Shortest Path Centrality and the APSP problem via VC-dimension and Rademacher Averages

TL;DR

This paper introduces a novel approach to the APSP problem by defining shortest path centrality and using VC-dimension and Rademacher averages to develop an efficient probabilistic algorithm with guarantees.

Contribution

It proposes a new centrality measure for shortest paths and an algorithm based on progressive sampling that efficiently computes distances for paths with high centrality.

Findings

Algorithm outputs data structures with size proportional to vertex diameter.

Expected running time is logarithmic in the number of vertices.

Sample size bounds are tighter using VC-dimension theory.

Abstract

In this paper we are interested in a version of the All-pairs Shortest Paths problem (APSP) that fits neither in the exact nor in the approximate case. We define a measure of centrality of a shortest path, related to the ``importance'' of such shortest path in the graph, and propose an algorithm based on the idea of progressive sampling that, for {\it any fixed constants} $0 < \epsilon$, $ \delta < 1$, given an undirected graph $G$ with non-negative edge weights, outputs with probability $1 - \delta$ a data structure of size $n \cdot \textrm{Diam}_V(G)$, where $\textrm{Diam}_V(G)$ is the vertex diameter of $G$, in expected time $\mathcal{O}(\lg n \max(m + n \log n, n \cdot \textrm{Diam}_V(G)))$ containing the (exact) distance and the shortest path between every pair of vertices $(u,v)$ that has centrality at least $\epsilon$. The progressive sampling technique is sensitive to the probability distribution of the input (if we assume that $G$ is chosen from a prescribed random distribution), but even in the case where we take no assumption about such distribution, we show an upper bound for the sample size using VC-dimension theory that is tighter than the bound given by standard Hoeffding and union bounds, since VC-dimension theory captures the combinatorial structure of the input graph.