Fully Dynamic All Pairs All Shortest Paths [article]

Matteo Pontecorvi, Vijaya Ramachandran
2022 arXiv   pre-print
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(ν^*^2 ·log^3 n) time per update, where n = |V|, and ν^* bounds the number of edges
more » ... 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.
arXiv:1412.3852v4 fatcat:7eget5wtrbeqbmm2yzwf4ckkea