely new type of data structure, though. More often, it will suffice to augment a textbook data structure by storing additional information in it. You can then program new operations for the data structure to support the desired applica- tion. Augmenting a data structure is not always straightforward, however, since the added information must be updated and maintained by the ordinary operations on the data structure. This chapter discusses two data structures that we construct by augmenting red- black trees. Section 14.1 describes a data structure that supports general order- statistic operations on a dynamic set. We can then quickly find the ith smallest number in a set or the rank of a given element in the total ordering of the set. Section 14.2 abstracts the process of augmenting a data structure and provides a theorem that can simplify the process of augmenting red-black trees. Section 14.3 uses this theorem to help design a data structure for maintaining a dynamic set of intervals, such as time intervals. Given a query interval, we can then quickly find an interval in the set that overlaps it. 14.1 Dynamic order statistics Chapter 9 introduced the notion of an order statistic. Specifically, the ith order statistic of a set of n elements, where i 2 f1;2;:::;ng, is simply the element in the set with the ith smallest key. -
Lecture 04 Linear Structures Sort
Algorithmics (6EAP) MTAT.03.238 Linear structures, sorting, searching, etc Jaak Vilo 2018 Fall Jaak Vilo 1 Big-Oh notation classes Class Informal Intuition Analogy f(n) ∈ ο ( g(n) ) f is dominated by g Strictly below < f(n) ∈ O( g(n) ) Bounded from above Upper bound ≤ f(n) ∈ Θ( g(n) ) Bounded from “equal to” = above and below f(n) ∈ Ω( g(n) ) Bounded from below Lower bound ≥ f(n) ∈ ω( g(n) ) f dominates g Strictly above > Conclusions • Algorithm complexity deals with the behavior in the long-term – worst case -- typical – average case -- quite hard – best case -- bogus, cheating • In practice, long-term sometimes not necessary – E.g. for sorting 20 elements, you dont need fancy algorithms… Linear, sequential, ordered, list … Memory, disk, tape etc – is an ordered sequentially addressed media. Physical ordered list ~ array • Memory /address/ – Garbage collection • Files (character/byte list/lines in text file,…) • Disk – Disk fragmentation Linear data structures: Arrays • Array • Hashed array tree • Bidirectional map • Heightmap • Bit array • Lookup table • Bit field • Matrix • Bitboard • Parallel array • Bitmap • Sorted array • Circular buffer • Sparse array • Control table • Sparse matrix • Image • Iliffe vector • Dynamic array • Variable-length array • Gap buffer Linear data structures: Lists • Doubly linked list • Array list • Xor linked list • Linked list • Zipper • Self-organizing list • Doubly connected edge • Skip list list • Unrolled linked list • Difference list • VList Lists: Array 0 1 size MAX_SIZE-1 3 6 7 5 2 L = int[MAX_SIZE] -
Augmentation
TB Schardl 6.046J/18.410J October 02, 2009 Recitation 4 Augmentation The idea of augmentation is to add information to parts of a standard data structure to allow it to perform more complex tasks. You’ve practiced this concept already on Problem set 2, in order to allow skip lists to support O(log n) time RANK and SELECT methods. We’ll look at a similar prob- lem of creating “order statistic trees” by augmenting 2-3-4 trees to support RANK and SELECT. (The same methodology works for augmenting BSTs.) Order Statistic Trees Problem: Augment an ordinary 2-3-4 tree to support efficient RANK and SELECT-RANK methods without hurting the asymptotic performance of the normal tree operations. Idea: In each node, store the rank of the node in its subtree. An example order statistic tree is shown in Figure 1. 30 8 10 17 37 45 77 3 5 2 4 6 3 7 14 20 24 33 42 50 84 1 2 1 1 2 1 1 1 1 Figure 1: An example order statistic tree. The original values stored in the tree are shown in the top box in a darker gray, while their associated rank values are shown in the bottom box in a lighter gray. RANK,SELECT: We will examine SELECT, noting that the procedure for RANK is similar. Suppose we are searching for the ith order statistic. Starting from the root of the tree, look at the rank associated with the current node, n. If n.rank = i, return n. If n.rank > i, recursively SELECT the ith order statistic from the left child of n. -
Assignment of Master's Thesis
CZECH TECHNICAL UNIVERSITY IN PRAGUE FACULTY OF INFORMATION TECHNOLOGY ASSIGNMENT OF MASTER’S THESIS Title: Approximate Pattern Matching In Sparse Multidimensional Arrays Using Machine Learning Based Methods Student: Bc. Anna Kučerová Supervisor: Ing. Luboš Krčál Study Programme: Informatics Study Branch: Knowledge Engineering Department: Department of Theoretical Computer Science Validity: Until the end of winter semester 2018/19 Instructions Sparse multidimensional arrays are a common data structure for effective storage, analysis, and visualization of scientific datasets. Approximate pattern matching and processing is essential in many scientific domains. Previous algorithms focused on deterministic filtering and aggregate matching using synopsis style indexing. However, little work has been done on application of heuristic based machine learning methods for these approximate array pattern matching tasks. Research current methods for multidimensional array pattern matching, discovery, and processing. Propose a method for array pattern matching and processing tasks utilizing machine learning methods, such as kernels, clustering, or PSO in conjunction with inverted indexing. Implement the proposed method and demonstrate its efficiency on both artificial and real world datasets. Compare the algorithm with deterministic solutions in terms of time and memory complexities and pattern occurrence miss rates. References Will be provided by the supervisor. doc. Ing. Jan Janoušek, Ph.D. prof. Ing. Pavel Tvrdík, CSc. Head of Department Dean Prague February 28, 2017 Czech Technical University in Prague Faculty of Information Technology Department of Knowledge Engineering Master’s thesis Approximate Pattern Matching In Sparse Multidimensional Arrays Using Machine Learning Based Methods Bc. Anna Kuˇcerov´a Supervisor: Ing. LuboˇsKrˇc´al 9th May 2017 Acknowledgements Main credit goes to my supervisor Ing. -
Balanced Binary Search Trees 1 Introduction 2 BST Analysis
csce750 — Analysis of Algorithms Fall 2020 — Lecture Notes: Balanced Binary Search Trees This document contains slides from the lecture, formatted to be suitable for printing or individ- ual reading, and with some supplemental explanations added. It is intended as a supplement to, rather than a replacement for, the lectures themselves — you should not expect the notes to be self-contained or complete on their own. 1 Introduction CLRS 12, 13 A binary search tree is a data structure that supports these operations: • INSERT(k) • SEARCH(k) • DELETE(k) Basic idea: Store one key at each node. • All keys in the left subtree of n are less than the key stored at n. • All keys in the right subtree of n are greater than the key stored at n. Search and insert are trivial. Delete is slightly trickier, but not too bad. You may notice that these operations are very similar to the opera- tions available for hash tables. However, data structures like BSTs remain important because they can be extended to efficiently sup- port other useful operations like iterating over the elements in order, and finding the largest and smallest elements. These things cannot be done efficiently in hash tables. 2 BST Analysis Each operation can be done in time O(h) on a BST of height h. Worst case: Θ(n) Aside: Does randomization help? • Answer: Sort of. If we know all of the keys at the start, and insert them in a random order, in which each of the n! permutations is equally likely, then the expected tree height is O(lg n). -
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 . -
CMSC 420: Lecture 7 Randomized Search Structures: Treaps and Skip Lists
CMSC 420 Dave Mount CMSC 420: Lecture 7 Randomized Search Structures: Treaps and Skip Lists Randomized Data Structures: A common design techlque in the field of algorithm design in- volves the notion of using randomization. A randomized algorithm employs a pseudo-random number generator to inform some of its decisions. Randomization has proved to be a re- markably useful technique, and randomized algorithms are often the fastest and simplest algorithms for a given application. This may seem perplexing at first. Shouldn't an intelligent, clever algorithm designer be able to make better decisions than a simple random number generator? The issue is that a deterministic decision-making process may be susceptible to systematic biases, which in turn can result in unbalanced data structures. Randomness creates a layer of \independence," which can alleviate these systematic biases. In this lecture, we will consider two famous randomized data structures, which were invented at nearly the same time. The first is a randomized version of a binary tree, called a treap. This data structure's name is a portmanteau (combination) of \tree" and \heap." It was developed by Raimund Seidel and Cecilia Aragon in 1989. (Remarkably, this 1-dimensional data structure is closely related to two 2-dimensional data structures, the Cartesian tree by Jean Vuillemin and the priority search tree of Edward McCreight, both discovered in 1980.) The other data structure is the skip list, which is a randomized version of a linked list where links can point to entries that are separated by a significant distance. This was invented by Bill Pugh (a professor at UMD!). -
CS302ES Regulations
DATA STRUCTURES Subject Code: CS302ES Regulations : R18 - JNTUH Class: II Year B.Tech CSE I Semester Department of Computer Science and Engineering Bharat Institute of Engineering and Technology Ibrahimpatnam-501510,Hyderabad DATA STRUCTURES [CS302ES] COURSE PLANNER I. CourseOverview: This course introduces the core principles and techniques for Data structures. Students will gain experience in how to keep a data in an ordered fashion in the computer. Students can improve their programming skills using Data Structures Concepts through C. II. Prerequisite: A course on “Programming for Problem Solving”. III. CourseObjective: S. No Objective 1 Exploring basic data structures such as stacks and queues. 2 Introduces a variety of data structures such as hash tables, search trees, tries, heaps, graphs 3 Introduces sorting and pattern matching algorithms IV. CourseOutcome: Knowledge Course CO. Course Outcomes (CO) Level No. (Blooms Level) CO1 Ability to select the data structures that efficiently L4:Analysis model the information in a problem. CO2 Ability to assess efficiency trade-offs among different data structure implementations or L4:Analysis combinations. L5: Synthesis CO3 Implement and know the application of algorithms for sorting and pattern matching. Data Structures Data Design programs using a variety of data structures, CO4 including hash tables, binary and general tree L6:Create structures, search trees, tries, heaps, graphs, and AVL-trees. V. How program outcomes areassessed: Program Outcomes (PO) Level Proficiency assessed by PO1 Engineeering knowledge: Apply the knowledge of 2.5 Assignments, Mathematics, science, engineering fundamentals and Tutorials, Mock an engineering specialization to the solution of II B Tech I SEM CSE Page 45 complex engineering problems. -
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. -
B-Treaps: a Uniquely Represented Alternative to B-Trees
B-Treaps: A Uniquely Represented Alternative to B-Trees Daniel Golovin? Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA [email protected] Abstract. We present the first uniquely represented data structure for an external memory model of computation, a B-tree analogue called a B-treap. Uniquely represented data structures represent each logical state with a unique machine state. Such data structures are strongly history-independent; they reveal no information about the historical sequence of operations that led to the current logical state. For example, a uniquely represented file-system would support the deletion of a file in a way that, in a strong information-theoretic sense, provably removes all evidence that the file ever existed. Like the B-tree, the B-treap has depth O(logB n), uses linear space with high probability, where B is the block transfer size of the external memory, and supports efficient one-dimensional range queries. 1 Introduction Most computer applications store a significant amount of information that is hidden from the application interface—sometimes intentionally but more often not. This information might consist of data left behind in memory or disk, but can also consist of much more subtle variations in the state of a structure due to previous actions or the ordering of the actions. To address the concern of releasing historical and potentially private information various notions of history independence have been derived along with data structures that support these notions [10, 12, 9, 6, 1]. Roughly, a data structure is history independent if someone with complete access to the memory layout of the data structure (henceforth called the “observer”) can learn no more information than a legitimate user accessing the data structure via its standard interface (e.g., what is visible on screen). -
Questions for Computer Algorithms
www.YoYoBrain.com - Accelerators for Memory and Learning Questions for computer algorithms Category: Default - (246 questions) Define: symmetric key encryption also known as secret-key encryption, is a cryptography technique that uses a single secret key to both encrypt and decrypt data. Define: cipher text encrypted text Define: asymmetric encryption also known as public-key encryption, relies on key pairs. In a key pair, there is one public key and one private key. Define: hash is a checksum that is unique to a specific file or piece of data. Can be used to verify that a file has not been modified after the hash was generated. Algorithms: describe insertion sort algorithm Start with empty hand and cards face down on the table. Remove one card at a time from the table and insert it into the correct position in left hand. To find the correct position for a card, we compare it with each of the cards already in the hand, from right to left. Algorithms: loop invariant A loop is a statement which allows code to be repeatedly executed;an invariant of a loop is a property that holds before (and after) each repetition Algorithms: selection sort algorithm Sorting n numbers stored in array A by first finding the smallest element of A and exchanging it with A[1], then the second smallest and exchanging it with A[2], etc. Algorithms: merge sort Recursive formula1.) Divide the n-element sequence into 2 sub sequences on n/2 elements each2.) Conquer - sort the 2 sub sequences using merge sort3.) Combine - merge the 2 sorted sub sequences to produce sorted answer.