A Range Space with Constant VC Dimension for All-pairs Shortest Paths in Graphs

Abstract

. Let G be an undirected graph with non-negative edge weights and let S be a subset of its shortest paths such that, for every pair ( u, v ) of distinct vertices, S contains exactly one shortest path between u and v . In this paper we define a range space associated with S and prove that its VC dimension is 2. As a consequence, we show a bound for the number of shortest paths trees required to be sampled in order to solve a relaxed version of the All-pairs Shortest Paths problem (APSP) in G . In this version of the problem we are interested in computing all shortest paths with a certain “importance” at least ε . Given any 0 < ε , δ < 1, we propose a O ( m + n log n + (diam V ( G ) ) 2 ) sampling algorithm that outputs with probability 1 − δ the (exact) distance and the shortest path between every pair of vertices ( u, v ) that appears as subpath of at least a proportion ε of all shortest paths in the set S , where diam V ( G ) is the vertex-diameter of G . The bound that we obtain for the sample size depends only on ε and δ , and do not depend on the size of the graph.