Fully Dynamic All Pairs Shortest Paths with Real Edge Weights

We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with real-valued edge weights. Given a dynamic directed graph G such that each edge can assume at most S different real values, we show how to support updates in O(n2.5 √ S log n ) amortized time and queries in optimal worst-case time. No previous fully dynamic algorithm was known for this problem. In the special case where edge weights can only be increased, we give a randomized algorithm with one-sided error which supports updates faster in O(S · n log n) amortized time. We also show how to obtain query/update trade-offs for this problem, by introducing two new families of algorithms. Algorithms in the first family achieve an update bound of Õ(S · k · n2) 1 and a query bound of Õ(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 Õ ( S · k · n2 ) and a query bound of Õ(n2/k2), and are competitive with the best known update bounds (first family included) for k in the range (n/S)1/6 ≤ k < (n/S)1/3. Work partially supported by the IST Programme of the EU under contract n. IST-199914.186 (ALCOM-FT) and by CNR, the Italian National Research Council, under contract n. 01.00690.CT26. Portions of this work have been presented at the 42nd Annual Symp. on Foundations of Computer Science (FOCS 2001) [8] and at the 29th International Colloquium on Automata, Languages, and Programming (ICALP’02) [9]. Email: [email protected]. URL: http://www.dis.uniroma1.it/~demetres. Email: [email protected]. URL: http://www.info.uniroma2.it/~italiano. Throughout the paper, we use Õ(f(n)) to denote O(f(n)polylog(n)).