Showing papers on "Prim's algorithm published in 1998"

Greedy Algorithms for Minimum Spanning Tree

Abstract: The glossary de nes a spanning tree for a connected graph with non-negative weights on its edges, and one problem: nd a max weight spanning tree. Remarkably, the greedy algorithm results in a solution. Here we present similar greedy algorithms due to Prim [3] and Kruskal [2], respectively, for the problem: nd a min weight spanning tree. Graham and Hell [1] gives a history of the problem, which originated with the work of Czekanowski in 1909. The material here is based on Rosen [4].

Approximation Algorithms for Some Optimum Communication Spanning Tree Problems

Abstract: Let G = (V, E,w) be an undirected graph with nonnegative edge weight w, and r be a nonnegative vertex weight. The product-requirement optimum communication spanning tree (PROCT) problem is to find a spanning tree T minimizing Σi,j∈V r(i)r(j)d(T, i, j), where d(T, i, j) is the distance between i and j on T. The sum-requirement optimum communication spanning tree (SROCT) problem is to minimize Σi,j∈V (r(i) + r(j))d(T, i, j). Both the two problems are special cases of the general optimum communication spanning tree problem, and are generalizations of the shortest total path length spanning tree problem. In this paper, we present an O(n5) time 1.577-approximation algorithm for the PROCT problem, and an O(n3) time 2-approximation algorithm for the SROCT problem, where n is the number of vertices.

Relation-Algebraic Derivation of Spanning Tree Algorithms

Abstract: We use Tarski's relational algebra to derive a series of algorithms for computing spanning trees of undirected graphs, including a variant of Prim's minimum spanning tree algorithm.

An Algorithm for Finding All the Spanning Trees in Undirected Graphs

Abstract: In this paper, we propose an algorithm for nding all the spanning trees in undirected graphs. The algorithm requires O(n +m + n) time and O(n + m) space, where the given graph has n vertices, m edges and spanning trees. For outputting all the spanning trees explicitly, this algorithm is optimal.

Communication-optimal parallel minimum spanning tree algorithms (extended abstract)

Abstract: Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first non-trivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of the input graph, and m the number of edges of the input graph. We show that in the worst case a total of Ω(k · min(m,pn)) bits need to be transmitted in order to solve the MST problem, where k is the number of bits required to represent a single edge weight. This implies that if a message can contain at most O(k) bits, any BSP algorithm for finding an MST requires communication time Ω(g ·min(m/p, n)), where g is the gap parameter of the BSP model. In addition, we present two algorithms whose running times match the lower bounds in different situations. Both algorithms perform linear work and use a number of supersteps independent of the input size. The first algorithm is simple but can employ at most m/n processors efficiently. Hence, it should be applied in situations where the input graph is relatively dense. The second algorithm is a randomized algorithm that performs linear work with high probability, provided that m ≥ n · log p. This is the first linear work BSP algorithm for finding an MST in sparse graphs. Department of Computer Science, University of Toronto, Canada. Email: micah@cs.toronto.edu. Supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada, and by ITRC, an Ontario Centre of Excellence. This research was conducted in part while the author was at the Heinz Nixdorf Institute Graduate College, Paderborn, Germany. Bosch Telecom GmbH, Dept. UC-ON/ERS, Backnang, Germany. Email: Wolfgang.Dittrich2@pcm.bosch.de. This research was done while the author was working at the University of Paderborn, Germany. Heinz Nixdorf Institute, Dept. of Computer Science, University of Paderborn, Germany. Email: {benj,inri}@uni-paderborn.de. This research was partially supported by DFG-SFB 376 “Massive Parallelität” and EU ESPRIT Long Term Research Project 20244 (ALCOM-IT). Computer Science Institute, University of Wroc law, Poland. Email: mirekk@tcs.uni.wroc.pl.

A fault-tolerant token passing algorithm on tree networks

Abstract: We present a self stabilizing token passing algorithm for a tree network. The algorithm is based on the 4 state mutual exclusion algorithm of E.W. Dijkstra (1974) and works under the distributed daemon model of execution. Although our algorithm relies upon an underlying tree network topology, it is not less general than the protocol by S. Huang and N. Chen (1993) since a spanning tree of a network can be obtained by a number of self stabilizing algorithms (A. Arora and M. Gouda, 1993; N. Chen et al., 1991; S. Huang and N. Chen, 1992; S. Sur and P.K. Srimani, 1992). Token passing on a spanning tree thus places no restriction on the topology of the underlying distributed system.

Finding the k Most Vital Edges with Respect to Minimum Spanning Trees for k=2 and 3.

Abstract: Let G(V,E) be a weighted, undirected, connected simple graph with n vertices and m edges. The k most vital edge problem with respect to minimum spanning trees is to find a set S of k edges from E such that the removal of all edges in S results in the greatest increase in the weight of the minimum spanning tree in the remaining graph G(V,E−S). In this paper we present a better algorithm to solve this problem for k = 2 and 3. The proposed algorithms run in times O(nα(3n, n)) for k = 2 and O(nα(4n, n)) for k = 3, which improve previously known results by O(n/α(3n, n)) and O(n/α(4n, n)) factors, respectively, where α is a functional inverse of Ackermann’s function. The algorithms can also be implemented on a CREW PRAM in which case they need O(log n log log n) time using O(m+n/ log n) processors, and O(log n log logn) time using O(n/ log n) processors, respectively.

Efficient associative algorithm to find the least spanning tree of a graph with a node degree constraint

Abstract: We have described an associative algorithm to find the smallest spanning tree of a graph with a degree constraint on one of the nodes. The associative algorithm is based on Gabow's algorithm. It is described as a STAR procedure that runs in timeO(@#@ nlogn). This is also the time it takes to construct a minimum spanning tree of a graph on an associative parallel processor [4]. The associative algorithm relies on a number of new constructions that are also useful for other problems. Note that, unlike [13], we use a very simple and natural data representation on an associative parallel processor in the form of two-dimensional binary-coded tables.

Spanning tree problem on a fuzzy network

Abstract: Given a connected graph with edge costs, we seek a spanning tree with total cost as small as possible. In this paper, we introduce the existence possibilities of edges to the graph and define "the tree reliabilities" by them. Then, we consider bicriteria, i.e., about minimization of the weights (in the original spanning tree problem) and maximization of the reliabilities, and formulate an extended model of the spanning tree problem as a bicriteria programming problem on a fuzzy network.