Showing papers on "Spanning tree published in 1980"
TL;DR: The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree and the Delaunay (Voronoi) triangulation.
1,213 citations
28 Apr 1980
TL;DR: This paper attempts to get at some of the fundamental properties of distributed computing by means of the following question: “How much does each processor in a network of processors need to know about its own identity, the identities of other processors, and the underlying connection network in order for the network to be able to carry out useful functions?
Abstract: This paper attempts to get at some of the fundamental properties of distributed computing by means of the following question: “How much does each processor in a network of processors need to know about its own identity, the identities of other processors, and the underlying connection network in order for the network to be able to carry out useful functions?” The approach we take is to require that the processors be designed without any knowledge (or only very broad knowledge) of the networks they are to be used in, and furthermore, that all processors with the same number of communication ports be identical. Given a particular network function, e.g., setting up a spanning tree, we ask whether processors may be designed so that when they are embedded in any connected network and started in some initial configuration, they are guaranteed to accomplish the desired function.
621 citations
TL;DR: Algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees can be implemented to solve practical problems very efficiently.
Abstract: : Geometric closest-point problems deal with the proximity relationships in k-dimensional point sets. Examples of closest-point problems include building minimum spanning trees, nearest neighbor searching, and triangulation construction. Shamos and Hoey (1975) have shown how the Voronoi diagram can be used to solve a number of planar closest-point problems in optimal worst-case time. In this paper we extend their work by giving optimal expected-time algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees. In addition to establishing theoretical bounds, the algorithms in this paper can be implemented to solve practical problems very efficiently.
332 citations
TL;DR: A primal and a dual heuristic procedure and a branch-and-bound algorithm are proposed to construct a DCMST, which is formulated as a linear 0–1 integer programming problem.
210 citations
TL;DR: A new parallel algorithm is studied that constructs an MST of an N-node graph in time proportional to N lg N, on an N(lg N)-processor computing system.
73 citations
TL;DR: In this paper the computational complexity of spanning tree problems is studied with the aim of identifying a borderline between easy and hard problems.
67 citations
TL;DR: In this article, the authors presented an 0 (n log n) heuristic algorithm for the Rectilinear Steiner Minimal Tree (RSMT) problem, which is based on a decomposition approach which first partitions the vertex...
Abstract: This paper presents an 0 (n log n) heuristic algorithm for the Rectilinear Steiner Minimal Tree (RSMT) problem. The algorithm is based on a decomposition approach which first partitions the vertex ...
42 citations
TL;DR: The question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles is answered in the negative by exhibiting graphs on n =2 m +1, m ≥5, points in which one point hasdegree 4, all others have degree 3, and only two spanning cycles exist.
33 citations
01 Jan 1980
TL;DR: The by now classical Held and Karp procedure for the travelling salesman problem (TSP) and the “3”/2-heuristic of Christofides for the Euclidian TSP are both based on the existence of good algorithms for the minimum spanning tree problem.
Abstract: The by now classical Held and Karp procedure for the travelling salesman problem (TSP) and the “3”/2-heuristic of Christofides for the Euclidian TSP are both based on the existence of good algorithms for the minimum spanning tree problem.
18 citations
TL;DR: In this paper, a new determinant formula for the number of spanning arborescences of a digraph is presented, which generalizes the determinant formulas given by Maurer (SIAM J. Appl. Math., 1976) for the spanning trees of an undirected graph.
Abstract: A new proof of Tutte’s trinity theorem (Proc. Cambridge Phil. Soc., 1948), (North-Holland, 1973) is presented. The proof is based on a new determinant formula for the number of spanning arborescences of a digraph. This formula generalizes the determinant formula given by Maurer (SIAM J. Appl. Math., 1976) for the number of spanning trees of an undirected graph.
16 citations
01 Jan 1980
TL;DR: An algorithm which generates the appropriate acyclic spanning subgraphs and is compared to two more standard algorithms based on sample space partitioning and a unified approach to the application of this decomposition which is quite general and easy to extend are presented.
Abstract: We consider the problem of determining the probability that a network will be able to fulfill its purpose at some instant in time given the joint probability of the states of the elements of the network for that instant. In particular, we consider computing the probabilities that a pair of terminals can communicate, all vertices can communicate, a root vertex can communicate with all others, a certain level of demands can be met in a transportation network, and the shortest path is less than a certain length. Two approaches based on combinatorial analysis of graphs are examined here.
Using the inclusion-exclusion principle on minimal success events is a standard technique which is considered to be cumbersome because of the large number of terms that must be computed. For the rooted communication problem in a directed graph, we show that by combining like terms, each term in the final expression is either +1 or -1 times the probability that all the edges in an acyclic spanning subgraph are working. This results in a significant reduction in the number of terms. Furthermore, it is easy to determine whether the coefficient is +1 or -1. To establish these results, we prove two strictly combinatorial properties of directed graphs associated with their directed spanning trees. We describe an algorithm which generates the appropriate acyclic spanning subgraphs and compare it to two more standard algorithms based on sample space partitioning.
The second approach discussed is that of using the decomposition tree of a network to divide the reliability problem into smaller subproblems. The decomposition tree represents a biconnected graph in terms of its triconnected components. Using the triconnected components has been suggested for many reliability problems and includes the special cases of parallel- and series-reductions. By using the tree and the concept of a return edge for each component, we present a unified approach to the application of this decomposition which is quite general and easy to extend. We discuss when decomposition should be desirable and consider its application to reliability problems which are not derived from graphs.
TL;DR: In this paper, it is shown that planar maps and their dual have equal complexity and that the same holds for hypermaps, and the definition of a spanning hypertree is also given.
Abstract: The number of spanning trees in a graph is often called it's complexity [1]. A tree is of course of complexity one and it is a classical result of Cayley that the complete graph Kn has complexity nn-2. Between these two numbers lies the complexity of a connected graph. In the case of planar maps it is well-known that a map and its dual have equal complexity (see for instance [2]). in this communication we show that a similar result holds for hypermaps. To prove this result we use a diagram containing six hypermaps which is very related to W.T. Tutte's Trinity [9]. All the needed definitions are recalled. We thus give a few details about hypermaps, their underlying hypergraphs and the maps associated to them. The definition of a spanning hypertree is also given.
TL;DR: This paper generalizes a Dirac type sufficient condition ensuring the existence of a Hamiltonian cycle to one ensuring theexistence of a closed spanning walk of length less than a specified value and presents an O (p2 log p) algorithm for finding such aclosed spanning walk in a graph with p vertices satisfying the authors' condition.
Abstract: A Hamiltonian walk of a graph is a closed spanning walk of minimum length. In this paper we generalize a Dirac type sufficient condition ensuring the existence of a Hamiltonian cycle to one ensuring the existence of a closed spanning walk of length less than a specified value. Furthermore, we present an O (p2 log p) algorithm for finding such a closed spanning walk in a graph with p vertices satisfying our condition.
TL;DR: Based on the suggested criteria formulated as a tradeoff between the satisfaction of user and company holders, a comparative evaluation is tabulated, which can be used as a qualitative guideline while forming a “reliable” as well as near optimal cost-allocation strategy depending upon the locally prevailing conditions also.
TL;DR: An approach to code motion and hoisting, a program optimization technique, is discussed and a linear algorithm is developed that provides sufficient but not necessary conditions for hoisting.
TL;DR: In this paper, the problem of finding the linear graph with the maximum number of spanning trees was studied, where only the number of nodes N and the number B are prescribed, and the problem was solved for D less than or equal to N. The problem is solved for connected graphs G (N, B ) obtained by deleting D branches from a complete graph K N.
Abstract: The problem is to determine the linear graph that has the maximum number of spanning trees, where only the number of nodes N and the number of branches B are prescribed. We deal with connected graphs G ( N , B ) obtained by deleting D branches from a complete graph K N . Our solution is for D less than or equal to N
TL;DR: It is shown that the complete bipartite graph Km,n, for any pair m, n, and all subgraphs of K2,n for any n, are uniquely reconstructable from their spanning trees.