New parallelism and heuristic approaches for generating tree t‐spanners in graphs

Abstract

SummaryThe ‐admissibility is a min‐max problem that concerns to determine whether a graph contains a spanning tree in which the distance between any two adjacent vertices of  is at most  in . The stretch index of , , is the smallest  for which is ‐admissible. This problem is in P for , NP‐complete for , , and remaining open for . In a very recent development, Couto et al. (Inf Process Lett, 2022; 177: 106265) introduced both sequential and parallel algorithms for constructing spanning trees. Additionally, they proposed two greedy heuristics for generating a candidate solution tree, but they left unresolved the issue of how to decide between two vertices when both have equal chances of being chosen in a greedy step. This criterion is important, since different branches can yield different stretch indexes. In response to this question, we developed nine new heuristics that use the concept of vertex importance in complex networks. Our research evaluates results on several types of graphs, including Barabási‐Albert, Erdős‐Rényi, Watts‐Strogatz, and bipartite graphs. Furthermore, we introduce a new parallel algorithm that employs a method using induced cycle of the graph to compare its performance with previously proposed algorithms. We develop a deep analysis on the proposed strategies (parallel and heuristics) comparing all of them to the other ones in the literature and as a result we obtain the best results so far in order to obtain exact values (or heuristics) of stretch indexes.