Planar graphs, negative weight edges, shortest paths, and near linear time

The authors present an O(n log³ n) time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by R. Lipton et al., (1978 )which ran in O(n³2/) time, and the best polynomial algorithm developed by M. Henzinger et al. (1994) which ran in O˜(n⁴3/) time. We also present significantly improved algorithms for query and dynamic versions of the shortest path problems.