Fully dynamic shortest paths in digraphs with arbitrary arc weights

  • Authors:

  • Daniele Frigioni;Alberto Marchetti-Spaccamela;Umberto Nanni

  • Affiliations:

  • Dipartimento di Ingegneria Elettrica, Università dell'Aquila, Monteluco di Roio, 1-67040 L'Aquila, Italy;Dipartimento di Informatica e Sistemistica, Università di Roma "La Sapienza," via Salaria 113, 00198 Roma, Italy;Dipartimento di Informatica e Sistemistica, Università di Roma "La Sapienza," via Salaria 113, 00198 Roma, Italy

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2003

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Abstract

We propose a new solution for the fully dynamic single source shortest paths problem in a directed graph G = (N, A) with arbitrary arc weights, that works for any digraph and has optimal space requirements and query time. If a negative-length cycle is introduced in the subgraph of G reachable from the source during an update operation, then it is detected by the algorithm. Zero-length cycles are explicitly handled. We evaluate the cost of the update operations as a function of a structural property of G and of the number of the output updates. We show that, if G has a k-bounded accounting function (as in the case of digraphs with genus, arboricity, degree, treewidth or page number bounded by k), then the update procedures require O (min{m, kċnA}ċlogn) worst case time for weight-decrease operations, and O(min{mċlogn, kċ (nA + nB) ċ logn + n}) worst case time for weight-increase operations. Here, n = |N|, m = |A|, nA is the number of nodes affected by the input update, that is the nodes changing either the distance or the parent in the shortest paths tree, and nB is the number of nonaffected nodes considered by the algorithm that also belong to zero-length cycles. If zero-length cycles are not allowed, then nB is zero and the bound for weight-increase operations is O(min{mċlogn, kċnA - logn + n}). Similar amortized bounds hold if we perform also insertions and deletions of arcs.