On the minimum diameter spanning tree problem (original) (raw)

A distributed algorithm for constructing a minimum diameter spanning tree

2004

We present a new algorithm, which solves the problem of distributively finding a mini-mum diameter spanning tree of any (non-negatively) real-weighted graph G = (V,E, ω). As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algo-rithm to find an absolute center of G. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V |) time complexity and O (|V | |E|) message complexity.

The Diameter of the Minimum Spanning Tree of a Complete Graph

Let {X1, . . . , X ( n 2 ) } be independent identically distributed weights for the edges of Kn. If Xi = Xj for i = j, then there exists a unique minimum weight spanning tree T of Kn with these edge weights. We show that the expected diameter of T is Θ(n 1/3 ). This settles a question of .

On central spanning trees of a graph

Lecture Notes in Computer Science, 1996

We consider the collection of all spanning trees of a graph with distance between them based on the size of the symmetric difference of their edge sets. A central spanning tree of a graph is one for which the maximal distance to all other spanning trees is minimal. We prove that the problem of constructing a central spanning tree is algorithmically difficult and leads to an NP-complete problem.

Approximating the Geometric Minimum-Diameter Spanning Tree

2003

Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. In this paper, we present an approximation algorithm that generates a tree whose diameter is no more than (1 + ) times that of a GMDST, for any > 0. Our algorithm reduces the problem to several grid-aligned versions of the problem and runs within time O( −3 + n) and space O(n) improving the result by Gudmundsson et al. [4].

Computing a (1+ε)-approximate geometric minimum-diameter spanning tree

2004

Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. For several years, the best upper bound on the time to compute a GMDST was cubic with respect to the number of points in the input set. Recently, Timothy Chan introduced a subcubic time algorithm. In this paper we present an algorithm that generates a tree whose diameter is no more than (1 + ε) times that of a GMDST, for any ε > 0. Our algorithm reduces the problem to several grid-aligned versions of the problem and runs within time O(ε −3 + n) and space O(n).

The Diameter and Maximum Link of the Minimum Routing Cost Spanning Tree Problem

Science and Technology Indonesia

The minimum routing cost spanning tree (MRCST) is a spanning tree that minimizes the sum of pairwise distances between its vertices given a weighted graph. In this study, we use Campos Algorithm with slight modifications on the coefficient of spanning potential. Those algorithms were implemented on a random table problem data of complete graphs of order 10 to 100 in increments of 10. The goal is to find the diameter (the largest shortest path distance) and the maximum link (the maximum number of edges connecting two vertices) in the spanning tree solution of MRCST. The result shows that a slight modification of the spanning potential coefficients gives better solutions.

Novel Deterministic Heuristics for Building Minimum Spanning Trees with Constrained Diameter

Lecture Notes in Computer Science, 2009

Given a connected, weighted, undirected graph G with n vertices and a positive integer bound D, the problem of computing the lowest cost spanning tree from amongst all spanning trees of the graph containing paths with at most D edges is known to be NP-Hard for 4 ≤ D < n-1. This is termed as the Diameter Constrained, or Bounded Diameter Minimum Spanning Tree Problem (BDMST). A well known greedy heuristic for this problem is based on Prim's algorithm, and computes BDMSTs in O(n 4) time. A modified version of this heuristic using a tree-center based approach runs an order faster. A greedy randomized heuristic for the problem runs in O(n 3) time and produces better (lower cost) spanning trees on Euclidean benchmark problem instances when the diameter bound is small. This paper presents two novel heuristics that compute low cost diameter-constrained spanning trees in O(n 3) and O(n 2) time respectively. The performance of these heuristics vis-à-vis the extant heuristics is shown to be better on a wide range of Euclidean benchmark instances used in the literature for the BDMST Problem.

A Distributed Algorithm for the Minimum Diameter Spanning Tree Problem

1998

We present a new algorithm, which solves the problem of distributively nding a minimum diameter spanning tree of any arbitrary positively real-weighted graph. We use a new fast linear time intermediate all-pairs shortest paths routing protocol. The resulting distributed algorithm is asynchronous, it works for arbitrary named network, and achieves O(n) time complexity and O(nm) message complexity.

Solving diameter-constrained minimum spanning tree problems by constraint programming

International Transactions in Operational Research, 2010

The Diameter Constrained Minimum Spanning Tree Problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with each other at a minimum cost, subject to a given quality requirement. We propose alternative formulations using constraint programming that circumvent weak lower bounds given by most mixed integer programming formulations. Computational results show that the proposed formulation combined with an appropriate search procedure solve larger instances and is faster than other approaches in the literature.

The directed minimum-degree spanning tree problem

2001

Consider a directed graph G= V, E) with n vertices and a root vertex r∈ V. The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasipolynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is at most O (Δ*)+ log n) where, Δ* is the degree of some optimal tree for the problem.