We obtain the following results related to dynamic versions of the shortest-paths problem: (i) Reductions that show that the incremental and dceremental singlesource 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 shortestpaths problem in directed unweighted graphs with an amortized update time of Õ(m√n) and a worst case query time is O(n3/4). (iii) A deterministic O(n2log n) time algorithm for constructing a (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.