Additive Spanners in Nearly Quadratic Time

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, δH (u, v) ≤ δG(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(n4/3 ) edges in O(mn2/3 ) time. It is unknown if there exists a constant C and an additive C-spanner with o(n4/3 ) edges. Moreover, for C ≤ 5 all known constructions require Ω(n3/2 ) edges. We give a significantly more efficient construction of an additive 6-spanner. The number of edges in our spanner is n4/3 polylog n, matching what was previously known up to a polylogarithmic factor, but we greatly improve the time for construction, from O(mn2/3 ) to n2 polylog n. Notice that mn2/3 ≤ n2 only if m ≤ n4/3 , 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 wi...

• Type: Conference
• Year: 2010
• Where: ICALP
• Authors: David P. Woodruff
• Keywords: Additive 6-spanner, Additive C-spanner, Graph G, ICALP 2010, Programming Languages