Let k ≥ 2 be an integer. We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G′ = (V, E′) with O(kn1+1/k) edges such that for any two vertices u, v ∈ V, if δG(u, v) = d, then δG′(u, v) = d + O(d1-1/k-1). Furthermore, we show that such subgraphs can be constructed in O(mn1/k) time, where m and n are the number of edges and vertices in the original graph. We also show that it is possible to construct a weighted graph G* = (V, E*) with O(kn 1+1/(2k-1)) edges such that for every u, v ∈ V, if δG(u, v) = d, then d ≤ δG*(u, v) = d + O(d1-1/k-1). These are the first such results with additive error terms. of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated.
|Number of pages||8|
|State||Published - 2006|
|Event||Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms - Miami, FL, United States|
Duration: 22 Jan 2006 → 24 Jan 2006
|Conference||Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms|
|Period||22/01/06 → 24/01/06|