Incremental single-source shortest paths in digraphs with arbitrary positive arc weights

Abstract

This paper studies the incremental single-source shortest paths (SSSP) problem in general digraphs with arbitrary positive arc weights. First, we examine several properties of single-source shortest paths in general digraphs with arbitrary positive arc weights, and devise a nontrivial local search algorithm LSA to handle a single arc weight increase in such a digraph, which takes at most O(n⋅max⁡{1,nlog⁡n/m}) expected update time where n is the number of nodes and m is the number of arcs in the digraph. LSA also works on undirected graphs.Furthermore, this paper analyzes the expected update time of LSA dealing with edge weight increases or edge deletions in Erdös–Rényi (a.k.a., G(n,p)) random graphs. For weighted G(n,p) random graphs with arbitrary positive edge weights, LSA takes at most O(h(Ts)) expected update time to deal with a single edge weight increase as well as O(pn2h(Ts)) total update time, where h(Ts) is the height of input SSSP tree Ts. For G(n,p) random graphs, LSA takes O(ln⁡n) expected update time to handle a single edge deletion as well as O(pn2ln⁡n) total update time when 20ln⁡n/n≤p<2ln⁡n/n, and O(1) expected update time to handle a single edge deletion as well as O(pn2) total update time when p>2ln⁡n/n. Specifically, LSA takes the least total update time of O(nln⁡nh(Ts)) for weighted G(n,p) random graphs with p=cln⁡n/n,c>1 as well as O(n3/2(ln⁡n)1/2) for G(n,p) random graphs with p=cln⁡n/n,c>2.