Edge-disjoint spanners in tori

Abstract

A spanning subgraph S = ( V , E ′ ) of a connected graph G = ( V , E ) is an ( x + c ) -spanner if for any pair of vertices u and v , d S ( u , v ) ≤ d G ( u , v ) + c where d G and d S are the usual distance functions in G and S , respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of two edge-disjoint spanners of delay approximately the size of the smaller dimension. Moreover, we show that this delay is close to the best possible. In three-dimensional tori, we find a set of three edge-disjoint spanners with delay approximately the sum of the sizes of the two smaller dimensions when all dimensions are of even size. Surprisingly, we also find a set of two edge-disjoint spanners in three-dimensional tori of constant delay. In d -dimensional tori, we show that for any k ≤ d / 5 , there is a set of k edge-disjoint spanners with delay depending only on k and the size of the smaller k dimensions.