Maintaining shortest paths under deletions in weighted directed graphs

We present an improved algorithm for maintaining all-pairs (1 + ε) approximate shortest paths under deletions and weight-increases. The previous state of the art for this problem was total update time O ̃(n ²√m/ε) for directed, unweighted graphs [2], and Õ(mn=ε) for undirected, unweighted graphs [12]. Both algorithms were randomized and had constant query time. Note that Õ(mn) is a natural bar- rier because even with a (1 + ε) approximation, there is no o(mn) combinatorial algorithm for the static all-pairs short- est path problem. Our algorithm works on directed, weighted graphs and has total (randomized) update time Õ(mnlog(R)=ε) where R is the ratio of the largest edge weight ever seen in the graph, to the smallest such weight (our query time is constant). Note that log(R) = O(log(n)) as long as weights are poly- nomial in n. Although Õ(mnlog(R)=ε) is the total time over all updates, our algorithm also requires a clearly unavoid- able constant time per update. Thus, we effectively expand the Õ (mn) total update time bound from undirected, un- weighted graphs to directed graphs with polynomial weights. This is in fact the first non-trivial algorithm for decremental all-pairs shortest paths that works on weighted graphs (pre- vious algorithms could only handle small integer weights). By a well known reduction from decremental algorithms to fully dynamic ones [9], our improved decremental algorithm leads to improved query-update tradeoffs for fully dynamic (1 + ε) approximate APSP algorithm in directed graphs.