Search Trees for Strings a Balanced Binary Search Tree Is a Powerful Data Structure That Stores a Set of Objects and Supports Many Operations Including
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 -
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. -
Lecture 26 Fall 2019 Instructors: B&S Administrative Details
CSCI 136 Data Structures & Advanced Programming Lecture 26 Fall 2019 Instructors: B&S Administrative Details • Lab 9: Super Lexicon is online • Partners are permitted this week! • Please fill out the form by tonight at midnight • Lab 6 back 2 2 Today • Lab 9 • Efficient Binary search trees (Ch 14) • AVL Trees • Height is O(log n), so all operations are O(log n) • Red-Black Trees • Different height-balancing idea: height is O(log n) • All operations are O(log n) 3 2 Implementing the Lexicon as a trie There are several different data structures you could use to implement a lexicon— a sorted array, a linked list, a binary search tree, a hashtable, and many others. Each of these offers tradeoffs between the speed of word and prefix lookup, amount of memory required to store the data structure, the ease of writing and debugging the code, performance of add/remove, and so on. The implementation we will use is a special kind of tree called a trie (pronounced "try"), designed for just this purpose. A trie is a letter-tree that efficiently stores strings. A node in a trie represents a letter. A path through the trie traces out a sequence ofLab letters that9 represent : Lexicon a prefix or word in the lexicon. Instead of just two children as in a binary tree, each trie node has potentially 26 child pointers (one for each letter of the alphabet). Whereas searching a binary search tree eliminates half the words with a left or right turn, a search in a trie follows the child pointer for the next letter, which narrows the search• Goal: to just words Build starting a datawith that structure letter. -
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. -
CSE 326: Data Structures Binary Search Trees
Announcements (1/23/09) CSE 326: Data Structures • HW #2 due now • HW #3 out today, due at beginning of class next Binary Search Trees Friday. • Project 2A due next Wed. night. Steve Seitz Winter 2009 • Read Chapter 4 1 2 ADTs Seen So Far The Dictionary ADT • seitz •Data: Steve • Stack •Priority Queue Seitz –a set of insert(seitz, ….) CSE 592 –Push –Insert (key, value) pairs –Pop – DeleteMin •ericm6 Eric • Operations: McCambridge – Insert (key, value) CSE 218 •Queue find(ericm6) – Find (key) • ericm6 • soyoung – Enqueue Then there is decreaseKey… Eric, McCambridge,… – Remove (key) Soyoung Shin – Dequeue CSE 002 •… The Dictionary ADT is also 3 called the “Map ADT” 4 A Modest Few Uses Implementations insert find delete •Sets • Unsorted Linked-list • Dictionaries • Networks : Router tables • Operating systems : Page tables • Unsorted array • Compilers : Symbol tables • Sorted array Probably the most widely used ADT! 5 6 Binary Trees Binary Tree: Representation • Binary tree is A – a root left right – left subtree (maybe empty) A pointerpointer A – right subtree (maybe empty) B C B C B C • Representation: left right left right pointerpointer pointerpointer D E F D E F Data left right G H D E F pointer pointer left right left right left right pointerpointer pointerpointer pointerpointer I J 7 8 Tree Traversals Inorder Traversal void traverse(BNode t){ A traversal is an order for if (t != NULL) visiting all the nodes of a tree + traverse (t.left); process t.element; Three types: * 5 traverse (t.right); • Pre-order: Root, left subtree, right subtree 2 4 } • In-order: Left subtree, root, right subtree } (an expression tree) • Post-order: Left subtree, right subtree, root 9 10 Binary Tree: Special Cases Binary Tree: Some Numbers… Recall: height of a tree = longest path from root to leaf. -
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. -
Efficient In-Memory Indexing with Generalized Prefix Trees
Efficient In-Memory Indexing with Generalized Prefix Trees Matthias Boehm, Benjamin Schlegel, Peter Benjamin Volk, Ulrike Fischer, Dirk Habich, Wolfgang Lehner TU Dresden; Database Technology Group; Dresden, Germany Abstract: Efficient data structures for in-memory indexing gain in importance due to (1) the exponentially increasing amount of data, (2) the growing main-memory capac- ity, and (3) the gap between main-memory and CPU speed. In consequence, there are high performance demands for in-memory data structures. Such index structures are used—with minor changes—as primary or secondary indices in almost every DBMS. Typically, tree-based or hash-based structures are used, while structures based on prefix-trees (tries) are neglected in this context. For tree-based and hash-based struc- tures, the major disadvantages are inherently caused by the need for reorganization and key comparisons. In contrast, the major disadvantage of trie-based structures in terms of high memory consumption (created and accessed nodes) could be improved. In this paper, we argue for reconsidering prefix trees as in-memory index structures and we present the generalized trie, which is a prefix tree with variable prefix length for indexing arbitrary data types of fixed or variable length. The variable prefix length enables the adjustment of the trie height and its memory consumption. Further, we introduce concepts for reducing the number of created and accessed trie levels. This trie is order-preserving and has deterministic trie paths for keys, and hence, it does not require any dynamic reorganization or key comparisons. Finally, the generalized trie yields improvements compared to existing in-memory index structures, especially for skewed data. -
Splay Trees Last Changed: January 28, 2017
15-451/651: Design & Analysis of Algorithms January 26, 2017 Lecture #4: Splay Trees last changed: January 28, 2017 In today's lecture, we will discuss: • binary search trees in general • definition of splay trees • analysis of splay trees The analysis of splay trees uses the potential function approach we discussed in the previous lecture. It seems to be required. 1 Binary Search Trees These lecture notes assume that you have seen binary search trees (BSTs) before. They do not contain much expository or backtround material on the basics of BSTs. Binary search trees is a class of data structures where: 1. Each node stores a piece of data 2. Each node has two pointers to two other binary search trees 3. The overall structure of the pointers is a tree (there's a root, it's acyclic, and every node is reachable from the root.) Binary search trees are a way to store and update a set of items, where there is an ordering on the items. I know this is rather vague. But there is not a precise way to define the gamut of applications of search trees. In general, there are two classes of applications. Those where each item has a key value from a totally ordered universe, and those where the tree is used as an efficient way to represent an ordered list of items. Some applications of binary search trees: • Storing a set of names, and being able to lookup based on a prefix of the name. (Used in internet routers.) • Storing a path in a graph, and being able to reverse any subsection of the path in O(log n) time. -
Binary Search Tree
ADT Binary Search Tree! Ellen Walker! CPSC 201 Data Structures! Hiram College! Binary Search Tree! •" Value-based storage of information! –" Data is stored in order! –" Data can be retrieved by value efficiently! •" Is a binary tree! –" Everything in left subtree is < root! –" Everything in right subtree is >root! –" Both left and right subtrees are also BST#s! Operations on BST! •" Some can be inherited from binary tree! –" Constructor (for empty tree)! –" Inorder, Preorder, and Postorder traversal! •" Some must be defined ! –" Insert item! –" Delete item! –" Retrieve item! The Node<E> Class! •" Just as for a linked list, a node consists of a data part and links to successor nodes! •" The data part is a reference to type E! •" A binary tree node must have links to both its left and right subtrees! The BinaryTree<E> Class! The BinaryTree<E> Class (continued)! Overview of a Binary Search Tree! •" Binary search tree definition! –" A set of nodes T is a binary search tree if either of the following is true! •" T is empty! •" Its root has two subtrees such that each is a binary search tree and the value in the root is greater than all values of the left subtree but less than all values in the right subtree! Overview of a Binary Search Tree (continued)! Searching a Binary Tree! Class TreeSet and Interface Search Tree! BinarySearchTree Class! BST Algorithms! •" Search! •" Insert! •" Delete! •" Print values in order! –" We already know this, it#s inorder traversal! –" That#s why it#s called “in order”! Searching the Binary Tree! •" If the tree is -
X-Fast and Y-Fast Tries
x-Fast and y-Fast Tries Outline for Today ● Bitwise Tries ● A simple ordered dictionary for integers. ● x-Fast Tries ● Tries + Hashing ● y-Fast Tries ● Tries + Hashing + Subdivision + Balanced Trees + Amortization Recap from Last Time Ordered Dictionaries ● An ordered dictionary is a data structure that maintains a set S of elements drawn from an ordered universe � and supports these operations: ● insert(x), which adds x to S. ● is-empty(), which returns whether S = Ø. ● lookup(x), which returns whether x ∈ S. ● delete(x), which removes x from S. ● max() / min(), which returns the maximum or minimum element of S. ● successor(x), which returns the smallest element of S greater than x, and ● predecessor(x), which returns the largest element of S smaller than x. Integer Ordered Dictionaries ● Suppose that � = [U] = {0, 1, …, U – 1}. ● A van Emde Boas tree is an ordered dictionary for [U] where ● min, max, and is-empty run in time O(1). ● All other operations run in time O(log log U). ● Space usage is Θ(U) if implemented deterministically, and O(n) if implemented using hash tables. ● Question: Is there a simpler data structure meeting these bounds? The Machine Model ● We assume a transdichotomous machine model: ● Memory is composed of words of w bits each. ● Basic arithmetic and bitwise operations on words take time O(1) each. ● w = Ω(log n). A Start: Bitwise Tries Tries Revisited ● Recall: A trie is a simple data 0 1 structure for storing strings. 0 1 0 1 ● Integers can be thought of as strings of bits. 1 0 1 1 0 ● Idea: Store integers in a bitwise trie. -
Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke
Trees Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke Department of Computer Science & Engineering University of Nebraska|Lincoln Lincoln, NE 68588, USA [email protected] http://cse.unl.edu/~cbourke 2015/01/31 21:05:31 Abstract These are lecture notes used in CSCE 156 (Computer Science II), CSCE 235 (Dis- crete Structures) and CSCE 310 (Data Structures & Algorithms) at the University of Nebraska|Lincoln. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License 1 Contents I Trees4 1 Introduction4 2 Definitions & Terminology5 3 Tree Traversal7 3.1 Preorder Traversal................................7 3.2 Inorder Traversal.................................7 3.3 Postorder Traversal................................7 3.4 Breadth-First Search Traversal..........................8 3.5 Implementations & Data Structures.......................8 3.5.1 Preorder Implementations........................8 3.5.2 Inorder Implementation.........................9 3.5.3 Postorder Implementation........................ 10 3.5.4 BFS Implementation........................... 12 3.5.5 Tree Walk Implementations....................... 12 3.6 Operations..................................... 12 4 Binary Search Trees 14 4.1 Basic Operations................................. 15 5 Balanced Binary Search Trees 17 5.1 2-3 Trees...................................... 17 5.2 AVL Trees..................................... 17 5.3 Red-Black Trees.................................. 19 6 Optimal Binary Search Trees 19 7 Heaps 19