Sparse Spanners with Small Distance and Congestion Stretches

Given a graph G, a classical problem in graph theory is the construction of a spanner H - a sparse subgraph of G that closely approximates the distances between nodes in G. The distance stretch alpha of H is the factor of how much the distances in H increase versus G. Here, we consider sparse spanner constructions that can also preserve the node congestion of routing problems in G. The congestion stretch beta of H is the factor of how much the (smallest) congestion of a routing problem increases in H versus G. We introduce the notion of (alpha, beta)-DC-spanner (i.e., a Distance-Congestion-spanner) that simultaneously controls the stretches for distance and congestion. We show that for expander graphs with n nodes, there is a (3, O(log n))-DC-spanner with O(n(5/3)) edges. We also examine-regular graphs with Delta >= n(2/3), where we show how to obtain a (3, O(root Lambda center dot log n))-DC-spanner with O(n(5/3) log(2) n) edges. Finally, we show that there is a graph such that any optimal size 3-distance spanner has Omega(n(7/6)) edges and is a (3, Omega(n(1/6)))-DC-spanner.