Additive Spanners in Nearly Quadratic Time

Abstract

We consider the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u,v, δ _H (u,v) ≤ δ _G (u,v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6-spanner with O(n ^4/3) edges in O(mn ^2/3) time. It is unknown if there exists a constant C and an additive C-spanner with o(n ^4/3) edges. Moreover, for C ≤ 5 all known constructions require Ω(n ^3/2) edges.

We give a significantly more efficient construction of an additive 6-spanner. The number of edges in our spanner is n ^4/3 polylog n, matching what was previously known up to a polylogarithmic factor, but we greatly improve the time for construction, from O(mn ^2/3) to $n^2 {\rm polylog} \ n$. Notice that mn ^2/3 ≤ n ² only if m ≤ n ^4/3, but in this case G itself is a sparse spanner. We thus provide both the fastest and the sparsest (up to logarithmic factors) known construction of a spanner with constant additive distortion.

We give similar improvements in the construction time of additive spanners under the assumption that the input graph has large girth, or more generally, the input graph has few edges on short cycles.