Improved Bounds and New Trade-Offs for Dynamic All Pairs Shortest Paths

Let G be a directed graph with n vertices, subject to dynamic updates, and such that each edge weight can assume at most S diierent arbitrary real values throughout the sequence of updates. We present a new algorithm for maintaining all pairs shortest n) amortized time per update and in O(1) worst-case time per distance query. This improves over previous bounds. We also show how to obtain query/update trade-oos for this problem, by introducing two new families of algorithms. Algorithms in the rst family achieve an update bound of e O(Skn 2) 1 and a query bound of e O(n=k), and improve over the best known update bounds for k in the range (n=S) 1=3 k < (n=S) 1=2. Algorithms in the second family achieve an update bound of e O ? S k n 2 and a query bound of e O(n 2 =k 2), and are competitive with the best known update bounds ((rst family included) for k in the range (n=S) 1=6 k < (n=S) 1=3 .