On Dynamic Shortest Paths Problems - Roditty, Liam; Zwick, Uri (Algorithmica, 2011)
We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of (we use to hide small poly-logarithmic factors) and a worst case query time is O ( n 3/4 ). A deterministic O ( n 2 log n ) time algorithm for constructing an O (log n )-spanner with O ( n ) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.