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Abstract

We study the problem of maintaining the distances and the shortest paths from a source node in a directed graph with arbitrary arc weights, when weight updates of arcs are performed. We propose algorithms that work for any digraph and have optimal space requirements and query time. If a negative--length cycle is introduced during weight decrease operations it is detected by the algorithms. The proposed algorithms explicitly deal with zero--length cycles. The cost of update operations depends on the class of the considered digraph and on the number of the output updates. We show that, if the digraph has a $k$-bounded accounting function (as in the case of digraphs with genus, arboricity, degree, treewidth or pagenumber bounded by $k$) the update procedures require $O(k\cdot n\cdot \log n)$ worst case time. In the case of digraphs with $n$ nodes and $m$ arcs $k=O(\sqrt{m})$, and hence we obtain $O(\sqrt{m}\cdot n \cdot \log n)$ worst case time per operation, which is better for a factor of $O(\sqrt{m} / \log n)$ than recomputing everything from scratch after each input update. If we perform also insertions and deletions of arcs all the above bounds become amortized.