Fully Dynamic All Pairs All Shortest Paths

TL;DR

This paper introduces two fully dynamic algorithms for maintaining all pairs shortest paths and betweenness centrality in directed graphs with positive weights, supporting edge weight updates efficiently.

Contribution

It presents novel algorithms that extend previous work to handle dynamic updates for all pairs shortest paths and betweenness centrality with improved amortized running times.

Findings

First algorithm runs in amortized O(ν*^2 log^3 n) time per update.

Second, faster algorithm reduces amortized cost by a logarithmic factor.

Algorithms generalize and improve upon previous static and decremental methods.

Abstract

We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph $G=(V,E)$ with a positive real weight on each edge. We present two fully dynamic algorithms for this problem in which an update supports either weight increases or weight decreases on a subset of edges incident to a vertex. Our first algorithm runs in amortized $O({\nu^*}^2 \cdot \log^3 n)$ time per update, where $n = |V|$, and $\nu^*$ bounds the number of edges that lie on shortest paths through any single vertex. Our APASP algorithm leads to the same amortized bound for the fully dynamic computation of betweenness centrality (BC), which is a parameter widely used in the analysis of large complex networks. Our method is a generalization and a variant of the fully dynamic algorithm of Demetrescu and Italiano [DI04] for unique shortest path, and it builds on our recent decremental APASP [NPR14]. Our second (faster) algorithm reduces the amortized cost per operation by a logarithmic factor, and uses new data structures and techniques that are extensions of methods in a fully dynamic algorithm by Thorup.