We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter c > 1 has a worst-case update time of O(cn 2+ 2 /3 log 4 /3 n) and answers distance queries correctly with probability 1 − 1/n c , against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time ofÕ(n 2+ 3 /4 ) and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.
Thorup [60] gave a fully dynamic all-pairs shortest paths structure with worst-case update timeÕ(n 2.75 ), and Abraham et al. REF gave a randomized worst-case update time boundÕ(n 2+2/3 ).
Fully dynamic all-pairs shortest paths with worst-case update-time revisited
ArXiv
ArXiv
Mathematics Computer Science