On Reducing Stretch in Spanning Trees

Abstract

A parameter crucial for preserving the underlying shortest path information in spanning tree construction is called stretch. It is the ratio of the distance of a pair of nodes in the spanning tree to their shortest distance in the graph. In this paper, we present a distributed heuristic LSTree that constructs a Minimum Average Stretch Spanning Tree of an $n-\text{node}$ undirected and unweighted graph in $𝒪\left(n\right)$ rounds of the CONGEST model, assuming the nodes know the size of the network. We like to stress that the LSTree protocol is the first use of Betweenness Centrality in constructing low-stretch trees. The heuristic outperforms the current benchmark algorithm of Alon et al. and other spanning tree construction techniques when tested against synthetic and real-world graph inputs. This paper concludes after giving a distributed edge addition technique for building an overlay while reducing the maximum stretch in the spanning tree generated by LSTree. The overlay is a relaxation in the topological requirement, albeit equivalent in functionality to the network backbone. Hence, in this way, the paper considers a holistic view towards building low-stretch spanning trees: reducing both average stretch and max stretch in a single approach.