Showing papers on "Spanning tree published in 1972"

On Solving Constrained Transportation Problems

Abstract: : The paper presents a computationally efficient method for solving transportation problems with several additional constraints. The method is basically the primal simplex method specialized to fully exploit the topological structure embedded in the problem. It couples the Poly-omega technique of Charnes and Cooper with the Row-Column Sum Method to yield an 'inverse compactification' which minimizes the basis information that has to be stored between successive iterations, and in addition minimizes the arithmetic calculations required in pivoting. In particular the solution procedure only requires the storage of a spanning tree and a (q + 1) x q matrix (where q is the number of additional constraints) for each basis. The steps of updating costs and finding representations reduce to a sequence of simpler operations which fully utilize the triangularity of the spanning tree. Procedures for obtaining basic primal 'feasible' starts are also presented. (Author)

Algorithm 422: minimal spanning tree [H]

Abstract: This algorithm generates a spanning tree of minimal total edge length for an undirected graph specified by an array of inter-node edge lengths using a technique suggested by Dijkstra [1]. Execution time is proportional to the square of the number of nodes of the graph; a minimal spanning tree for a graph of 50 nodes is generated in 0.1 seconds on an IBM System 360/67. Previous algorithms [2, 3, 4, 5] require an amount of computation which depends on the graph topology and edge lengths and are best suited to graphs with few edges.

Cellular arrays for the solution of graph problems

Abstract: A cellular array is a two-dimensional, checkerboard type interconnection of identical modules (or cells), where each cell contains a few bits of memory and a small amount of combinational logic, and communicates mainly with its immediate neighbors in the array. The chief computational advantage offered by cellular arrays is the improvement in speed achieved by virtue of the possibilities for parallel processing. In this paper it is shown that cellular arrays are inherently well suited for the solution of many graph problems. For example, the adjacency matrix of a graph is easily mapped onto an array; each matrix element is stored in one cell of the array, and typical row and column operations are readily implemented by simple cell logic. A major challenge in the effective use of cellular arrays for the solution of graph problems is the determination of algorithms that exploit the possibilities for parallelism, especially for problems whose solutions appear to be inherently serial. In particular, several parallelized algorithms are presented for the solution of certain spanning tree, distance, and path problems, with direct applications to wire routing, PERT chart analysis, and the analysis of many types of networks. These algorithms exhibit a computation time that in many cases grows at a rate not exceeding log2n, where n is the number of nodes in the graph. Straightforward cellular implementations of the well-known serial algorithms for these problems require about n steps, and noncellular implementations require from n2 to n3 steps.

Number of Spanning Trees in Multigraph Wheels

Abstract: Taking the graph called the wheel W_{n} as the join of K_{1} and C_{n} , we consider it along with three multigraph variations. A recursion formula for the number of spanning trees in W_{n}, T_{n} = 3T_{n-1}-T_{n-2} + 2 , and other pertinent formulas are presented. A related resistance network problem is also discussed briefly.

Improved algorithm for the construction of minimal spanning trees

Abstract: An improved algorithm for the construction of minimal spanning trees is proposed which is particularly suitable for use with data sets containing repeated elements. The algorithm may be applied to cluster-detection and pattern-segmentation problems, such as arise in automatic speech recognition.

Generalized m series in tree enumeration

Abstract: Consideration of a particular series formed by deleting certain branches of the complete graph K sub t with t vertices A class of graphs which contains the m series as a special case is considered, and a new formula is given for the m series itself (an m series graph is obtained from K sub t by removing m branches forming a closed loop) A formula is derived for the number of labeled spanning trees in the graph obtained by deleting certain branches from K sub t

Number of Time of Vertices (in Example Vertices in Set P seconds) SOA MST "M"s"T/

Abstract: In Table II, examples of larger graphs were solved using the sub- optimal algorithm. The solutions are compared with the correspond- ing minimum spanning trees (MST) whose total weights are given in the column MST. The last column indicates the improvement of SOA over MST. Gilbert and Pollack (l ) conjectured that the ratio of the total weight of edges of an optimal solution to that of a minimum spanning tree, for Steiner's problem with Euclidian distance, has a lower bound of 0.866. However, the result in Table II indicates even lower ratios are achieved by suboptimal solutions in the case of recti- linear distance. The computer time required by this algorithm is much less than that by the optimal algorithm of Dreyfus and Wagner. For example, according to their estimation, example 4 would require 6.5 min using their algorithm. Hence, the suboptimal algo- rithm appears to be a good compromise between computer time and quality of solution.