Certain height-balanced subtrees of hypercubes
26 May 2016-Vol. 1, Iss: 1, pp 32-41
TL;DR: This paper considers two subclasses of height-balanced trees and proves that every tree in the classes and is a subtree of the hypercube.
Abstract: A height-balanced tree is a desired data structure for performing operations such as search, insert and delete, on high-dimensional external data storage. Its preference is due to the fact that it always maintains logarithmic height even in worst cases. It is a rooted binary tree in which for every vertex the difference (denoted as balance factor) in the heights of the subtrees, rooted at the left and the right child of the vertex, is at most one. In this paper, we consider two subclasses of height-balanced trees and . A tree in is such that all the vertices up to (a predetermined) level t has balance factor one and the remaining vertices have balance factor zero. A tree in is such that all the vertices at alternate levels up to t has balance factor one and the remaining vertices have balance factor zero. We prove that every tree in the classes and is a subtree of the hypercube.
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TL;DR: In this article , a new way of classifying numerical sets (division into abstraction classes, called cuts hereafter in this paper) after imposing a constraint on the sum of elements of their subsets was presented.
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01 Jan 1973
4,243 citations
Book•
01 Sep 1991
TL;DR: This chapter discusses sorting on a Linear Array with a Systolic and Semisystolic Model of Computation, which automates the very labor-intensive and therefore time-heavy and expensive process of manually sorting arrays.
Abstract: Preface Acknowledgments Notation 1 Arrays and Trees 1.1 Elementary Sorting and Counting 1.1.1 Sorting on a Linear Array Assessing the Performance of the Algorithm Sorting N Numbers with Fewer Than N Processors 1.1.2 Sorting in the Bit Model 1.1.3 Lower Bounds 1.1.4 A Counterexample-Counting 1.1.5 Properties of the Fixed-Connection Network Model 1.2 Integer Arithmetic 1.2.1 Carry-Lookahead Addition 1.2.2 Prefix Computations-Segmented Prefix Computations 1.2.3 Carry-Save Addition 1.2.4 Multiplication and Convolution 1.2.5 Division and Newton Iteration 1.3 Matrix Algorithms 1.3.1 Elementary Matrix Products 1.3.2 Algorithms for Triangular Matrices 1.3.3 Algorithms for Tridiagonal Matrices -Odd-Even Reduction -Parallel Prefix Algorithms 1.3.4 Gaussian Elimination 1.3.5 Iterative Methods -Jacobi Relaxation -Gauss-Seidel Relaxation Finite Difference Methods -Multigrid Methods 1.4 Retiming and Systolic Conversion 1.4.1 A Motivating Example-Palindrome Recognition 1.4.2 The Systolic and Semisystolic Model of Computation 1.4.3 Retiming Semisystolic Networks 1.4.4 Conversion of a Semisystolic Network into a Systolic Network 1.4.5 The Special Case of Broadcasting 1.4.6 Retiming the Host 1.4.7 Design by Systolic Conversion-A Summary 1.5 Graph Algorithms 1.5.1 Transitive Closure 1.5.2 Connected Components 1.5.3 Shortest Paths 1.5.4 Breadth-First Spanning Trees 1.5.5 Minimum Weight Spanning Trees 1.6 Sorting Revisited 1.6.1 Odd-Even Transposition Sort on a Linear Array 1.6.2 A Simple Root-N(log N + 1)-Step Sorting Algorithm 1.6.3 A (3 Root- N + o(Root-N))-Step Sorting Algorithm 1.6.4 A Matching Lower Bound 1.7 Packet Routing 1.7.1 Greedy Algorithms 1.7.2 Average-Case Analysis of Greedy Algorithms -Routing N Packets to Random Destinations -Analysis of Dynamic Routing Problems 1.7.3 Randomized Routing Algorithms 1.7.4 Deterministic Algorithms with Small Queues 1.7.5 An Off-line Algorithm 1.7.6 Other Routing Models and Algorithms 1.8 Image Analysis and Computational Geometry 1.8.1 Component-Labelling Algorithms -Levialdi's Algorithm -An O (Root-N)-Step Recursive Algorithm 1.8.2 Computing Hough Transforms 1.8.3 Nearest-Neighbor Algorithms 1.8.4 Finding Convex Hulls 1.9 Higher-Dimensional Arrays 1.9.1 Definitions and Properties 1.9.2 Matrix Multiplication 1.9.3 Sorting 1.9.4 Packet Routing 1.9.5 Simulating High-Dimensional Arrays on Low-Dimensional Arrays 1.10 problems 1.11 Bibliographic Notes 2 Meshes of Trees 2.1 The Two-Dimensional Mesh of Trees 2.1.1 Definition and Properties 2.1.2 Recursive Decomposition 2.1.3 Derivation from KN,N 2.1.4 Variations 2.1.5 Comparison With the Pyramid and Multigrid 2.2 Elementary O(log N)-Step Algorithms 2.2.1 Routing 2.2.2 Sorting 2.2.3 Matrix-Vector Multiplication 2.2.4 Jacobi Relaxation 2.2.5 Pivoting 2.2.6 Convolution 2.2.7 Convex Hull 2.3 Integer Arithmetic 2.3.1 Multiplication 2.3.2 Division and Chinese Remaindering 2.3.3 Related Problems -Iterated Products -Rooting Finding 2.4 Matrix Algorithms 2.4.1 The Three-Dimensional Mesh of Trees 2.4.2 Matrix Multiplication 2.4.3 Inverting Lower Triangular Matrices 2.4.4 Inverting Arbitrary Matrices -Csanky's Algorithm -Inversion by Newton Iteration 2.4.5 Related Problems 2.5 Graph Algorithms 2.5.1 Minimum-Weight Spanning Trees 2.5.2 Connected Components 2.5.3 Transitive Closure 2.5.4 Shortest Paths 2.5.5 Matching Problems 2.6 Fast Evaluation of Straight-Line Code 2.6.1 Addition and Multiplication Over a Semiring 2.6.2 Extension to Codes with Subtraction and Division 2.6.3 Applications 2.7 Higher-Dimensional meshes of Trees 2.7.1 Definitions and Properties 2.7.2 The Shuffle-Tree Graph 2.8 Problems 2.9 Bibliographic Notes 3 Hypercubes and Related Networks 3.1 The Hypercube 3.1.1 Definitions and Properties 3.1.2 Containment of Arrays -Higher-Dimensional Arrays -Non-Power-of-2 Arrays 3.1.3 Containment of Complete Binary Trees 3.1.4 Embeddings of Arbitrary Binary Trees -Embeddings with Dilation 1 and Load O(M over N + log N) -Embeddings with Dilation O(1) and Load O (M over N + 1) -A Review of One-Error-Correcting Codes -Embedding Plog N into Hlog N 3.1.5 Containment of Meshes of Trees 3.1.6 Other Containment Results 3.2 The Butterfly, Cube-Connected-Cycles , and Benes Network 3.2.1 Definitions and Properties 3.2.2 Simulation of Arbitrary Networks 3.2.3 Simulation of Normal Hypercube Algorithms 3.2.4 Some Containment and Simulation Results 3.3 The Shuffle-Exchange and de Bruijn Graphs 3.3.1 Definitions and Properties 3.3.2 The Diaconis Card Tricks 3.3.3 Simulation of Normal Hypercube Algorithms 3.3.4 Similarities with the Butterfly 3.3.5 Some Containment and Simulation Results 3.4 Packet-Routing Algorithms 3.4.1 Definitions and Routing Models 3.4.2 Greedy Routing Algorithms and Worst-Case Problems 3.4.3 Packing, Spreading, and Monotone Routing Problems -Reducing a Many-to-Many Routing Problem to a Many-to-One Routing Problem -Reducing a Routing Problem to a Sorting Problem 3.4.4 The Average-Case Behavior of the Greedy Algorithm -Bounds on Congestion -Bounds on Running Time -Analyzing Non-Predictive Contention-Resolution Protocols 3.4.5 Converting Worst-Case Routing Problems into Average-Case Routing Problems -Hashing -Randomized Routing 3.4.6 Bounding Queue Sizes -Routing on Arbitrary Levelled Networks 3.4.7 Routing with Combining 3.4.8 The Information Dispersal Approach to Routing -Using Information Dispersal to Attain Fault-Tolerance -Finite Fields and Coding Theory 3.4.9 Circuit-Switching Algorithms 3.5 Sorting 3.5.1 Odd-Even Merge Sort -Constructing a Sorting Circuit with Depth log N(log N +1)/2 3.5.2 Sorting Small Sets 3.5.3 A Deterministic O(log N log log N)-Step Sorting Algorithm 3.5.4 Randomized O(log N)-Step Sorting Algorithms -A Circuit with Depth 7.45 log N that Usually Sorts 3.6 Simulating a Parallel Random Access Machine 3.6.1 PRAM Models and Shared Memories 3.6.2 Randomized Simulations Based on Hashing 3.6.3 Deterministic Simulations using Replicated Data 3.6.4 Using Information Dispersal to Improve Performance 3.7 The Fast Fourier Transform 3.7.1 The Algorithm 3.7.2 Implementation on the Butterfly and Shuffle-Exchange Graph 3.7.3 Application to Convolution and Polynomial Arithmetic 3.7.4 Application to Integer Multiplication 3.8 Other Hypercubic Networks 3.8.1 Butterflylike Networks -The Omega Network -The Flip Network -The Baseline and Reverse Baseline Networks -Banyan and Delta Networks -k-ary Butterflies 3.8.2 De Bruijn-Type Networks -The k-ary de Bruijn Graph -The Generalized Shuffle-Exchange Graph 3.9 Problems 3.10 Bibliographic Notes Bibliography Index Lemmas, Theorems, and Corollaries Author Index Subject Index
2,895 citations
1,438 citations
14 Jan 1963
TL;DR: The organization of information placed in the points of an automatic computer is discussed and the role of memory, storage and retrieval in this regard is discussed.
Abstract: : The organization of information placed in the points of an automatic computer is discussed. (Author)
846 citations
"Certain height-balanced subtrees of..." refers background in this paper
...AnAVL tree is also called a height-balanced tree and is formally defined in [1] as below....
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...The fact that the worst height of a height-balanced tree is logarithmic (see [1, 12]) has motivated the author to study the embedding of height-balanced trees into hypercubes....
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TL;DR: The author, a well-known researcher in paralle l computing, once again has proved his expertise and authority on the materials covered and this book will certainly have an impact to the psychology of students and researchers alike.
Abstract: In the ever-expanding field of parallel computing, we have seen a number of textbooks , some emphasizing the design aspects of parallel algorithms based on abstract models of paralle l machines (such as PRAMs) and some others focusing on the topological properties of paralle l architectures . What is needed in this area is a book which provides a linkage between th e topological properties of a parallel network and its computational capabilities or limitations , as well as comparative analyses of parallel architectures, not only among the proposed ones but also in view of a desirable general-purpose parallel machine which is yet to be built . The book under review comes closest to this goal . The author, a well-known researcher in paralle l computing, once again has proved his expertise and authority on the materials covered . This book will certainly have an impact to the psychology of students and researchers alike, on ho w to correlate parallel architectures and algorithms . Physically, this book is organized around three categories of parallel architectures : Arrays and Trees, Meshes of Trees, and Hypercubic networks . Each category covers not only th e basic type of architectures but also other variants or related models . For example, Chapter 1 on Arrays and Trees encompasses linear arrays, two-dimensional arrays, trees, ring, torus, X tree, pyramid, multigrid networks, systolic and semisystolic networks, and higher-dimensional arrays as well . Similarly, Chapter 2 on Meshes of Trees shows different ways of looking at two-dimensional meshes of trees at the beginning and further extends to higher-dimensiona l meshes of trees, and shuffle-tree graphs at the end . The third chapter, Hypercubes and Related Networks, covers butterfly, cube-connected-cycles, Benes network, shuffle-exchange, de Bruij n network, butterfly-like networks (Omega network, flip network, baseline and reverse baselin e networks, Banyan and delta networks, and k-ary butterfy), and de Bruijn-type networks (k-ar y de Bruijn network, and generalized shuffle-exchange network) . Whereas the above parallel networks constitute the architectural domain of the hook as th e basis, the application domain — parallel computation problems and algorithms — threads th e chapters together and helps a reader to view the similarities and differences of each network , from algorithm design standpoint . In addition to the definitions and characterizations of th e topological properties of the parallel architectures, each chapter examines a carefully-chose n subset of fundamental computational problems such as integer arithmetic, prefix computation , list ranking, sorting and counting, matrix arithmetic, graph problems, Fast Fourier Transfor m and Discrete Fourier Transform, computational geometry, and image analysis etc . The solution s to these problems are explored from simple algorithms to more complicated ones until it achieve s optimality. This approach seems to be adequate to reveal the capability and limitations of eac h network . The problems and algorithms are not treated in an isolated context but provokes a reader to capture what is achievable in terms of speedup and efficiency, and what is the limi t in terms of lower hounds, in a particular parallel network under focus . The author pays special attention to the routing problem . Considering that routing is a common vehicle for solving most of the regular and irregular parallel computation problem s in a fixed-connection network, the general capability of each network against an abstract parallel machine model is properly exposed via routing problem . Also discussed are the containment/embedding of one network in another, i .e . mapping between networks and the simulatio n
665 citations