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#### Fully dynamic shortest paths and negative cycle detection on diagraphs with Arbitrary Arc Weights

##### Fulltext (public)

MPI-I-98-1-009.pdf

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##### Supplementary Material (public)

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##### Citation

Frigioni, D., Marchetti-Spaccamela, A., & Nanni, U.(1998). *Fully
dynamic shortest paths and negative cycle detection on diagraphs with Arbitrary Arc Weights* (MPI-I-1998-1-009).
Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-7BD9-A

##### 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.