Computing a Diameter-Constrained Minimum Spanning Tree in Parallel

Abstract

Abstract.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 DiameterConstrained 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.

Bounded-Diameter MST Instances with Hybridization of Multi-Objective EA

“…Deo and Abdalla [8] proposed One Time Tree Construction (OTTC), based on Prim's algorithm. It builds a spanning tree, starting from each node and connecting the nearest neighbor that does not violate the diameter constraint.…”

“…BDMST has many applications in real-world [2,4,22]; it is an NP-hard problem within diameter bound (D) ranges 4 ≤ D < |V| -1 [9], where diameter bound (D) is a constraint, the maximum feasible longest path between two vertices of a connected, undirected, weighted graph G to generate feasible MSTs and V is the set of vertices of G. Well-known heuristics which are evolved to provide solutions to BDMST problem, are: e.g., one time tree construction (OTTC) [8], iterative refinement (IR) [8], randomized greedy heuristics (RGH) [21], random tree construction (RTC) [10], center based tree construction (CBTC) [10] and center-based recursive clustering (CBRC) [20]. Initially, algorithmic complexity directed the development of various heuristics.…”

“…Initially, algorithmic complexity directed the development of various heuristics. The complexity of OTTC, IR, CBTC and CBRC is O(n 3 ) [8,10,20]. Depending on initial vertex, generated spanning tree (ST) differs for OTTC, IR and CBTC heuristics; therefore, one needs to execute the heuristics with each of the vertices as initial vertex with the expectation to get better low-weight spanning tree and then select the best generated tree which makes the complexity O(n 4 ).…”

Diameter Constrained Fuzzy Minimum Spanning Tree Problem

“…This problem has also been applied to data compression by Bookstien and Klien, 9 and distributed mutual exclusion in parallel computing by Raymond, 10 and Deo and Abdalla. 11 Different formulation of the DCMST has been found in the literature. 8,11,12 These formulations implicitly use a property of feasible diameter constrained spanning tree, pointed out by Handler.…”

“…11 Different formulation of the DCMST has been found in the literature. 8,11,12 These formulations implicitly use a property of feasible diameter constrained spanning tree, pointed out by Handler. 13 He pointed out that, when D is even, a central vertex i ∈ V must exist in a feasible tree T , such that no other vertex of T is more than D/2 edges away from i and when D is odd, a central edge e = (i, j) ∈ E must exist in T , such that no vertex of T is more than (D − 1)/2 edges away from the closest extremity of (i, j).…”