Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

Abstract

AbstractWe consider the all pairs all shortest paths (APASP) problem, which maintains the shortest path dag rooted at every vertex in a directed graph \(G=(V,E)\) with positive edge weights. For this problem we present a decremental algorithm (that supports the deletion of a vertex, or weight increases on edges incident to a vertex). Our algorithm runs in amortized \(O({\nu ^*}^2 \cdot \log n)\) time per update, where \(n = |V| \), and \({\nu ^*}\) bounds the number of edges that lie on shortest paths through any given vertex. Our APASP algorithm can be used for the decremental computation of betweenness centrality (BC), which is widely used in the analysis of large complex networks. No nontrivial decremental algorithm for either problem was known prior to our work. Our method is a generalization of the decremental algorithm of Demetrescu and Italiano [3] for unique shortest paths, and for graphs with \({\nu ^*}= O(n)\), we match the bound in [3]. Thus for graphs with a constant number of shortest paths between any pair of vertices, our algorithm maintains APASP and BC scores in amortized time \(O(n^2 \cdot \log n)\) under decremental updates, regardless of the number of edges in the graph.