Showing papers on "Spanning tree published in 1979"

A Distributed Algorithm for Minimum Weight Spanning Trees. Revision

Abstract: : A distributed algorithm is presented that constructs the minimum weight spanning tree in a connected undirected graph with distinct edge weights. A processor exists at each node of the graph, knowing initially only the weights of the adjacent edges. The processors obey the same algorithm and exchange messages with neighbors until the tree is constructed. The total number of messages required for a graph of N nodes and E edges is at most 5N log of N to the base 2 + 2E and a message contains at most one edge weight plus log of 8N to the base 2 bits. The algorithm can be initiated spontaneously at any node or at any subset of nodes.

Combinatorial Optimization with Rational Objective Functions

Abstract: Let A be the problem of minimizing c1, x1, +... + cnxn subject to certain constraints on x = x1,..., xn, and let B be the problem of minimizing a0 + a1x1 +... + anxn/b0 + b1x1 +... + bnxn subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[pn] comparisons and O[qn] additions, then B is solvable in time O[pnqn + pn]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio simple paths, maximum ratio weighted matchings, etc., can be computed withing polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O|E| · |V|2 · log|V| for a minimum ratio cycle and O|E| · log2|V| · log log |V| for a minimum ratio spanning tree are developed.

Applications of Path Compression on Balanced Trees

Abstract: We devise a method for computing functions defined on paths in trees. The method is based on tree manipulation techniques first used for efficiently representing equivalence relations. It has an almost-linear running time. We apply the method to give O(m $\alpha$(m,n)) algorithms for two problems. A. Verifying a minimum spanning tree in an undirected graph (best previous bound: O(m log log n) ). B. Finding dominators in a directed graph (best previous bound: O(n log n + m) ). Here n is the number of vertices and m the number of edges in the problem graph, and $\alpha$(m,n) is a very slowly growing function which is related to a functional inverse of Ackermann''s function. The method is also useful for solving, in O(m $\alpha$(m,n)) time, certain kinds of pathfinding problems on reducible graphs. Such problems occur in global flow analysis of computer programs and in other contexts. A companion paper will discuss this application.

An O(n log n) Algorithm for Rectilinear Minimal Spanning Trees

Abstract: Let P be a set of pomts m the plane with rectdmear distance An O(n log n) a lgonthm for the construeUon o f a Voronot diagram for P ts gwen By using prewously known results, a mmmaal spammuag tree for P can be derived from a Voronot diagram for P m hnear tmae.

Enhancements of Spanning Tree Labeling Procedures for Network Optimization.

Abstract: : New labeling techniques are provided for accelerating the basis exchange step of specialized linear programming methods for network problems. Computational results are presented which show that these techniques substantially reduce the amount of computation involved in updating operations. (Author)

A graph theoretic approach to the development of minimal phylogenetic trees

Abstract: The problem of determining the minimal phylogenetic tree is discussed in relation to graph theory. It is shown that this problem is an example of the Steiner problem in graphs which is to connect a set of points by a minimal length network where new points can be added. There is no reported method of solving realistically-sized Steiner problems in reasonable computing time. A heuristic method of approaching the phylogenetic problem is presented, together with a worked example with 7 mammalian cytochrome c sequences. It is shown in this case that the method develops a phylogenetic tree that has the smallest possible number of amino acid replacements. The potential and limitations of the method are discussed. It is stressed that objective methods must be used for comparing different trees. In particular it should be determined how close a given tree is to a mathematically determined lower bound. A theorem is proved which is used to establish a lower bound on the length of any tree and if a tree is found with a length equal to the lower bound, then no shorter tree can exist.

Two papers on a tree-structured parallel computer

Abstract: : This report consists of two papers describing various aspects of a new tree-structured parallel computer. The first paper, 'A tree machine for searching problems' by J. L. Bentley and H. T. Kung, describes the basic architecture of the machine. A set of N elements can be maintained on an N-processor version of the machine such that insertions, deletions, queries and updates can all be processed in 2 lg N time units. The queries can be very complex, including problems arising in ordered set manipulation, data bases, and statistics. The machine is pipelined so that M successive operations can be performed in M-1 + 2 lg N time units. The paper studies both the basic machine structure and a VLSI implementation of the machine. The second paper, 'A parallel algorithm for constructing minimum spanning trees' by J. L. Bentley, shows how an (N/lg N)-processor version of the machine can solve the problem of constructing minimum spanning trees in time proportional to N lg N. This algorithm is an improvement over existing algorithms in several ways. (Author)

Minimum-length fundamental cycle set

Abstract: Significance of a spanning tree which yields a fundamental set of cycles with minimum total length is discussed. It is shown through a counter example that for this problem a locally optimal spanning tree need not be globally optimal, as conjectured earlier in the literature.

On the spanning trees of self‐dual maps

Abstract: Summary A central reflex is a map on the sphere that is transformed into its dual map by reflection in the center. It is shown in [2] that the number T(M) of spanning trees of a central reflex M is a perfect square {X(M)}2. In the present paper we determine X(M) directly as the number of spanning trees of M in which no edge is equivalent, under central reflection, to the dual edge of another. The paper is constructed as an expository article on the theory of dual planar maps. The proof that T(M) = {X(M)}2 is presented as an exercise in this theory.

The Complexity of Restricted Minimum Spanning Tree Problems (Extended Abstract)

Abstract: We examine the complexity of finding in a given finite metric the shortest spanning tree which satisfies a property P. Most problems discussed in the mathematical programming literature—including the minimum spanning tree problem, the matching problem matroid intersection, the travelling salesman problem, and many others—can be thus formulated. We study in particular isomonphism properties—those that are satisfied by at most one tree with a given number of nodes. We show that the complexity of these problems is captured by the rate of growth of a rather unexpected—and easy to calculate—parameter.

Efficient algorithms for simple matroid intersection problems

Abstract: Given a matroid, where each element has a realvalued cost and is colored red or green; we seek a minimum cost base with exactly q red elements. This is a simple case of the matroid intersection problem. A general algorithm is presented. Its efficiency is illustrated in the special case of finding a minimum spanning tree with q red edges; the time is O(m log log n + n α (n,n) log n). Efficient algorithms are also given for job scheduling matroids and partition matroids. An algorithm is given for finding a minimum spanning tree where a vertex r has prespecified degree; it shows this problem is equivalent to finding a minimum spanning tree, without the degree constraint. An algorithm is given for finding a minimum spanning tree on a directed graph, where the given root r has prespecified degree; the time is O(m log n), the same as for the problem without the degree constraint.

A graph-theoretical approach to region detection

Abstract: This paper presents a general and computationally inexpensive algorithmic scheme that falls in a region detection category. In the scheme, a minimal spanning tree is used as a path of a sequential region grower. The algorithm traverses the spatially adjacent graph while maintaining the structural organization of an image. The graph-theoretical evaluation of heuristics is described and examples of implementation are given.

Compact storage schemes for formatted files by spanning trees

Abstract: The use of spanning trees in the compression of data files is studied. A new upper bound for the length of the minimal spanning tree, giving the size of the compressed file, is derived. A special front compression technique is proposed for unordered files. The space demands are compared to an information theoretical lower bound of the file size.

Storage reduction in relational database systems using spanning trees and other graph theoretic methods

Abstract: There are three major configurations for data base systems: the network approach, the hierarchical approach, and the relational approach [4]. At present two of these are commercially available. The network approach (as proposed and developed by CODASYL, the conference on data systems languages) has been implemented in Sperry Univac's DMS ii00, Honeywell's IDS, Cullinane's IDMS, Cincom's TOTAL and others. The hierarchical approach has been implemented by IBM in its Information Management System (IMS), MRI's System 2000, and others. The third approach is essentially in the research and exploratory development stage. Some prototypes of the relational approach, IBM's System R and INGRES, have been developed and tested [2], but mainly for research and study purposes.