Sparse Spanners with Small Distance and Congestion Stretches

Costas Busch, Dariusz R. Kowalski, Peter Robinson

Abstract: 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~α 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 β 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 (α, β)-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(n5/3) edges. We also examine Δ-regular graphs with Δ ≥ n2/3, where we show how to obtain a (3, O(√Δ ⋅ log n))-DC-spanner with O(n5/3 log2n) edges. Finally, we show that there is a graph such that any optimal size 3-distance spanner has Ω(n7/6) edges and is a (3, Ω(n1/6))-DC-spanner.