Improved Bounds for Shortest Paths in Dense Distance Graphs

We study the problem of computing shortest paths in so-called dense distance graphs. Every planar graph G on n vertices can be partitioned into a set of O(n/r) edge-disjoint regions (called an r-division) with O(r) vertices each, such that each region has O( √ r) vertices (called boundary vertices) in common with other regions. A dense distance graph of a region is a complete graph containing all-pairs distances between its boundary nodes. A dense distance graph of an r-division is the union of the O(n/r) dense distance graphs of the individual pieces. Since the introduction of dense distance graphs by Fakcharoenphol and Rao [4], computing single-source shortest paths in dense distance graphs has found numerous applications in fundamental planar graph algorithms. Fakcharoenphol and Rao [4] proposed an algorithm (later called FR-Dijkstra) for computing singlesource shortest paths in a dense distance graph in O ( n √ r logn log r ) time. We show an