A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. The Diameter-Constrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 ≤ k ≤ (n - 2). Therefore, one has to depend on heuristics and live with approximate solutions. In this paper, we explore two heuristics for the DCMST problem: First, we present a one-time-treeconstruction algorithm that constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting edges to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. It is particularly suited when the specified values for k are small--independent of n. The second algorithm starts with an unconstrained MST and iteratively refines it by replacing edges, one by one, in long paths until there is no path left with more than k edges. This heuristic was found to be better suited for larger values of k. We discuss convergence, relative merits, and parallel implementation of these heuristics on the MasPar MP-1 -- a massively parallel SIMD machine with 8192 processors. Our extensive empirical study shows that the two heuristics produce good solutions for a wide variety of inputs.
[1]
C. R. Muthukrishnan,et al.
A Note on Raymond's Tree Based Algorithm for Distributed Mutual Exclusion
,
1992,
Inf. Process. Lett..
[2]
Thomas E. Stern,et al.
Multicasting in a linear lightwave network
,
1993,
IEEE INFOCOM '93 The Conference on Computer Communications, Proceedings.
[3]
S. Wang,et al.
A tree-based distributed algorithm for the K-entry critical section problem
,
1994,
Proceedings of 1994 International Conference on Parallel and Distributed Systems.
[4]
David S. Johnson,et al.
Computers and Intractability: A Guide to the Theory of NP-Completeness
,
1978
.
[5]
E. Palmer.
Graphical evolution: an introduction to the theory of random graphs
,
1985
.
[6]
K. Phua,et al.
Optimization : techniques and applications
,
1992
.
[7]
Ayman M. Abdalla,et al.
Computing a diameter-constrained minimum spanning tree
,
2001
.
[8]
Narsingh Deo,et al.
Constrained spanning tree problems: Approximate methods and parallel computation
,
1997,
Network Design: Connectivity and Facilities Location.
[9]
Randy Chow,et al.
Distributed Operating Systems & Algorithms
,
1997
.
[10]
Kerry Raymond,et al.
A tree-based algorithm for distributed mutual exclusion
,
1989,
TOCS.