Shortest Paths in Planar Graphs with Negative Weight Edges

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Keywords and Synonyms

Shortest paths in planar graphs with general arc weights; Shortest paths in planar graphs with arbitrary arc weights   

Problem Definition

This problem is to find shortest paths in planar graphs with general edge weights. It is known that shortest paths exist only in graphs that contain no negative weight cycles. Therefore, algorithms that work in this case must deal with the presence of negative cycles, i. e., they must be able to detect negative cycles.

In general graphs, the best known algorithm, the Bellman‐Ford algorithm, runs in time O(mn) on graphs with n nodes and m edges, while algorithms on graphs with no negative weight edges run much faster. For example, Dijkstra's algorithm implemented with the Fibonacchi heap runs in time \( O(m + n\log n) \), and, in case of integer weights Thorup's algorithm runs in linear time. Goldberg [5] also presented an \( O(m\sqrt{n}\log L) \)-time algorithm where L denotes the ...