Advertisement

Novel Deterministic Heuristics for Building Minimum Spanning Trees with Constrained Diameter

  • C. Patvardhan
  • V. Prem Prakash
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5909)

Abstract

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(n4) 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(n3) 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(n3) and O(n2) 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.

Keywords

Heuristics Diameter Constrained Minimum Spanning Tree Bounded Diameter Greedy 

References

  1. 1.
    Abdalla, A., Deo, N., Gupta, P.: Random-tree Diameter and the diameter constrained MST. In: Congressus Numerantium, vol. 144, pp. 161–182. Utilitas Mathematica (2000)Google Scholar
  2. 2.
    Bala, K., Petropoulos, K., Stern, T.E.: Multicasting in a linear lightwave network. In: IEEE INFOCOM 1993, pp. 1350–1358 (1993)Google Scholar
  3. 3.
    Bookstein, A., Klein, S.T.: Compression of correlated bit-vectors. Information Systems 16(4), 110–118 (1996)Google Scholar
  4. 4.
    Raymond, K.: A tree-based algorithm for distributed mutual exclusion. ACM Transactions on Computer Systems 7(1), 61–77 (1989)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Kortsarz, G., Peleg, D.: Approximating shallow-light trees. In: Proc. Eighth ACM-SIAM Symposium on Discrete Algorithms, pp. 103–110 (1997)Google Scholar
  7. 7.
    Julstrom, B.A., Raidl, G.R.: A permutation-coded EA for the BDMST problem. In: GECCO 2003 Workshops Proc., Workshop on Analysis & Design of Representations, pp. 2–7 (2003)Google Scholar
  8. 8.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957)Google Scholar
  9. 9.
    Julstrom, B.A.: Greedy Heuristics for the Bounded Diameter Minimum Spanning Tree Problem. ACM J. Exp. Algor. 14, Article 1.1 (2009)Google Scholar
  10. 10.
    Achuthan, N.R., Caccetta, L., Cacetta, P., Geelen, J.F.: Algorithms for the minimum weight spanning tree with bounded diameter problem. Optimization: Techniques and Applications, 297–304 (1992)Google Scholar
  11. 11.
    Singh, A., Gupta, A.K.: Improved heuristics for the bounded diameter minimum spanning tree problem. Soft Computing 11(10), 911–921 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • C. Patvardhan
    • 1
  • V. Prem Prakash
    • 2
  1. 1.Faculty of EngineeringDayalbagh Educational InstituteAgraIndia
  2. 2.Department of Information TechnologyAnand Engineering CollegeAgraIndia

Personalised recommendations