Showing papers on "Spanning tree published in 1991"

Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes

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

Choosing a Spanning Tree for the Integer Lattice Uniformly

Abstract: Consider the nearest neighbor graph for the integer lattice $\mathbf{Z}^d$ in $d$ dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for $\mathbf{Z}^d$. This is shown to be a tree if and only if $d \leq 4$. In this case, the tree has only one topological end, that is, there are no doubly infinite paths. When $d \geq 5$ the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

Distributed program checking: a paradigm for building self-stabilizing distributed protocols

Abstract: The notion of distributed program checking as a means of making a distributed algorithm self-stabilizing is explored. A compiler that converts a deterministic synchronous protocol pi for static networks into a self-stabilizing version of pi for dynamic networks is described. If T/sub pi / is the time complexity of pi and D is a bound on the diameter of the final network, the compiled version of pi stabilizes in time O(D+T/sub pi /) and has the same space complexity as pi . The general method achieves efficient results for many specific noninteractive tasks. For instance, solutions for the shortest paths and spanning tree problems take O(D) to stabilize, an improvement over the previous best time of O(D/sup 2/). >

Spanning trees with many leaves

Abstract: A connected graph having large minimum vertex degree must have a spanning tree with many leaves. In particular, let $l( n,k )$ be the maximum integer m such that every connected n-vertex graph with minimum degree at least k has a spanning tree with at least m leaves. Then $l( n,3 )\geqq n/4 + 2, l( n,4 )\geqq ( 2n + 8 )/5,$ and $l( n,k )\leqq n - 3\lfloor n/( k + 1 ) \rfloor + 2$ for all k. The lower bounds are proved by an algorithm that constructs a spanning tree with at least the desired number of leaves. Finally, $l( n,k )\geqq ( 1 - b \ln k/k )n$ for large k, again proved algorithmically, where b is any constant exceeding 2.5.

Euclidean minimum spanning trees and bichromatic closest pairs

Abstract: We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inEd in timeO(Fd(N,N) logdN), whereFd(n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inEd. IfFd(N,N)=Ω(N1+?), for some fixed ?>0, then the running time improves toO(Fd(N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)2/3+m log2n+n log2m) inE3, which yields anO(N4/3 log4/3N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE3. Ind?4 dimensions we obtain expected timeO((nm)1?1/([d/2]+1)+?+m logn+n logm) for the bichromatic closest pair problem andO(N2?2/([d/2]+1)?) for the Euclidean minimum spanning tree problem, for any positive ?.

Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees

Abstract: Ambivalent data structures are presented for several problems on undirected graphs. They are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log beta (m,n)+min(k/sup 3/2/, km/sup 1/2/)) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n+k(log n)/sup 3/) time. Ambivalent data structures are also used to maintain dynamically 2-edge-connectivity information. Edges and vertices can be inserted or deleted in O(m/sup 1/2/) time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Again, the techniques are extended to maintain an embedded planar graph so that edges and vertices can be inserted or deleted in O((log n)/sup 3/) time, and a query answered in O(log n) time. >

On the core of network synthesis games

Abstract: We use polynomial formulations to show that several rational and discrete network synthesis games, including the minimum cost spanning tree game, satisfy the assumptions of Owen's linear production game model. We also discuss computational issues related to finding and recognizing core points for these classes of games.

Minimum diameter spanning trees and related problems

Abstract: The problem of finding a minimum diameter spanning tree (MDST) of a set of n points in the Euclidean space is considered. The diameter of a spanning tree is the maximum distance between any two poi...

Constructing multidimensional spanner graphs

Abstract: Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing edges in time. We apply this spanner to the construction of approximate minimum spanning trees.

Communication network and a method of regulating the transmission of data packets in a communication network

Abstract: A two-phase fairness algorithm for regulating the transmission of data packets in a communication network having a multitude of nodes connected together to form a spanning tree. In a first or broadcast phase of the fairness algorithm, a first control signal is transmitted over the spanning tree from a root node or a root edge of the tree. This first control signal indicates to each node of the tree a first number of data packets that the node is permitted to transmit in a corresponding time interval. Each intermediate node of the tree transmits the first signal to each of its children nodes only when one or more various conditions relating to the transmission of data packets are satisfied. In a second or merge phase of the fairness algorithm, a second control signal is transmitted from the leaves of the tree to the root node or root edge of the tree. Each intermediate node of the tree transmits the second signal to its parent node only after receiving the second signal from all of its children nodes and after one or more various conditions relating to the transmission of data packets from the intermediate node to its parent node are satisfied. After the root node or root edge of the tree receives the second signal from all of its or their children nodes, the algorithm is repeated.

Optimizing equijoin queries in distributed databases where relations are hash partitioned

Abstract: Consider the class of distributed database systems consisting of a set of nodes connected by a high bandwidth network. Each node consists of a processor, a random access memory, and a slower but much larger memory such as a disk. There is no shared memory among the nodes. The data are horizontally partitioned often using a hash function. Such a description characterizes many parallel or distributed database systems that have recently been proposed, both commercial and academic. We study the optimization problem that arises when the query processor must repartition the relations and intermediate results participating in a multijoin query. Using estimates of the sizes of intermediate relations, we show (1) optimum solutions for closed chain queries; (2) the NP-completeness of the optimization problem for star, tree, and general graph queries; and (3) effective heuristics for these hard cases.Our general approach and many of our results extend to other attribute partitioning schemes, for example, sort-partitioning on attributes, and to partitioned object databases.

An approach for proving lower bounds: Solution of Gilbert-Pollak's conjecture on Steiner ratio

Abstract: A family of finitely many continuous functions on a polytope X, namely (g/sub i/(x))/sub i in I/, is considered, and the problem of minimizing the function f(x)=max/sub i in I/g/sub i/(x) on X is treated. It is shown that if every g/sub i/(x) is a concave function, then the minimum value of f(x) is achieved at finitely many special points in X. As an application, a long-standing problem about Steiner minimum trees and minimum spanning trees is solved. In particular, if P is a set of n points on the Euclidean plane and L/sub s/(P) and L/sub m/(P) denote the lengths of a Steiner minimum tree and a minimum spanning tree on P, respectively, it is proved that, for any P, L/sub S/(P)>or= square root 3L/sub m/(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak (1968). >

Intersection and Decomposition Algorithms for Planar Arrangements

Abstract: This book presents a study of various problems related to arrangements of lines, segments, or curves in the plane. The first problem is a proof of almost tight bounds on the length of (n,s)-Davenport-Schinzel sequences, a technique for obtaining optimal bounds for numerous algorithmic problems. Then the intersection problem is treated. The final problem is improving the efficiency of partitioning algorithms, particularly those used to construct spanning trees with low stabbing numbers, a very versatile tool in solving geometric problems. A number of applications are also discussed.

An adaptation of the interior point method for solving the global routing problem

Abstract: A linear-programming (LP) solution technique is applied to solve the global routing problem. Minimal spanning trees that were found in the subgrid by generating vertical and horizontal lines about the points of a given net and a modified interior point approach that reduces the number of arc constraints which were considered in the routing problem were used to reduce the problem site. An interior point algorithm approach is developed to solve the resulting LP problem. A FORTRAN code ROUTFAST is tested on several problems of varying arc capacities. ROUTFAST is compared with MINOS 5.0 which is a Simplex-based code on the same set of problems. ROUTFAST is at least 8-20 times faster than MINOS on the largest problems tested and seems to be getting faster as the problem size and arc capacity increases. Using a modification (ROUTLP) to solve four benchmark gate-array layout problems, ROUTFAST yielded competitive global routing results to TimberWolfSC Version 5.4 and in a fraction of TimberWolfSC's running time. >

Dynamic trees and dynamic point location

Abstract: This paper describes new methods for maintaining a point-location data structure for a dynamically changing monotone subdivisionS. The main approach is based on the maintenance of two interlaced spanning trees, one forS and one for the graph-theoretic planar dual ofS. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan (J. Comput. System Sci., 26 (1983), pp. 362{381), leading to a scheme that achieves vertex insertion/deletion in O(logn) time, insertion/deletion of k-edge monotone chains in O(logn+k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(logn) time, leading to an improved method for spatial point location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point location (where one builds S incrementally), improving the query bound to O(logn log logn) time and the update bounds to O(1) amortized time in this case. This appears to be the rst on-line method to achieve a polylogarithmic query time and constant update time.

Applications of a new space partitioning technique

Abstract: We present several applications of a recent space partitioning technique of Chazelle, Sharir and Welzl [8]. Our results include efficient algorithms for output-sensitive hidden surface removal, for ray shooting in two and three dimensions, and for constructing spanning trees with low stabbing number.

An empirical analysis of algorithms for constructing a minimum spanning tree

Abstract: We compare algorithms for the construction of a minimum spanning tree through largescale experimentation on randomly generated graphs of different structures and different densities In order to extrapolate with confidence, we use graphs with up to 130,000 nodes (sparse) or 750,000 edges (dense) Algorithms included in our experiments are Prim's algorithm (implemented with a variety of priority queues), Kruskal's algorithm (using presorting or demand sorting), Cheriton and Tarjan's algorithm, and Fredman and Tarjan's algorithm We also ran a large variety of tests to investigate low-level implementation decisions for the data structures, as well as to enable us to eliminate the effect of compilers and architectures

Spanning trees with bounded degrees

Abstract: Lets andk be positive integers. We prove that ifG is ak-connected graph containing no independent set withks+2 vertices thenG has a spanning tree with maximum degree at mosts+1. Moreover ifs≥3 and the independence number α(G) is such that α(G)≤1+k(s−1)+c for some0≤c≤k thenG has a spanning tree with no more thanc vertices of degrees+1.

Performance-driven global routing for cell based ICs

Abstract: Advances in VLSI technology and the increased complexity of circuit designs cause performance to become an increasingly important constraint for layout. The issue of delay optimization during the global routing phase is addressed. This problem is formulated as the construction of a bounded-radius spanning tree for a given pointset in the plane, and a family of effective heuristics is presented. This approach has very good empirical performance with respect to total wirelength, and can be smoothly tuned between the competing requirements of minimum delay and minimum total netlength, as confirmed by extensive computational results which confirm this. Extensions can be made to the graph and Steiner versions of the problem, and a number of open problems are described. >

Optimal Broadcasting in Mesh-Connected Architectures

Abstract: In this paper, we disprove the common assumption that the time for broadcasting in a mesh is at best proportional to the square root of the number of processors, at least in the presence of worm-hole routing. We present an optimal algorithm for broadcasting in mesh-connected distributed-memory architectures with worm-hole routing. By organizing the processing nodes in a logical spanning tree, the algorithm executes in time proportional to the logarithm of the number of nodes without inducing contention in the communication network. We restrict the number of nodes in each dimension of the processor mesh to be a power of two. Our method provides insight into how to avoid and/or reduce network contention on meshes for other communication operations. Experimental results on the Intel Touchstone Delta system are included.

Transitions in geometric minimum spanning trees (extended abstract)

Abstract: 2. We study some combinatorial and algorithmic problems related to transitions in Euclidean minimum spanning trees arising from an arbitrary motion of one or more points of the input set.

Construction of multidimensional spanner graphs, with applications to minimum spanning trees

Abstract: Given a connected graph G = (V,E) with positive edge weights, define the dis~ce dc(~,v) between vertices u and v to be the length of a shortest path from u to v in G. A subgraph G’ of G is said to be a t-spanner for G if, for every pair of vertices u and v, dc?(u,v) < t“&(U,V). We show how to construct a (1 + &)-spanner for a complete Euclidean graph in O (n log n+~) time; this algorithm Ed works in any LP metric. We use this spanner to approximate minimum spanning trees in any fixed dimension and obtain a (1+&)-approximation in time O(n log n + ~ log ~ (~,n)). This algorithm is slightly faster than the analogius appmximaf.e minimum spanning tree result of Vaidya [13].

Offline algorithms for dynamic minimum spanning tree problems

Abstract: We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log2n) work per modification. No previous dynamic geometric MST algorithm was known.

Spanning trees with low crossing number

Abstract: Let P be a point set in the plane and T a spanning tree on P, whose edges are realized by segments. We define the crossing number of T as the maximum number of edges of T intersected by a single Une. We give a A(n) deterministic algorithm finding a spanning tree with crossing number O(fh) on a given n point set {this crossing number is asymptotically optimal), and a A(«) randomized (Las Vegas) algorithm finding a spanning tree with crossing number O(^Jnlogn) (hère f (n) = A (g (n)) means f(n) = O(g(n)\\o%n) for a constante). This improves results of Welzl and Edelsbrunner et al. We also consider the construction of a family of OQogri) spanning trees, such thatfor every Une X there is a tree in this family such that X crosses only O(fn.\\o%n) ofits edges. We obtain a A (H) Monte Carlo algorithm for this problem, improving a resuit of Edelsbrunner é tal . This resuit has numerous conséquences for the construction offurther randomized algorithms, using the above problems as a subroutine. Résumé. Soit P un ensemble de points du plan et soit T un arbre recouvrant de P, dont les arêtes sont des segments. Le nombre de croisements de T est le nombre maximal d'arêtes de T intersectées par une même droite. Si f et g sont deux fonctions, on pose f(n) = A(g(n)) s'il existe une constante c telle que ƒ (n) = O(g(n)\\off(ri)). Nous donnons un algorithme déterministe en A (AÏ') pour construire un arbre recouvrant dont le nombre de croisement est O(/n), où n est le nombre de points (ce nombre de croisement est asymptotiquement optimal); on donne également un algorithme probabiliste en A(«) pour construire un arbre recouvrant dont le nombre de croisement est O(fn\\ogn). Ceci améliore des résultats de Welzl, Edelsbrunner et al. On considère également la construction d'une famille de O(\\ogn) arbres recouvrants tels que, pour chaque droite X il existe un arbre de la famille tel que X intersecte seulement O(fiï log M) arêtes de l'arbre. On obtient un algorithme probabiliste en A (n) pour ce problème, ce qui améliore un résultat de Edelsbrunner et al. Ce résultat a de nombreuses conséquences pour la construction d'autres algorithmes probabilistes, qui utilisent alors les solutions des problèmes ci-dessus comme sous-programmes.