Abstract
In this paper, we present an 0(n\log ^3 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 Lipton, Rose, and Tarjan in 1978 which ran in 0(n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} ) time, and the best polynomial algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which ran in \widetilde0(n^{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}} ) time. We also present significantly improved algorithms for query and dynamic versions of the shortest path problems.