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

Abstract

In this paper, we present an O(n log 3 n) time algorithm for finding shortest paths in an n-node 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 runs in O(n 3/2 ) time, and the best polynomial time algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which runs inÕ(n 4/3 ) time. We also present significantly improved data structures for reporting distances between pairs of nodes and algorithms for updating the data structures when edge weights change.

Finding Shortest Non-separating and Non-contractible Cycles for Topologically Embedded Graphs

“…The recursive subdivision that Henzinger et al [14] require can then be obtained using the division by Eppstein in the first level and then continue in each planar subpiece using their approach Djidjev [4] and Fakcharoenphol and Rao [11] (slightly improved by Klein [17] for nonnegative edge-lengths) describe data structures for shortest path queries in planar graphs. We will need the following special case.…”

Shortest Paths in Planar Graphs with Real Lengths in O(nlog2 n/loglogn) Time

“…Henzinger, Klein, Rao, and Subramanian [4] obtained a (not strongly) polynomial bound ofÕ(n 4/3 ). Later, Fakcharoenphol and Rao [3] showed how to solve the problem in O(n log 3 n) time and O(n log n) space. Recently, Klein, Mozes, and Weimann [7] presented a linear space O(n log 2 n) time recursive algorithm.…”

“…To deal with this, we show a new technique for using the Monge property in graphs that do not necessarily posses that property. Both [3] and [7] showed how to partition a set of distances that are not Monge, into subsets, each of which is Monge. Exploiting this property, the distances within each subset can be processed efficiently.…”

Engineering Route Planning Algorithms

“…Hence, techniques developed for planar graphs will often also work for road networks. Using O(n log 2 n) space and preprocessing time, query time O( √ n log n) can be achieved [22,41] for directed planar graphs without negative cycles. Queries accurate within a factor (1 + ) can be answered in near constant time using O((n log n)/ ) space and preprocessing time [70].…”