Implementation and Performance Analysis of Exponential Tree Sorting
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:Submit/0396732
Dynamic 3-sided Planar Range Queries with Expected Doubly Logarithmic Time⋆ Gerth Stølting Brodal1, Alexis C. Kaporis2, Apostolos N. Papadopoulos4, Spyros Sioutas3, Konstantinos Tsakalidis1, Kostas Tsichlas4 1 MADALGO⋆⋆, Department of Computer Science, Aarhus University, Denmark gerth,tsakalid @madalgo.au.dk 2 { } Computer Engineering and Informatics Department, University of Patras, Greece [email protected] 3 Department of Informatics, Ionian University, Corfu, Greece [email protected] 4 Department of Informatics, Aristotle University of Thessaloniki, Greece [email protected] Abstract. This work studies the problem of 2-dimensional searching for the 3-sided range query of the form [a,b] ( , c] in both main × −∞ and external memory, by considering a variety of input distributions. We present three sets of solutions each of which examines the 3-sided problem in both RAM and I/O model respectively. The presented data structures are deterministic and the expectation is with respect to the input distribution: (1) Under continuous µ-random distributions of the x and y coordinates, we present a dynamic linear main memory solution, which answers 3- sided queries in O(log n + t) worst case time and scales with O(log log n) expected with high probability update time, where n is the current num- ber of stored points and t is the size of the query output. We external- ize this solution, gaining O(logB n + t/B) worst case and O(logB logn) amortized expected with high probability I/Os for query and update operations respectively, where B is the disk block size. (2)Then, we assume that the inserted points have their x-coordinates drawn from a class of smooth distributions, whereas the y-coordinates are arbitrarily distributed. -
Exponential Structures for Efficient Cache-Oblivious Algorithms
Exponential Structures for Efficient Cache-Oblivious Algorithms Michael A. Bender1, Richard Cole2, and Rajeev Raman3 1 Computer Science Department, SUNY Stony Brook, Stony Brook, NY 11794, USA. [email protected]. 2 Computer Science Department, Courant Institute, New York University, New York, NY 10012, USA. [email protected]. 3 Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, UK. [email protected] Abstract. We present cache-oblivious data structures based upon exponential structures. These data structures perform well on a hierarchical memory but do not depend on any parameters of the hierarchy, including the block sizes and number of blocks at each level. The problems we consider are searching, partial persistence and planar point location. On a hierarchical memory where data is transferred in blocks of size B, some of the results we achieve are: – We give a linear-space data structure for dynamic searching that supports searches and updates in optimal O(logB N) worst-case I/Os, eliminating amortization from the result of Bender, Demaine, and Farach-Colton (FOCS ’00). We also consider finger searches and updates and batched searches. – We support partially-persistent operations on an ordered set, namely, we allow searches in any previous version of the set and updates to the latest version of the set (an update creates a new version of the set). All operations take an optimal O(logB (m+N)) amortized I/Os, where N is the size of the version being searched/updated, and m is the number of versions. – We solve the planar point location problem in linear space, taking optimal O(logB N) I/Os for point location queries, where N is the number of line segments specifying the partition of the plane. -
Section Handout
CS106B Handout 37 Autumn 2012 November 26th, 2012 Section Handout Discussion Problem 1: Quadtrees A quadtree is a rooted tree structure where each internal node has precisely four children. Every node in the tree represents a square, and if a node has children, each encodes one of that square’s four quadrants. Quadtrees have many applications in computer graphics, because they can be used as in- memory models of images. That they can be used as in-memory versions of black and white images is easily demonstrated via the following (borrowed from Wikipedia.org): The 8 by 8 pixel image on the left is modeled by the quadtree on the right. Note that all leaf nodes are either black or white, and all internal nodes are shaded gray. The internal nodes are gray to reflect the fact that they contain both black and white pixels. When the pixels covered by a particular node are all the same color, the color is stored in the form of a Boolean and all four children are set to NULL. Otherwise, the node’s sub-region is recursively subdivided into four sub-quadrants, each represented by one of four children. Given a Grid<bool> representation of a black and white image, implement the gridToQuadtree function, which reads the image data, constructs the corresponding quadtree, and returns its root. Frame your implementation around the following data structure: struct quadtree { int lowx, highx; // smallest and largest x value covered by node int lowy, highy; // smallest and largest y value covered by node bool isBlack; // entirely black? true. -
Suggested Final Project Topics
CS166 Handout 10 Spring 2019 April 25, 2019 Suggested Final Project Topics Here is a list of data structures and families of data structures we think you might find interesting topics for a final project. You're by no means limited to what's contained here; if you have another data structure you'd like to explore, feel free to do so! If You Liked Range Minimum Queries, Check Out… Range Semigroup Queries In the range minimum query problem, we wanted to preprocess an array so that we could quickly find the minimum element in that range. Imagine that instead of computing the minimum of the value in the range, we instead want to compute A[i] ★ A[i+1] ★ … ★ A[j] for some associative op- eration ★. If we know nothing about ★ other than the fact that it's associative, how would we go about solving this problem efficiently? Turns out there are some very clever solutions whose run- times involve the magical Ackermann inverse function. Why they're worth studying: If you really enjoyed the RMQ coverage from earlier in the quarter, this might be a great way to look back at those topics from a different perspective. You'll get a much more nuanced understanding of why our solutions work so quickly and how to adapt those tech- niques into novel settings. Lowest Common Ancestor and Level Ancestor Queries Range minimum queries can be used to solve the lowest common ancestors problem: given a tree, preprocess the tree so that queries of the form “what node in the tree is as deep as possible and has nodes u and v as descendants?” LCA queries have a ton of applications in suffix trees and other al- gorithmic domains, and there’s a beautiful connection between LCA and RMQ that gave rise to the first ⟨O(n), O(1)⟩ solution to both LCA and RMQ. -
Functional Data Structures and Algorithms
Charles University in Prague Faculty of Mathematics and Physics DOCTORAL THESIS Milan Straka Functional Data Structures and Algorithms Computer Science Institute of Charles University Supervisor of the thesis: doc. Mgr. Zdeněk Dvořák, Ph.D. Study programme: Computer Science Specialization: Discrete Models and Algorithms (4I4) Prague 2013 I am grateful to Zdeněk Dvořák for his support. He was very accommodative during my studies. He quickly discovered any errors in my early conjectures and suggested areas of interest that prove rewarding. I would also like to express my sincere gratitude to Simon Peyton Jones for his supervision and guidance during my internship in Microsoft Research Labs, and also for helping me with one of my papers. Our discussions were always very intriguing and motivating. Johan Tibell supported my work on data structures in Haskell. His enthusiasm encouraged me to overcome initial hardships and continues to make the work really enjoyable. Furthermore, I would like to thank to Michal Koucký for comments and dis- cussions that improved the thesis presentation considerably. Finally, all this would not be possible without my beloved wife and my parents. You make me very happy, Jana. iii iv I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act. -
Sequential Nonlinear Learning
SEQUENTIAL NONLINEAR LEARNING a thesis submitted to the graduate school of engineering and science of bilkent university in partial fulfillment of the requirements for the degree of master of science in electrical and electronics engineering By Nuri Denizcan Vanlı August, 2015 Sequential Nonlinear Learning By Nuri Denizcan Vanlı August, 2015 We certify that we have read this thesis and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. S. Serdar Kozat (Advisor) Prof. Dr. A. Enis C¸etin Assoc. Prof. Dr. C¸a˘gatay Candan Approved for the Graduate School of Engineering and Science: Prof. Dr. Levent Onural Director of the Graduate School ii ABSTRACT SEQUENTIAL NONLINEAR LEARNING Nuri Denizcan Vanlı M.S. in Electrical and Electronics Engineering Advisor: Assoc. Prof. Dr. S. Serdar Kozat August, 2015 We study sequential nonlinear learning in an individual sequence manner, where we provide results that are guaranteed to hold without any statistical assump- tions. We address the convergence and undertraining issues of conventional non- linear regression methods and introduce algorithms that elegantly mitigate these issues using nested tree structures. To this end, in the second chapter, we intro- duce algorithms that adapt not only their regression functions but also the com- plete tree structure while achieving the performance of the best linear mixture of a doubly exponential number of partitions, with a computational complexity only polynomial in the number of nodes of the tree. In the third chapter, we propose an incremental decision tree structure and using this model, we introduce an online regression algorithm that partitions the regressor space in a data driven manner. -
Dynamic Data Structures for Document Collections and Graphs
Dynamic Data Structures for Document Collections and Graphs J. Ian Munro∗ Yakov Nekrichy Jeffrey Scott Vitterz Abstract In the dynamic indexing problem, we must maintain a changing collection of text documents so that we can efficiently support insertions, deletions, and pattern matching queries. We are especially interested in developing efficient data structures that store and query the documents in compressed form. All previous compressed solutions to this problem rely on answering rank and select queries on a dynamic sequence of symbols. Because of the lower bound in [Fredman and Saks, 1989], answering rank queries presents a bottleneck in compressed dynamic indexing. In this paper we show how this lower bound can be circumvented using our new framework. We demonstrate that the gap between static and dynamic variants of the indexing problem can be almost closed. Our method is based on a novel framework for adding dynamism to static compressed data structures. Our framework also applies more generally to dynamizing other problems. We show, for example, how our framework can be applied to develop compressed representations of dynamic graphs and binary relations. 1 Introduction Motivated by the preponderance of massive data sets (so-called \big data"), it is becoming increas- ingly useful to store data in compressed form and moreover to manipulate and query the data while in compressed form. For that reason, such compressed data structures have been developed in the context of text indexing, graph representations, XML indexes, labeled trees, and many other ap- plications. In this paper we describe a general framework to convert known static compressed data structures into dynamic compressed data structures. -
Dynamic Data Structures: Orthogonal Range Queries and Update Efficiency Konstantinos Athanasiou Tsakalidis
Dynamic Data Structures: Orthogonal Range Queries and Update Efficiency Konstantinos Athanasiou Tsakalidis PhD Dissertation Department of Computer Science Aarhus University Denmark Dynamic Data Structures: Orthogonal Range Queries and Update Efficiency A Dissertation Presented to the Faculty of Science of Aarhus University in Partial Fulfilment of the Requirements for the PhD Degree by Konstantinos A. Tsakalidis August 11, 2011 Abstract (English) We study dynamic data structures for different variants of orthogonal range reporting query problems. In particular, we consider (1) the planar orthogonal 3-sided range reporting problem: given a set of points in the plane, report the points that lie within a given 3-sided rectangle with one unbounded side, (2) the planar orthogonal range maxima reporting problem: given a set of points in the plane, report the points that lie within a given orthogonal range and are not dominated by any other point in the range, and (3) the problem of designing fully persistent B-trees for external memory. Dynamic problems like the above arise in various applications of network optimization, VLSI layout design, computer graphics and distributed computing. For the first problem, we present dynamic data structures for internal and external memory that support planar orthogonal 3-sided range reporting queries, and insertions and deletions of points efficiently over an average case sequence of update operations. The external memory data structures find applications in constraint and temporal databases. In particular, we assume that the coor- dinates of the points are drawn from different probabilistic distributions that arise in real-world applications, and we modify existing techniques for updating dynamic data structures, in order to support the queries and the updates more efficiently than the worst-case efficient counterparts for this problem. -
Binary Trees, While 3-Ary Trees Are Sometimes Called Ternary Trees
Rooted Tree A rooted tree is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the root: In some contexts, in which only a rooted tree would make sense, the term tree is often used. Infinite Tree A rooted tree is infinite if it contains a countably infinite number of nodes. Finite Tree Similarly, a rooted tree is finite if it contains a finite number of nodes. Parent Consider a rooted tree T whose root is r T Let t be a node of T From Paths in Trees are Unique, there is only one path from t to r T Let π:T−{r T }→T be the mapping defined as: π(t)= the node adjacent to t on the path to r T Then π(t) is known as the parent (or parent node) of t , and π as the parent function or parent mapping. Root Node The root node, or just root, is the one node in a rooted tree which, by definition, has no parent. Ancestor An ancestor (or ancestor node) of a node t of a rooted tree T whose root is r T is a node in the path from t to r T . Thus, the root of a rooted tree T is the ancestor of every node of T (including itself). Proper Ancestor A proper ancestor of a node t is an ancestor of t which is not t itself. Children The children (or child nodes) of a node t in a rooted tree T are the elements of the set: {s∈T:π(s)=t} That is, the children of t are all the nodes of T of which t is the parent. -
Comparison of Advance Tree Data Structures
International Journal of Computer Applications (0975 – 8887) Volume 41– No.2, March 2012 Comparison of Advance Tree Data Structures Parth Patel Deepak Garg Student - Master of Engineering Associate Professor CSED, Thapar University CSED, Thapar University Patiala, India Patiala, India ABSTRACT 2. B-TREE AND R-TREE B-tree and R-tree are two basic index structures; many The data structure which was proposed by Rudolf Bayer for different variants of them are proposed after them. Different the Organization and Maintenance of large ordered database variants are used in specific application for the performance was B-tree [12]. B-tree has variable number of child node optimization. In this paper different variants of B-tree and R- with predefined range. Each time node is inserted or deleted tree are discussed and compared. Index structures are different internal nodes may join and split because range is fix. Each in terms of structure, query support, data type support and internal node of B-tree contains number of keys. Number of application. Index structure’s structures are discussed first. B- keys chosen between d and 2d, d is order of B-tree. Number tree and its variants are discussed and them R-tree and its of child node of any node is d+1 to 2d+1. B-tree keeps record variants are discussed. Some structures example is also shown in sorted order for traversing. The index is adjusted with for the more clear idea. Then comparison is made between all recursive algorithm. It can handle any no of insertion and structure with respect to complexity, query type support, data deletion. -
Insert a New Element Before Or After the Kth Element – Delete the Kth Element of the List
Algorithmics (6EAP) MTAT.03.238 Linear structures, sorting, searching, Recurrences, Master Theorem... Jaak Vilo 2020 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 > 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 Data structures • https://en.wikipedia.org/wiki/List_of_data_structures • https://en.wikipedia.org/wiki/List_of_terms_relating_to_algo rithms_and_data_structures 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] L[2]=7 Lists: Array 0 1 size MAX_SIZE-1 3 6 7 5 2 L = int[MAX_SIZE] L[2]=7 L[size++] -
Onacci Sequence and the Time Complexity of Generating the Conway Polynomial and Related Topological Invariants
THE FIBONACCI SEQUENCE AND THE TIME COMPLEXITY OF GENERATING THE CONWAY POLYNOMIAL AND RELATED TOPOLOGICAL INVARIANTS Phillip G. B r a d f o r d University of Kansas, Lawrence, KS 66045 (Submitted July 1988) 1. Introduction In this discussion, only tame embeddings of S^ in S3 that are oriented will be considered. All knots will be represented as regular projections and any projection is assumed to be regular. The reader is expected to know "big 0" notation; see [2] for example. Some knowledge of NP-Completeness is also useful. The algorithm analysis of the complexities is relative to the number of crossings of a given knot projection. Also, the analytical creation of the Conway polynomial is in the Class P. This is shown by the presentation of a well-known algorithm used for computing the Alexander polynomial which can be easily suited to generate the Conway polynomial in better than 0(n3) time. The proof of the Conway algorithm having exponential worst case time com- plexity is based on showing the existence of n crossing knot projections. Given these particular projections, the Conway algorithm may perform <9(((1 + /5)/2)n) operations on the various knot projections which the algorithm derives in order to calculate the Conway polynomial. Definition 1.1: The crossing number of a knot K is the minimum number of cross- ings for any regular projection of the knot K. 3 Definition 1.2: A split link (see [10], [11]) L c S is a link L = £]_ u L2 where 3 Li and L2 are nonempty sublinks and there exist two open balls, 5j and B2 in S such that Bx n B2 = 0 ; and LY C B1 and L2 C B2.