Fully dynamic all pairs shortest paths with real edge weights

  • Authors:

  • Camil Demetrescu;Giuseppe F. Italiano

  • Affiliations:

  • Dipartimento di Informatica e Sistemistica, Università di Roma "La Sapienza", Roma, Italy;Dipartimento di Informatica, Sistemi e Produzione, Università di Roma "Tor Vergata", Roma, Italy

  • Venue:
  • Journal of Computer and System Sciences - Special issue on FOCS 2001
  • Year:
  • 2006

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Abstract

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√Slog3n) amortized time and queries in optimal worst-case time. This algorithm is deterministic: no previous fully dynamic algorithm was known before for this problem. In the special case where edge weights can only be increased, we give a randomized algorithm with one-sided error that supports updates faster in O(S ċ n log3 n) amortized time. We also show how to obtain query/update trade-offs for this problem, by introducing two new families of randomized algorithms. Algorithms in the first family achieve an update bound of Õ(S ċ k ċ n2)1 and a query bound of Õ(n/k), and improve over the previous best known update bounds for k in the range (n/S)1/3≤kn/S)1/2. Algorithms in the second family achieve an update bound of Õ(S ċ k ċ n2) and a query bound of Õ(n2/k2), and are competitive with the previous best known update bounds (first family included) for k in the range (n/S)1/6 ≤ k n/S)1/3.