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Acta Applicandae Mathematicae

, Volume 102, Issue 2–3, pp 219–236 | Cite as

On the Minimum Average Distance Spanning Tree of the Hypercube

  • Maurice Tchuente
  • Paulin Melatagia Yonta
  • Jean-Michel NlongII
  • Yves Denneulin
Article

Abstract

Given an undirected and connected graph G, with a non-negative weight on each edge, the Minimum Average Distance (MAD) spanning tree problem is to find a spanning tree of G which minimizes the average distance between pairs of vertices. This network design problem is known to be NP-hard even when the edge-weights are equal. In this paper we make a step towards the proof of a conjecture stated by A.A. Dobrynin, R. Entringer and I. Gutman in 2001, and which says that the binomial tree B n is a MAD spanning tree of the hypercube H n . More precisely, we show that the binomial tree B n is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H n , attempts to improve the average distance between pairs of vertices, by adding an edge e of H n -T and removing an edge e′ from the unique cycle created by e. We also present a greedy algorithm which produces good solutions for the MAD spanning tree problem on regular graphs such as the hypercube and the torus.

Keywords

Minimum average distance Spanning tree Hypercube Binomial tree 1-move heuristic Local optimality Greedy algorithm Torus 

Mathematics Subject Classification (2000)

05C12 05C05 05C90 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Maurice Tchuente
    • 1
    • 2
    • 3
  • Paulin Melatagia Yonta
    • 1
  • Jean-Michel NlongII
    • 4
    • 5
  • Yves Denneulin
    • 5
  1. 1.Laboratoire MAT-IRD, Département d’Informatique, Faculté des SciencesUniversité de Yaoundé IYaoundéCameroun
  2. 2.Ecole Normale Supérieure de LyonInstitut des Systèmes ComplexesLyon Cedex 07France
  3. 3.Institut de Recherche pour le Développement (IRD)UR GEODESBondy CedexFrance
  4. 4.Département de Mathématique-Informatique, Faculté des SciencesUniversité de NgaoundéréNgaoundéréCameroun
  5. 5.Laboratoire Informatique et DistributionUMR CNRS, INPG, INRIA, UJF 5132Montbonnot Saint MartinFrance

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