Showing papers on "Spanning tree published in 1976"

Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem

Abstract: : An O(n sup 3) heuristic algorithm is described for solving n-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition. The algorithm involves as substeps the computation of a shortest spanning tree of the graph G defining the TSP, and the finding of a minimum cost perfect matching of a certain induced subgraph of G. A worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3/2. This represents a 50% reduction over the value 2 which was the previously best known such ratio for the performance of other polynomial-growth algorithms for the TSP.

Finding Minimum Spanning Trees

Abstract: This paper studies methods for finding minimum spanning trees in graphs. Results include 1. several algorithms with $O(m\log \log n)$ worst-case running times, where n is the number vertices and m is the number of edges in the problem graph; 2. an $O(m)$ worst-case algorithm for dense graphs (those for which m is $\Omega (n^{1 + \varepsilon } )$ for some positive constant $\varepsilon $); 3. an $O(n)$ worst-case algorithm for planar graphs; 4. relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.

On cost allocation for a spanning tree: A game theoretic approach

Abstract: Cooperative game theory solution concepts are used to allocate costs in a spanning tree network. Stable cost allocations are related to the core of a cooperative game and it is proved that every game generated from a minimum cost spanning tree with an immovable source has a core. A refinement of the core, called the irreducible core, is introduced and the extreme points of the solution can be characterized by permutations of the minimal cost spanning tree. Points in the irreducible core are shown to be stable under unions of additional players. A weighted Shapley value is used to obtain a unique allocation of costs. This value coincides with the marginal costs of the spanning tree when there is only one minimal spanning tree. When multiple sources are allowed, counterexamples to the existence of a core are presented unless extra taxes are levied on the users.

An upper bound for the probability of a union

Abstract: The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

Edge-disjoint spanning trees and depth-first search

Abstract: This paper presents an algorithm for finding two edge-disjoint spanning trees rooted at a fixed vertex of a directed graph. The algorithm uses depthfirst search and an efficient method for computing disjoint set unions. It requires O (e?(e, n)) time and O(e) space to analyze a graph with n vertices and e edges, where ? (e, n) is a very slowly growing function related to a functional inverse of Ackermann's function.

The Optimal Partitioning of Graphs

Abstract: The problem considered in this paper is that of partitioning a link-weighted graph G into two parts, each of which is constrained in size by the (given) maximum number of vertices that the part can contain. This is a special case of the general partitioning problem of a graph into k parts with size constraints, which appears in a number of very diverse problem areas.The paper presents a tree search method for solving the “partition in two” problem, with the objective of minimizing the sum of the costs of the cut links. The method employs a maximum flow subalgorithm for establishing bounds during the search, and uses the concept of the longest spanning tree of a graph to direct the forward branching steps. The size of the search is reduced further by a test (based on the solution of a special “knapsack” type problem), which is designed to detect potential size-infeasibility early in the search.Computational results are given for approximately 300 graphs of various sizes (up to 40 vertices), and for various...

A distributed algorithm for constructing minimal spanning trees in computer-communication networks

Abstract: This paper presents a distributed algorithm for constructing minimal spanning trees in computer-communication networks. The algorithm can be executed concurrently and asynchronously by the different computers of the network. This algorithm is also suitable for constructing minimal spanning trees using a multiprocessor computer system. There are many reasons for constructing minimal spanning trees in computer-communication networks since minimal spanning tree routing is useful in distributed operating systems for performing broadcast, in adaptive routing algorithms for transmitting delay estimates, and in other networks like the Packet Radio Network.

Topological conditions for the solvability of linear active networks

Abstract: The solvability problem of a linear active network is approached from a purely topological point of view using the two-graph method. It can be said that a topological condition for the solvability is the existence of a common tree of the voltage and current graphs. A few conditions for the existence of a common tree are derived. If there exists no common tree, subgraphs which cause the nonexistence can be distinguished, and a partition of two-graphs can be introduced. The partition has similar properties to the principal partition of a graph or the canonical form of a bipartite graph, and a structure of two-graphs represented by a partial ordering of sets of edges can be defined. An algorithm to find the partition and a common tree, if one exists, or if no common tree exists, a tree of one of the graphs which has as many common edges as possible with a tree of the other graph, is given. The decomposition of the coefficient matrix accompanying the structure is discussed, and algorithms to determine the decomposition is given.

Another algorithm for reducing bandwidth and profile of a sparse matrix

Abstract: The paper describes a new bandwidth reduction method for sparse matrices which promises to be both fast and effective in comparison with known methods. The algorithm operates on the undirected graph corresponding to the incidence matrix induced by the original sparse matrix, and separates into three distinct phases: (1) determination of a spanning tree of maximum length, (2) modification of the spanning tree into a free level structure of small width, (3) level-by-level numbering of the level structure. The final numbering produced corresponds to a renumbering of the rows and columns of a sparse matrix so as to concentrate non-zero elements of the matrix in a band about the main diagonal.

A Loss Function Minimization Strategy for Grouping from Dendrograms

Abstract: Lefkovitch, L. P. (Agricult. Canada, Statistical Research Service, Room E-266, Sir John Carling Bldg., Ottawa, Ontario KIA 0C5, Canada) 1976. A loss ftunction minimization strategy for grouping from dendrograms. Syst. Zool. 25:41-48.-This paper gives a method for obtaining subgroups of objects related by a dendrogram. After showing the relationship between spanning trees and dendrograms, the proposal is made to use the former to obtain a subset of all possible partitions. To decide which of the subsets is best, it is necessary to use a measure of clustering; the desirable attributes of such a measure lead to one which is based on a comparison between an object in its original disposition and on its conjectured group membership. The minimization of this measure, called a loss function, is proposed as the criterion for choosing the "best" groups. A small example illustrates the method. Many clustering methods, especially the disjoint hierarchical clustering procedures, do not cluster but produce dendrograms and make no decision on the level (the phenon line; Sneath and Sokal, 1973) at which to cut the dendrograms so as to form groups. The purpose of this paper is to indicate how clusters can be obtained from dendrograms, and from related structures, in some "best" sense; and if there are several locally "best" solutions, how these may be used to choose phenon lines and give rise to candidates for taxa of different rank. It is convenient to classify clustering methods somewhat differently from that adopted by Sneath and Sokal (1973), distinguished by the type of cluster (disjoint versus overlapping) and also by the relationship of the clusters to each other (hierarchical versus non-hierarchical) (Table 1). In this paper, the class of clustering procedures which are hierarchical, and which can be considered in the framework of the Lance and Williams (1967) general formulation are considered: let dcj be the distance between clusters i and j; let (ij) be the cluster formed by fusing clusters i and j; ai, aj, A and y are parameters. Then d ($j= k adik + a1djk + fd11 + y I dk -djk I The general algorithm for all members of this class of clustering procedures is Step (i) Find the pair of clusters, i and j which in some sense best satisfy the clustering criterion (e.g., clusters i and j are closest). Step (ii) Fuse the two clusters; let i represent the fused cluster; let the jth value of a list, c, take the value i, and also let the jth value of another list, t, take the value of dxj, the distance between clusters i and j. Step (iii) Compute the values required (e.g. d(J) k distances) between the new cluster and the remaining ones according to the appropriate rule. Step (iv) If all objects are not in one cluster, return to Step (i). The lists of pointers and lengths, namely the vectors c and t, describe a spanning tree of minimum length in the semi-metric space (see Appendix I) of Step (iii). Many, but not all, spanning trees can be formed into dendrograms representable on a plane without crossed lines; those clustering procedures which may give rise to non-monotonically increasing minimum distances (e.g. the weighted and unweighted centroid methods) cannot always be formed into such dendrograms without some compromise. All dendrograms, however, are representable as spanning trees, so that dividing

General solution to the spanning tree enumeration problem in arbitrary multigraph joins

Abstract: The number of spanning trees in an arbitrary graph or multigraph is obtained via a general formula involving eigenvalues of an associated matrix, This is shown to be particularly useful in the case of graphs (or multigraphs) which are joins, and a method for deriving the appropriate eigenvaiues of joins is given. As applications of this, concise general expressions are derived for the number of spanning trees in any wheel or top. For the ordinary n spoke wheel W_{n+1} = K_{1} + C_{n} the simple formula {\Pi|_{r=1}^{n-1} (3-2 \cos (2r{\pi}/n)) is derived. A more general concept of join of multigraphs is introduced, and this is applied to obtain a simple formula in terms of integers and cosines for the number of spanning trees in general multigraph wheels.

On maximally distant spanning trees of a graph

Abstract: A set ofk spanning trees of a graphG is calledmaximally distant if their union contains the maximum number of edges ofG. We present a necessary and sufficient condition for a set of spanning trees to be maximally distant. We also give an efficient algorithm which actually findsk maximally distant spanning trees in a given graph.

A combinatorial ranking problem

Abstract: An algorithm is presented for finding annth-best spanning tree of an edge-weighted graphG. In sharp contrast to related ranking algorithms, the number of steps is a linear function of the parametern. The results apply as well to ranking the bases of an abstract matroid.

Decomposition of a Graph Realizing a Degree Sequence into Disjoint Spanning Trees

Abstract: Let $(d_1 ,d_2 , \cdots ,d_n )$ be a sequence which is realizable by a simple graph, each $d_i \geqq k$, and such that the total number of edges (half the sum of the degrees) is at least $k(n - 1)$. Then there exists a graph having degree sequence $\{ {d_i } \}$ that contains k disjoint spanning trees. This result is proved and an efficient construction algorithm is presented.

Remark on “Algorithm 479: A Minimal Spanning Tree Clustering Method [Z]”

Abstract: The algorithm as given generally yields a large number of clusters containing only one point. These are not likely to be of much use. Clusters not containing at least MINPTS points can be eliminated by making the following changes to the subroutine CLUTR.