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Abstract:

We present here an algorithm for detecting (and outputting, if exists) a negative cycle in an $n$-vertex planar digraph $G$ with real edge weights. Its running time ranges from $O(n)$ up to $O(n^{1.5}\log n)$ as a certain topological measure of $G$ varies from $1$ up to $\Theta(n)$. Moreover, an efficient CREW PRAM implementation is given. Our algorithm applies also to digraphs whose genus $\gamma$ is $o(n)$.

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Or copy and paste:
   Dimitrios Kavvadias, Grammati E. Pantziou, Paul G. Spirakis, and Christos D. Zaroliagis, "Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem." Dartmouth Computer Science Technical Report PCS-TR94-206, 1994.

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