Multidimensional and Spatial Structures
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Application of TRIE Data Structure and Corresponding Associative Algorithms for Process Optimization in GRID Environment
Application of TRIE data structure and corresponding associative algorithms for process optimization in GRID environment V. V. Kashanskya, I. L. Kaftannikovb South Ural State University (National Research University), 76, Lenin prospekt, Chelyabinsk, 454080, Russia E-mail: a [email protected], b [email protected] Growing interest around different BOINC powered projects made volunteer GRID model widely used last years, arranging lots of computational resources in different environments. There are many revealed problems of Big Data and horizontally scalable multiuser systems. This paper provides an analysis of TRIE data structure and possibilities of its application in contemporary volunteer GRID networks, including routing (L3 OSI) and spe- cialized key-value storage engine (L7 OSI). The main goal is to show how TRIE mechanisms can influence de- livery process of the corresponding GRID environment resources and services at different layers of networking abstraction. The relevance of the optimization topic is ensured by the fact that with increasing data flow intensi- ty, the latency of the various linear algorithm based subsystems as well increases. This leads to the general ef- fects, such as unacceptably high transmission time and processing instability. Logically paper can be divided into three parts with summary. The first part provides definition of TRIE and asymptotic estimates of corresponding algorithms (searching, deletion, insertion). The second part is devoted to the problem of routing time reduction by applying TRIE data structure. In particular, we analyze Cisco IOS switching services based on Bitwise TRIE and 256 way TRIE data structures. The third part contains general BOINC architecture review and recommenda- tions for highly-loaded projects. -
Trees • Binary Trees • Newer Types of Tree Structures • M-Ary Trees
Data Organization and Processing Hierarchical Indexing (NDBI007) David Hoksza http://siret.ms.mff.cuni.cz/hoksza 1 Outline • Background • prefix tree • graphs • … • search trees • binary trees • Newer types of tree structures • m-ary trees • Typical tree type structures • B-tree • B+-tree • B*-tree 2 Motivation • Similarly as in hashing, the motivation is to find record(s) given a query key using only a few operations • unlike hashing, trees allow to retrieve set of records with keys from given range • Tree structures use “clustering” to efficiently filter out non relevant records from the data set • Tree-based indexes practically implement the index file data organization • for each search key, an indexing structure can be maintained 3 Drawbacks of Index(-Sequential) Organization • Static nature • when inserting a record at the beginning of the file, the whole index needs to be rebuilt • an overflow handling policy can be applied → the performance degrades as the file grows • reorganization can take a lot of time, especially for large tables • not possible for online transactional processing (OLTP) scenarios (as opposed to OLAP – online analytical processing) where many insertions/deletions occur 4 Tree Indexes • Most common dynamic indexing structure for external memory • When inserting/deleting into/from the primary file, the indexing structure(s) residing in the secondary file is modified to accommodate the new key • The modification of a tree is implemented by splitting/merging nodes • Used not only in databases • NTFS directory structure -
IR2 Trees for Spatial Database Keyword Search NSF Award
IR2 Trees for Spatial Database Keyword Search NSF Award: CNS-0220562 Mil: Infrastructure for Research and Training in Database Management for Web-based Geospatial Data Visualization with Applications to Aviation PI: Naphtali Rishe Florida International University School of Computing and Information Sciences Co-PI: Boonserm Wongsaroj Florida Memorial University Department of Computer Sciences and Mathematics ABSTRACT Our Mil-provided infrastructure and research support has enabled us to perform basic research in keyword search. Many applications require finding objects closest to a specified location that contains a set of keywords. For example, online yellow pages allow users to - specify an address and a set of keywords. In return, the user obtains a list of businesses whose description contains these keywords, ordered by their distance from the specified address. The problems of nearest neighbor search on spatial data and keyword search on text data have been extensively studied separately. However, to the best of our knowledge there is no efficient method to answer spatial keyword queries, that is, queries that specify both a location and a set of keywords. In this work, we present an efficient method to answer top-k spatial keyword queries. To do so, we introduce an indexing structure called IR2-Tree (Information Retrieval R-Tree) which combines an R-Tree with superimposed text signatures. We present algorithms that construct and maintain an IR2-Tree, and use it to answer top-k spatial keyword queries. Our algorithms are experimentally and analytically compared to current methods and are shown to have superior performance and excellent scalability. 1. INTRODUCTION An increasing number of applications require the efficient execution of nearest neighbor queries constrained by the properties of the spatial objects. -
Spatial Databases and Spatial Indexing Techniques
Spatial Databases and Spatial Indexing Techniques Timos Sellis Computer Science Division Department of Electrical and Computer Engineering National Technical University of Athens Zografou 15773, GREECE Tel: +30-1-772-1601 FAX: +30-1-772-1659 e-mail: [email protected] Spatial Database Systems Timos Sellis Spatial Databases and Spatial Indexing Techniques Timos Sellis National Technical University of Athens e-mail: [email protected] Aalborg, June 1998 Outline • Data Models • Algebra • Query Languages • Data Structures • Query Processing and Optimization • System Architecture • Open Research Issues Spatial Database Systems 1 1 Spatial Database Systems Timos Sellis Introduction : Spatial Database Management Systems (SDBMS) QUESTION “What is a Spatial Database Management System ?” ANSWER • SDBMS is a DBMS • It offers spatial data types in its data model and query language • support of spatial relationships / properties / operations • It supports spatial data types in its implementation • efficient indexing and retrieval • support of spatial selection / join Spatial Database Systems 2 Applications of SDBMS Traditional GIS applications • Socio-Economic applications • Urban planning • Route optimization, market analysis • Environmental applications • Fire or Pollution Monitoring • Administrative applications • Public networks administration • Vehicle navigation Spatial Database Systems 3 2 Spatial Database Systems Timos Sellis Applications of SDBMS (cont'd) Novel applications • Image and Multimedia databases • shape configuration -
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 = # -
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. -
Balanced Trees Part One
Balanced Trees Part One Balanced Trees ● Balanced search trees are among the most useful and versatile data structures. ● Many programming languages ship with a balanced tree library. ● C++: std::map / std::set ● Java: TreeMap / TreeSet ● Many advanced data structures are layered on top of balanced trees. ● We’ll see several later in the quarter! Where We're Going ● B-Trees (Today) ● A simple type of balanced tree developed for block storage. ● Red/Black Trees (Today/Thursday) ● The canonical balanced binary search tree. ● Augmented Search Trees (Thursday) ● Adding extra information to balanced trees to supercharge the data structure. Outline for Today ● BST Review ● Refresher on basic BST concepts and runtimes. ● Overview of Red/Black Trees ● What we're building toward. ● B-Trees and 2-3-4 Trees ● Simple balanced trees, in depth. ● Intuiting Red/Black Trees ● A much better feel for red/black trees. A Quick BST Review Binary Search Trees ● A binary search tree is a binary tree with 9 the following properties: 5 13 ● Each node in the BST stores a key, and 1 6 10 14 optionally, some auxiliary information. 3 7 11 15 ● The key of every node in a BST is strictly greater than all keys 2 4 8 12 to its left and strictly smaller than all keys to its right. Binary Search Trees ● The height of a binary search tree is the 9 length of the longest path from the root to a 5 13 leaf, measured in the number of edges. 1 6 10 14 ● A tree with one node has height 0. -
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. -
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. -
Assignment 3: Kdtree ______Due June 4, 11:59 PM
CS106L Handout #04 Spring 2014 May 15, 2014 Assignment 3: KDTree _________________________________________________________________________________________________________ Due June 4, 11:59 PM Over the past seven weeks, we've explored a wide array of STL container classes. You've seen the linear vector and deque, along with the associative map and set. One property common to all these containers is that they are exact. An element is either in a set or it isn't. A value either ap- pears at a particular position in a vector or it does not. For most applications, this is exactly what we want. However, in some cases we may be interested not in the question “is X in this container,” but rather “what value in the container is X most similar to?” Queries of this sort often arise in data mining, machine learning, and computational geometry. In this assignment, you will implement a special data structure called a kd-tree (short for “k-dimensional tree”) that efficiently supports this operation. At a high level, a kd-tree is a generalization of a binary search tree that stores points in k-dimen- sional space. That is, you could use a kd-tree to store a collection of points in the Cartesian plane, in three-dimensional space, etc. You could also use a kd-tree to store biometric data, for example, by representing the data as an ordered tuple, perhaps (height, weight, blood pressure, cholesterol). However, a kd-tree cannot be used to store collections of other data types, such as strings. Also note that while it's possible to build a kd-tree to hold data of any dimension, all of the data stored in a kd-tree must have the same dimension. -
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 -
L11: Quadtrees CSE373, Winter 2020
L11: Quadtrees CSE373, Winter 2020 Quadtrees CSE 373 Winter 2020 Instructor: Hannah C. Tang Teaching Assistants: Aaron Johnston Ethan Knutson Nathan Lipiarski Amanda Park Farrell Fileas Sam Long Anish Velagapudi Howard Xiao Yifan Bai Brian Chan Jade Watkins Yuma Tou Elena Spasova Lea Quan L11: Quadtrees CSE373, Winter 2020 Announcements ❖ Homework 4: Heap is released and due Wednesday ▪ Hint: you will need an additional data structure to improve the runtime for changePriority(). It does not affect the correctness of your PQ at all. Please use a built-in Java collection instead of implementing your own. ▪ Hint: If you implemented a unittest that tested the exact thing the autograder described, you could run the autograder’s test in the debugger (and also not have to use your tokens). ❖ Please look at posted QuickCheck; we had a few corrections! 2 L11: Quadtrees CSE373, Winter 2020 Lecture Outline ❖ Heaps, cont.: Floyd’s buildHeap ❖ Review: Set/Map data structures and logarithmic runtimes ❖ Multi-dimensional Data ❖ Uniform and Recursive Partitioning ❖ Quadtrees 3 L11: Quadtrees CSE373, Winter 2020 Other Priority Queue Operations ❖ The two “primary” PQ operations are: ▪ removeMax() ▪ add() ❖ However, because PQs are used in so many algorithms there are three common-but-nonstandard operations: ▪ merge(): merge two PQs into a single PQ ▪ buildHeap(): reorder the elements of an array so that its contents can be interpreted as a valid binary heap ▪ changePriority(): change the priority of an item already in the heap 4 L11: Quadtrees CSE373,