Results 1  10
of
13
An optimal minimum spanning tree algorithm
 J. ACM
, 2000
"... Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is ..."
Abstract

Cited by 46 (10 self)
 Add to MetaCart
Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edgeweight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T ∗ are T ∗ (m, n) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edgeexposure martingale for Gn,p.
Increasing the Weight of Minimum Spanning Trees
, 1996
"... The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omeg ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omega\Gamma/ = log k)approximation algorithm is presented for it, where k is the number of edges to be removed. The second problem is studied assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n 3 m 2 log(n 2 =m)) time algorithm is presented to solve it. 1 Introduction Consider a communication network in which information is broadcast over a minimum spanning tree. There are applications for which it is important to determine the maximum degradation in the performance of the broadcasting protocol that can be expected as a result of traffic fluctuations and link failures [25]. Also, there are several combinatorial op...
Cost Monotonicity, Consistency And Minimum Cost Spanning Tree Games
 GAMES AND ECONOMIC BEHAVIOR
, 2002
"... We propose a new cost allocation rule for minimum cost spanning tree games. The new rule is a core selection and also satisfies cost monotonicity. We also give characterization theorems for the new rule as well as the muchstudied Bird allocation. We show that the principal difference between these ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
We propose a new cost allocation rule for minimum cost spanning tree games. The new rule is a core selection and also satisfies cost monotonicity. We also give characterization theorems for the new rule as well as the muchstudied Bird allocation. We show that the principal difference between these two rules is in terms of their consistency properties.
On the History of Combinatorial Optimization (till 1960)
"... Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Introduction As a coherent mathematical discipline, combinatorial optimization is relatively young. When studying the history of the field, one observes a number of independent lines of research, separately considering problems like optimum assignment, shortest spanning tree, transportation, and the traveling salesman problem. Only in the 1950's, when the unifying tool of linear and integer programming became available and the area of operations research got intensive attention, these problems were put into one framework, and relations between them were laid. Indeed, linear programming forms the hinge in the history of combinatorial optimization. Its initial conception by Kantorovich and Koopmans was motivated by combinatorial applications, in particular in transportation and transshipment. After the formulation of linear programming as generic problem, and the development in 1947 by Dantzig of the simplex method as a tool, one has tried to attack about all combinatorial opti
Combinatorial Optimization: A Survey
, 1993
"... This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which the ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.
Dynamic degree constrained network design: a genetic algorithm approach
 Proceedings GECCO99. Genetic and Evolutionary Computation Conference. Eighth International Conference on Genetic Algorithms (ICGA99) and the Fourth Annual Genetic Programming Conference (GP99
, 1999
"... The design and development ofnetwork infrastructure to support missioncritical operations has become a critical and complicated issue. In this study we explore the use of genetic algorithms (GA) for the design of a degree constrained minimal spanning tree (DCMST) problem with varied degrees on each ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
The design and development ofnetwork infrastructure to support missioncritical operations has become a critical and complicated issue. In this study we explore the use of genetic algorithms (GA) for the design of a degree constrained minimal spanning tree (DCMST) problem with varied degrees on each node. The performance of GA was compared with two popular heuristics. The results indicate that GA provide better solution quality compared to heuristics, but is worse than heuristics in terms of computation time. 1
Counting and Constructing Minimal Spanning Trees
, 2000
"... . We revisit the minimal spanning tree problem in order to develop a theory of construction and counting of the minimal spanning trees in a network. The theory indicates that the construction of such trees consists of many different choices, all independent of each other. These results suggest a ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
. We revisit the minimal spanning tree problem in order to develop a theory of construction and counting of the minimal spanning trees in a network. The theory indicates that the construction of such trees consists of many different choices, all independent of each other. These results suggest a block approach to the construction of all minimal spanning trees in the network, and an algorithm to that effect is outlined as well as a formula for the number of minimal spanning trees. Key words and Phrases: Minimal spanning trees, equal edge replacements, equivalent edges, choices. 1991 Mathematics Subject Classification. Primary 05C05, 68R10. 1. Introduction. Let G be a connected loopfree nondirected graph, let T be a spanning tree of G, and let f be an edge of G not in T . We define P (f; T ), the path of f in T , to be the unique simple path in T that joins the vertices of f . Consider an ordered pair (e; f) of edges such that e 2 T , f 62 T and e is on P (f; T ). Define S = (T...
An O(log m) parallel algorithm for the most vital edge problem
"... : In this paper we study a parallel algorithm for the computation of the most vital edge (MVE) problem with respect to a given minimum spanning tree. Our algorithm runs on O(m) processors and has running time of O(log m), where m is the number of edges and n is the number of vertices of the given gr ..."
Abstract
 Add to MetaCart
: In this paper we study a parallel algorithm for the computation of the most vital edge (MVE) problem with respect to a given minimum spanning tree. Our algorithm runs on O(m) processors and has running time of O(log m), where m is the number of edges and n is the number of vertices of the given graph. Keywords: Parallel algorithms, graph algorithms, complexity, minimum spanning tree, most vital edge problem.  2  2. Preliminaries Here we collect some of the known results on the characteristics of the minimum spanning trees and the MVE problem. These will be stated in the form of lemmas without proofs, and wherever possible, we will give references to the original sources. The following design criteria have been used by different MST algorithms, see for example Graham and Hell [4]. 1. Add a shortest edge which joins different fragments. 2. Choose a vertex, v arbitrarily. Add a shortest edge which joins the fragment containing v to another fragment. 3. For every fragment add the ...