Abstract We obtain the following results related to dynamic versions of the shortestpaths problem: (i) 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. (ii) A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time ofÕ(m √ n) (we useÕ to hide small poly-logarithmic factors) and a worst case query time is O(n 3/4 ). (iii) 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.
Roditty and Zwick REF consider the incremental and decremental versions of the single source shortest path problem in weighted and unweighted directed graphs.
On Dynamic Shortest Paths Problems
Algorithmica
Algorithmica
Mathematics Computer Science