Showing papers on "Spanning tree published in 1987"
TL;DR: In this paper, a new combinatorial formulation of the Jones polynomial of a link is used to establish some basic properties of this polynomials, and a striking consequence of these properties is the result that a link admitting an alternating diagram with m crossings and with no "nugatory" crossing cannot be projected with fewer than m crossings.
510 citations
01 Jan 1987
TL;DR: New linear time distributed algorithms for a class of problems in an asynchronous communication network, including Minimum-Weight Spanning Tree, Leader Election, and computing a sensitive decomposable function are developed.
Abstract: This paper develops linear time distributed algorithms for a class of problems in an asynchronous communication network Those problems include Minimum-Weight Spanning Tree (MST), Leader Election, counting the number of network nodes, and computing a sensitive decomposable function (eg majority, parity, maximum, OR, AND) The main problem considered is the problem of finding the MST This problem, which has been known for at least 9 years, is one of the most fundamental and the most studied problems in the field of distributed network algorithms Any algorithm for any one of the problems above requires at least O(E + VlogV) communication and O(V) time in the general network In this paper, we present new algorithms, which achieve those lower bounds The best previous algorithm requires T(E + VlogV) in communication and T(V log V) in time Our result enables to improve algorithms for many other problems in distributed computing, achieving lower bounds on their communication and time complexities
329 citations
TL;DR: This work uses the track graph, a suitably defined grid-like structure, to obtain efficient solutions for rectilinear shortest paths and minimum spanning tree (MST) problems for a set of points in the plane in the presence of rectilInear obstacles.
Abstract: We study the rectilinear shortest paths and minimum spanning tree (MST) problems for a set of points in the plane in the presence of rectilinear obstacles. We use the track graph, a suitably defined grid-like structure, to obtain efficient solutions for both problems. The track graph consists of rectilinear tracks defined by the obstacles and the points for which shortest paths and a minimum spanning tree are sought. We use a growth process like Dijkstra's on the track graph to find shortest paths from any point in the set to all other points (the one-to-all shortest paths problem). For the one-to-all shortest paths problem for n points we derive an O(n min {log n, log e} + (e + k) log t) time algorithm, where e is the total number of edges of all obstacles, t is the number of extreme edges of all obstacles, and k is the number of intersections among obstacle tracks (all bounds are for the worst case). The MST for the points is constructed also in time O(n log n + (e + k) log t) by a hybrid method of searching for shortest paths while simultaneously constructing an MST. An interesting application of the MST algorithm is the approximation of Steiner trees in graphs.
125 citations
TL;DR: Parallel algorithms for finding the connected components (CC) and a minimum spanning FOREST of an undirected graph are presented and the PRAM algorithm is a simplification of the one appearing in [17].
Abstract: Parallel algorithms for finding the connected components (CC) and a minimum spanning FOREST (MSF) of an undirected graph are presented. The primary model of computation considered is that called "shuffle-exchange network" in which each processor has its own local memory, no memory is shared, and communication among processors is done via a fixed degree network. This model is very convenient for actual realization. Both algorithms have depth of O(log2 n) while using n2 processors. Here n is the number of vertices in the graph. The algorithms are first presented for the PRAM (parallel RAM) model, which is not realizable, but much more convenient for the design and presentation of algorithms. The CC and MSF algorithms are no exceptions. The CC PRAM algorithm is a simplification of the one appearing in [17]. A modification of this algorithm yields a simple and efficient MSF algorithm. Both have depth of O(log m) and they use m processors, where m is the number of edges in the graph.
124 citations
TL;DR: It is proved that a large class of distributed tasks cannot be solved in the presence of faulty processors, and the notion of the decision graph of a task is introduced, and it is shown that every problem whose decision graph is disconnected cannot be unsolved by reducing the unsolvability of this problem to the unsoliability of the consensus problem.
100 citations
TL;DR: This work considers the special case of the problem in which all costs are zero or one for arborescences and shows that a ‘continuity’ property is prossessed similar to that possessed by matroids, which enables it to determine in polynomial time the complete set of values of k for which a solution exists.
75 citations
01 Aug 1987-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: This paper proposes that a particular geometric object called the minimal spanning Voronoi tree captures the essence of the problem of finding a simple polygon through a given set of points in the plane that is “natural” in some perceptual sense.
Abstract: The problem considered in this paper is that of finding a simple polygon through a given set of points in the plane that is “natural” in some perceptual sense. We propose that a particular geometric object called the minimal spanning Voronoi tree captures the essence of the problem. Despite the fact that we can neither prove the existence of this geometric object nor design an exact algorithm for finding it, a search heuristic results in remarkably pleasing solutions to the problem.
68 citations
TL;DR: A new approach to study order and disorder in biological membranes and more generally in biological structures is developed based on a graph constructed on the set points representing the position of particles, called the minimal spanning tree.
61 citations
TL;DR: In this paper, it was proved that n − 1 V k,n converges with probability 1 to a constant α k,d, where k is the number of vertices of degree k in the Euclidean minimal spanning tree of X i,,, where the X i are independent, continuous random variables with values in R d.
Abstract: Let V k,n be the number of vertices of degree k in the Euclidean minimal spanning tree of X i , , where the X i are independent, absolutely continuous random variables with values in R d . It is proved that n –1 V k,n converges with probability 1 to a constant α k,d . Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of V k,n .
58 citations
TL;DR: The length of the minimal spanning tree on the complete graph on n vertices with edge weights determined by independent non-negative random variables with distribution F is proved to converge in probability to χ(3)/F′(0) , provided only that F have a non-zero derivative at the origin.
46 citations
TL;DR: An O(n2) algorithm for finding the point on the plane which, if added to a given set of n points, will result in the shortest possible spanning tree.
TL;DR: This work shows that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases, and shows that the construction of a minimum weight spanning tree requires, in the worst case, at least $\Omega (n^2 )$ messages.
Abstract: In a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in $O(n\log n)$ messages. We show in this work that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: First we show that the construction of a minimum weight spanning tree requires, in the worst case, at least $\Omega (n^2 )$ messages, and then we show that the construction of a spanning tree where the maximum degree is at most k may require at least $\Omega ({{n^2 } / k})$ messages in the worst case. Actually, in both cases the lower bounds are shown for the number of edges used in the worst case. Moreover, the results are valid for both asynchronous and synchronous networks, and are independent of the lengths of the messages. On the other hand, there are algorithms for the above tasks which achieve these lower bounds, up to a constant factor, and use messages of $O(\log n)$ length.
Proceedings Article•
01 Jan 1987
TL;DR: This paper analyzes the emulation of two-dimensional meshes, butterfly networks, and spanning trees on meshes, Boolean cubes, and Cube Connected Cycles and presents novel layouts for hypercubes and CCCs that offer better performance for some problems, while essentially maintaining the performance for other problems.
Abstract: In this paper we analyze the emulation of two-dimensional meshes, butterfly networks, and spanning trees on meshes, Boolean cubes, and Cube Connected Cycles (CCC) networks. We consider three timing models for signal propagation dong a wire: constant delay, capacitive delay, and resistive delay. We ais0 present novel layouts for hypercubes and CCCs that offer better performance for some problems, while essentially maintainingthe performance for other problems. The mesh interconnection performs better on all emulations for all delay models,if the communication throughput determines the performance. With resistive delay model, meshes also offer the best latency for all emulations. The hypercube and CCC layouts yield lower latency for emulating butterlly networks and spanning trees for the constant delay and capacitive delay models.
TL;DR: This paper describes the first distributed algorithm for constructing minimal spanning trees and the principles and techniques underlying its design will find application in large communication networks and large multiprocessor computer systems.
Abstract: Most algorithms for constructing minimal spanning trees are sequential in operation. Distributed algorithms for constructing these trees operate both concurrently and asynchronously, and are useful in store-and-forward packet-switching computer-communication networks where there is typically no single source of control. The difficulties in designing such algorithms arise from communication and synchronization problems. This paper discusses these problems and describes the first distributed algorithm for constructing minimal spanning trees. This algorithm and the principles and techniques underlying its design will find application in large communication networks and large multiprocessor computer systems.
TL;DR: A linear time algorithm for the Steiner problem in Halin networks is presented, providing another example where the recursive structure of the underlying network leads to an efficient algorithm.
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TL;DR: In this article, the authors define the number of spanning trees of a connected graph and the resistance of a spanning tree to a given edge if each edge of the tree is a one ohm resistor.
Abstract: Let s and t be distinct vertices of a connected graph G. The notation and definitions used here follow the excellent text of Bollabas [1]. Next let N be the number of spanning trees of G while F(s, t) = F is the number of spanning forests with two components, one containing s and one containing t. Call such a forest a thicket. The lemma in question states that Rs,t = F/N where Rt is the resistance of G between s and t if each edge of G represents a one ohm resistor. Since the right-hand side is a strictly graphical quantity this is a fascinating result which deserves to be better known. By way of illustration, look at the smallest example not immediately solved by the series and parallel laws.
TL;DR: An efficient algorithm for generating all the maximum spanning trees of a weighted graph and a polynomial time algorithm for counting them are presented.
TL;DR: A hierarchical graph model is defined that allows the exploitation of the hierarchy for the more efficient solution of graph problems on very large graphs and it is shown how to efficiently find minimum spanning forests in this graph model.
TL;DR: In this paper, the authors derived the following simple formula for t(Pn ) the number of spanning trees in the prism Pn, defined as the graph obtained by adding to the disjoint cycles all edges of the form ViWi The prism is sometimes denoted by K 2×Cn.
Abstract: Let the vertices of two disjoint, equal length cycles be labelled in one cycle and in the other. The prism Pn is defined as the graph obtained by adding to the disjoint cycles all edges of the form ViWi The prism is sometimes denoted by K 2×Cn . In this work we derive the following simple formula for t(Pn ) the number of spanning trees in .
Proceedings Article•
01 Jan 1987
TL;DR: It is proved that there exists a bijectionψ from A/P into P/A such that for any edgea∈A/P, (P/ψ(a)) ∪a is a spanning tree ofG whose weight is greater than or equal to that ofP.
Abstract: LetA be a maximum spanning tree andP be an arbitrary spanning tree of a connected weighted graphG. Then we prove that there exists a bijectionψ fromA/P intoP/A such that for any edgea∈A/P, (P/ψ(a)) ∪a is a spanning tree ofG whose weight is greater than or equal to that ofP. We apply this theorem to some problems concerning spanning trees of a weighted graph.
TL;DR: The paper describes two approaches to limit the communication requirements for solving the problems, a divide-and-conquer strategy applied to Sollin's algorithm for finding the minimum spanning tree of a graph and a novel data-reduction technique that constructs an auxiliary graph with no more than 2n − 2 edges.
Abstract: Parallel algorithms for computing the minimum spanning tree of a weighted undirected graph, and the bridges and articulation points of an undirected graphs on a fixed-size linear array of processors are presented. For a graph of n vertices, the algorithms operate on a linear array of p processors and require O(n2/p) time for all p, 1 ≤ p ≤ n. In particular, using n processors the algorithms require O(n) time which is optimal on this model. The paper describes two approaches to limit the communication requirements for solving the problems. The first is a divide-and-conquer strategy applied to Sollin's algorithm for finding the minimum spanning tree of a graph. The second uses a novel data-reduction technique that constructs an auxiliary graph with no more than 2n − 2 edges, whose bridges and articulation points are the bridges and articulation points of the original graph.
TL;DR: This paper presents a linear-time algorithm to embed an outerplanar graphG into a spanning tree with cost at most maxdegree(G) + 1, and shows that this problem isNP-complete even whenG is planar; it is easily solved when G is a tree; and there is a simple characterization for all graphs with cost 2 or less.
Abstract: The Min Cut Linear Arrangement problem asks, for a given graphG and a positive integerk, if there exists a linear arrangement ofG's vertices so that any line separating consecutive vertices in the layout cuts at mostk of the edges. A variation of this problem insists that the arrangement be made on a (fixed-degree) tree instead of a line. We show that (1) this problem isNP-complete even whenG is planar; (2) it is easily solved whenG is a tree; and (3) there is a simple characterization for all graphs with cost 2 or less. Our main result is a linear-time algorithm to embed an outerplanar graphG into a spanning tree with cost at most maxdegree(G) + 1. This result is important because it extends to an approximation algorithm for the standard Min Cut Linear Arrangement Problem on outerplanar graphs.
TL;DR: The chapter discusses various other parallel graph algorithms for shortest path, maximum matching, planarity testing, and maximal independent set and describes parallel algorithms for various nongraph-theoretic problems like arithmetic expression and polynomial evaluation, string matching, tree balancing, and alpha-beta search.
Abstract: Publisher Summary The chapter presents a survey of parallel algorithms for finding the connected and biconnected components of a graph The chapter classifies the various parallel algorithms for finding the connected components of undirected graphs according to two major criteria: the basic technique employed and the format of the input The basic techniques used in these algorithms are breadth-first search, transitive closure, and vertex collapse The most common form of input is adjacency matrix The chapter presents several parallel minimum spanning tree algorithms for different types of parallel computational models A minimum spanning tree of a weighted, connected, and undirected graph is defined as a set of edges of the graph that connects all vertices and whose total edge weight is minimum The chapter discusses various other parallel graph algorithms for shortest path, maximum matching, planarity testing, and maximal independent set It describes parallel algorithms for various nongraph-theoretic problems like arithmetic expression and polynomial evaluation, string matching, tree balancing, and alpha-beta search
19 Feb 1987
TL;DR: This paper investigates the problem of making existing spanning tree algorithms fault-resilient, and still overcome these difficulties, and introduces amortized message complexity as a tool for analyzing the message complexity.
Abstract: We study distributed algorithms for networks with undetectable fail-stop failures, assuming that all of them had occurred before the execution started (It was proved that distributed agreement cannot be reached when a node may fail during execution) Failures of this type are encountered, for example, during a recovery from a crash in the network We study the problems of leader election and spanning tree construction, that have been characterized as fundamental for this environment We point out that in presence of faults just duplicating messages in an existing algorithm does not suffice to make it resilient; actually, this redundancy gives rise to synchronization problems and also might increase the message complexity In this paper we investigate the problem of making existing spanning tree algorithms fault-resilient, and still overcome these difficulties Several lower bounds and optimal fault-resilient algorithms are presented for the first timeHowever, we believe that the main contribution of the paper is twofold: First, in designing the algorithms we use tools that thus argued to be rather general (for example, we extend the notion of token algorithms to multiple-token algorithms) In fact we are able to use them on several different algorithms, for several different families of networks Second, following the amortized computational complexity, we introduce amortized message complexity as a tool for analyzing the message complexity
Proceedings Article•
01 Jan 1987
TL;DR: In this article, the problem of maximizing a linear function of n binary variables subject to order constraints is considered. But the problem is equivalent to finding a maximum weight closure of a directed graph.
Abstract: Publisher Summary This chapter presents old and new results concerning two important preorders: reflexive and transitive relations in the set of variables of a 0–1 linear or nonlinear programming problem. It discusses the generation and the use of order constraints xi ≤ xi in linear and nonlinear 0–1 programming. The chapter discusses the problem of maximizing a linear function of n binary variables subject to order constraints. This problem is equivalent to finding a maximum weight closure of a directed graph. A direct reduction of the maximum weight closure problem is exhibited to a minimum cut one to provide a linear-time algorithm for the special case when the graph is a rooted tree. The chapter introduces a class of directed spanning trees of a graph—the palm trees—that generalize depth-first-search trees and shows that there is a natural one-to-one correspondence between palm trees and minimum cardinality sets of order constraints linearizing f. The chapter provides two different characterizations of palm trees and extends these results to arbitrary pseudo-boolean functions.
TL;DR: The paper presents a review of the IEEE 802 recommendations for MAC layer bridges and a full technical description of the Salford/Daresbury bridge protocol.
Abstract: This paper details the design of a MAC layer bridge, developed jointly by the University of Salford and S.E.R.C. Daresbury Laboratory which achieves the interconnection of IEEE 802 LANs and supports a bridge protocol allowing bridges to negotiate a loop free or spanning tree topology from an arbitrary extended LAN environment. The paper presents a review of the IEEE 802 recommendations for MAC layer bridges and a full technical description of the Salford/Daresbury bridge protocol.
TL;DR: In this article, the number of spanning trees on a large lattice is evaluated exactly for the 3D simple cubic lattice graph, and similar similarities to the evaluation of the lattice Green function are pointed out.
Abstract: The number of spanning trees on a large lattice is evaluated exactly for the 3D simple cubic lattice graph. Similarities to the evaluation of the lattice Green function are pointed out.