### Fast Heuristics for Large Instances of the Euclidean Bounded Diameter Minimum Spanning Tree Problem

#### Abstract

Given a connected, undirected graph G = (V, E) on n = jV j vertices, an integer bound D>=2 and non-zero

edge weights associated with each edge e 2 E, a bounded diameter minimum spanning tree (BDMST) on G

is defined as a spanning tree T E on G of minimum edge cost w(T) = P w(e), 8 e 2 T and tree diameter no

greater than D. The Euclidean BDMST Problem aims to find the minimum cost BDMST on graphs whose

vertices are points in Euclidean space and whose edge weights are the Euclidean distances between the

corresponding vertices. The problem of computing BDMSTs is known to be NP-Hard for 4 D n -1,

where D the diameter bound. Furthermore, the problem is known to be hard to approximate. Heuristics

are extant in the literature which build low cost, diameter-constrained spanning trees in O(n3) time. This

paper presents some fast and effective heuristic strategies for the Euclidean BDMST Problem and compares

their performance with that of the best known existing heuristics. Two of the proposed heuristics run in

O(n2pn) time and another faster heuristic runs in O(n2), thereby allowing them to quickly build low cost

BDSTs on larger sized problems than have been attempted hitherto. The proposed heuristics are shown to

perform better over a wide range of benchmark instances used in the literature for the Euclidean BDMST

Problem. Further, a new test suite of much larger problem sizes than attempted hitherto in the literature is

designed and results presented.

edge weights associated with each edge e 2 E, a bounded diameter minimum spanning tree (BDMST) on G

is defined as a spanning tree T E on G of minimum edge cost w(T) = P w(e), 8 e 2 T and tree diameter no

greater than D. The Euclidean BDMST Problem aims to find the minimum cost BDMST on graphs whose

vertices are points in Euclidean space and whose edge weights are the Euclidean distances between the

corresponding vertices. The problem of computing BDMSTs is known to be NP-Hard for 4 D n -1,

where D the diameter bound. Furthermore, the problem is known to be hard to approximate. Heuristics

are extant in the literature which build low cost, diameter-constrained spanning trees in O(n3) time. This

paper presents some fast and effective heuristic strategies for the Euclidean BDMST Problem and compares

their performance with that of the best known existing heuristics. Two of the proposed heuristics run in

O(n2pn) time and another faster heuristic runs in O(n2), thereby allowing them to quickly build low cost

BDSTs on larger sized problems than have been attempted hitherto. The proposed heuristics are shown to

perform better over a wide range of benchmark instances used in the literature for the Euclidean BDMST

Problem. Further, a new test suite of much larger problem sizes than attempted hitherto in the literature is

designed and results presented.

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