Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

TL;DR

This paper introduces a decremental algorithm for maintaining all pairs all shortest paths and betweenness centrality in directed graphs with positive weights, efficiently handling vertex deletions and edge weight increases.

Contribution

It presents the first nontrivial decremental algorithms for APASP and betweenness centrality, generalizing previous methods to broader graph classes.

Findings

Runs in amortized O(vstar^2  log n) time per update

Applicable to graphs with a constant number of shortest paths between pairs

Maintains APASP and BC scores efficiently under decremental updates

Abstract

We 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(\vstar^2 \cdot \log n) time per update, where n=|V|, and \vstar 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), a graph parameter that 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 [DI04] for unique shortest paths, and for graphs with \vstar =O(n), we match the bound in [DI04]. 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 \log n) under decremental updates, regardless of the number of edges in the graph.