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
2log 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.
