Shortest Paths in Directed Planar Graphs with Negative Lengths: a Linear-Space O(n log² n)-Time Algorithm

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

“…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.…”

“…The speed-up comes from a reduction of the recursion depth of the algorithm in [7] from O(log n) to O(log n/ log log n) levels. Each recursive step now becomes more involved.…”

“…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.…”

“…From observations in [7], our algorithm can be used to solve bipartite perfect matching, feasible flow, and feasible circulation in planar graphs in O(n log 2 n/ log log n) time. Chambers et al claimed [1] that the algorithm in [7] generalizes to bounded genus graphs.…”

“…Chambers et al claimed [1] that the algorithm in [7] generalizes to bounded genus graphs. However, the proof (Theorem 3.2 in [1]) is not detailed, and it seems that our new technique is required for its correctness [2].…”