Greedy heuristics for the bounded diameter minimum spanning tree problem

Abstract

Given a connected, weighted, undirected graph G and a bound D, the bounded diameter minimum spanning tree problem seeks a spanning tree on G of minimum weight among the trees in which no path between two vertices contains more than D edges. In Prim's algorithm, the diameter of the growing spanning tree can always be known, so it is a good starting point from which to develop greedy heuristics for the bounded diameter problem. Abdalla, Deo, and Gupta described such an algorithm. It imitates Prim's algorithm but avoids edges whose inclusion in the spanning tree would violate the diameter bound. Running the algorithm from one start vertex requires time that is O(n 3 ).A modification of this approach uses the start vertex as the center of the spanning tree (if D is even) or as one of the two center vertices (if D is odd). This yields a simpler algorithm whose time is O(n 2 ). A further modification chooses each next vertex at random rather than greedily, though it still connects each vertex to the growing tree with the lowest-weight feasible edge. On Euclidean problem instances with small diameter bounds, the randomized heuristic is superior to the two fully greedy algorithms, though its advantage fades as the diameter bound grows. On instances whose edge weights have been chosen at random, the fully greedy algorithms outperform the randomized heuristic. ACM Reference Format:Julstrom. B. A. 2009. Greedy heuristics for the bounded diameter minimum spanning tree problem. ACM

Efficient execution plans for distributed skyline query processing

“…This problem is NP-Hard [12], therefore we employ a heuristic algorithm to produce the k-hop execution plan efficiently. Specifically, we adapt a solution proposed by Abdalla et al [1] for the diameter-constrained minimum spanning tree problem, since the hop-constrained maximum spanning tree problem is a simplification of the bounded-diameter minimum spanning tree problem [15]. Notice that the dummy node added to the SD-graph ensures that there always exists at least one spanning tree with maximum height k (k ≥ 1).…”

Solving the Euclidean Bounded Diameter Minimum Spanning Tree Problem by Clustering–Based (Meta–)Heuristics

“…They are primarily based on Prim's minimum spanning tree (MST) algorithm and grow a height-restricted tree from a chosen center. One such example is the center based tree construction (CBTC) [8]. This approach works reasonably well on instances with random edge costs, but on Euclidean instances this leads to a backbone (the edges near the center) of relatively short edges.…”

“…In the randomized tree construction approach (RTC, Fig. 1(b)) from [8] not the overall cheapest node is added to the partial spanning tree but a random one which then is connected by the cheapest feasible edge. Thus at least the possibility to include longer edges into the backbone at the beginning of the algorithm is increased.…”

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

“…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 ).…”