Showing papers on "Spanning tree published in 1982"

The complexity of restricted spanning tree problems

Abstract: The complexity of the foUowmg class of problems Is investigated: Given a distance matrix, fred the shortest spanning tree that is isomorphic to a given prototype. Several classical combinatorial problems, both easy and hard, fall into this category for an appropriate choice of the family of prototypes, for example, taking the family to be the set of all paths gives the traveling salesman problem or taking the family to be the set of all 2-stars gives the weighted matching problem It is shown that the complexity of these problems depends explicitly on the rate of growth of a sLmple parameter of the family of prototypes.

Message routing in a computer network

Abstract: An adaptive distributed message routing algorithm that may be implemented in a computer program to control the routing of data messages in a packet message switching digital computer network. Network topology information is exchanged only between neighbor nodes in the form of minimum spanning trees, referred to as exclusionary trees. An exclusionary tree is formed by excluding the neighbor node and its links from the tree. From the set of exclusionary trees received a route table and transmitted exclusionary trees are constructed.

Power System Topological Observability Using a Direct Graph-Theoretic Approach

Abstract: The topological observability of a Power System State Estimation (PSSE) problem is considered. The PSSE observability is decoupled into two separate problems using the well-known P-?/Q-V decouple characteristic of power systems. Using a linear decoupled model for the measurements, the proposed method tries to find an observable spanning tree of the P-?measurement graph (and subsequently for the Q-V measurement graph) using an algorithm for the search of matroid intersections. The method is first applied to six well- known examples of the technical literature, and then is used to study six different cases of a realistic system comprising 121 buses.

A Data Structure for Parallel L/U Decomposition

Abstract: Some new results are presented concerning the pivoting of large systems of linear equations with respect to parallel processing techniques. It will be assumed that the processing of a pivot takes one time slot. The pivoting problem is studied by means of an associated graph model. Given a triangulated graph a set of label classes is established. Class k contains all pivots which may be processed in parallel during the kth time slot. The label classes are used to establish the elimination-tree (e-tree). The e-tree is a spanning tree for the given graph. The critical path in the e-tree indicates the minimum number of time slots necessary to complete the L/U-decomposition. Furthermore, the earliest and latest admissible time slot for the processing of every pivot may be derived, such that the critical path is not affected. The e-tree can be seen as a data structure to guide parallel processing based on sparsity.

A Cutset Approach to Reliability Evaluation in Communication Networks

Abstract: A global reliability measure called 'network reliability' (NR) is the probability that a call entering a probabilistic network at any originating node can reach every other node. This concept is quite useful in multiterminal networks such as computer networks and parallel processors, etc. A simple technique is presented for evaluating NR in symbolic form. The method is based on cutsets and is computationally advantageous with respect to the spanning tree approach. It requires fewer cutsets to be manipulated in the process of determining the NR expression. The number of cutsets is approximately half that of spanning trees even for a small sized computer communication network and there is a further improvement in the situation for larger networks.

A primal algorithm for finding minimum-cost flows in capacitated networks with applications

Abstract: Algorithms for finding a minimum-cost, single-commodity flow in a capacitated network are based on variants of the simplex method of linear programming. We describe an implementation of a primal algorithm which is fast and can solve large problems. The major ideas incorporated are (i) the sparsity of the network is used to reduce the time and computer storage space requirements; (ii) basic solutions are stored compactly as spanning trees of the network; (iii) a candidate stack is used to allow flexible strategies in choosing an arc to enter the basis tree; (iv) the predecessor and thread data structures are used to efficiently traverse the tree and to update the solution at each iteration; (v) rules are implemented to avoid cycling or stalling caused by degeneracy; and (vi) piecewise-linear, convex arc costs are handled implicitly. The Primal Network Flow Convex (PNFC) code implements this algorithm and three examples, from communication networks, that can be solved with PNFC are discussed: (i) solving the area transfer problem; (ii) scheduling the collection of traffic data records; and (iii) planning the placement of pair-gain systems.

The Bounded Path Tree Problem

Abstract: The subject of this paper is the bounded path tree (BPT) problem: An undirected graph $G ( V,E )$ is given whose edges have nonnegative lengths; two subsets I and J of V are also given, and nonnegative constants $U_i $, $W_i $ are associated with each $i \in I$, $j \in J$. The BPT problem asks for a tree of G whose vertex set contains $I \cup J$ and whose path joining vertices i and j is not longer than $U_i + W_j $, for each $i \in I$, $j \in J$. This problem generalizes the shortest path and the minimum longest path spanning tree problem. It complements standard min–max location problems, as it asks for a tree given the facility locations, instead of locating facilities in a given network. In this paper we propose some applications of the BPT problem for the design of emergency and communication networks, show its equivalence to an extension of the absolute center location problem and give an algorithm for its solution. This algorithm requires time $O( k| E | + k | V | \log k )$, where $k = | I \cup J |...

The Expected Time Complexity of Parallel Graph and Digraph Algorithms.

Abstract: : This paper determines upper bounds on the expected time complexity for a variety of known parallel algorithms for graph problems. For connectivity of both undirected and directed graphs, transitive closure and all pairs minimum cost paths, we prove the expected time is O(loglog n) for a parallel RAM model (RP-RAM) which allows random resolution of write conflicts, and expected time O(log n loglog n) for the P-RAM of (Wyllie, 79), which allows no write conflicts. We show that the expected parallel time for biconnected components and minimum spanning trees is O(loglog n)(2) for the RP-RAM and O(log n. (loglog n) (2)) for the P-RAM. Also we show that the problem of random graph isomorphism has expected parallel time O(loglog n) and O(log n) for the above parallel models, respectively. Our results also improve known upper bounds on the expected space required tor sequential graph algorithms. For example, we show that the problems of finding strong components, transitive closure and minimum cost paths have expected sequential space O(log-loglog n) with n (O)(1) time on a Turing Machine given random graphs as inputs.

Efficient Tree Handling Procedures for Allocation/Processing Networks.

Abstract: : Processing network problems are minimum cost network flow problems in which the flow entering (or leaving) a node may be constrained to do so in given proportions. These problems include many practical large-scale linear programming applications. In this paper a special class of processing network problems called allocation/processing (AP) network problems is described. AP network problems model a realistic situation in which a single raw material is allocated to factories and the resulting finished products are distributed to customers. A special partitioning method (PPR algorithm) is introduced for the solution of AP network problems. This algorithm maintains a working basis as well as a so-called representative spannng tree. A method for updating the representative spanning tree is discussed which avoids tracing a number of cycles at each iteration. A FORTRAN implementation (PPRNET) of the PPR algorithm is described. Comparative computational results are presented for randomly generated AP network problems which were solved by PPRNET and MINOS. These results indicate that significant computational gains are possible with the use of special purpose processing network codes. (Author)

Algorithms for Generating Cycles in a Graph

Abstract: The following problem is considered: Given an undirected, connected graph G, find a spanning tree in G such that the sum of the lengths of the fundamental cycles (with respect to this tree) is minimum. This problem, besides being interesting in its own right, is useful in a variety of situations It is shown that this problem is NP-complete. A number of polynomial-time, heuristic algorithms which yield "good" suboptimal solutions are presented and their performances are discussed. Finally, it is shown that for regular graphs of order n the expected value of the total length of a minimum fundamentalcycle set does not exceed O(n2).