Girth of a Planar Digraph with Real Edge Weights in O(n(log n)^3) Time

TL;DR

This paper presents an efficient algorithm for computing the shortest cycle (girth) in weighted planar directed graphs with real weights, improving previous bounds and applicable to arbitrary weights.

Contribution

It introduces a new O(n(log n)^3) time algorithm for girth computation in weighted planar digraphs, extending previous non-negative weight results.

Findings

Achieves faster girth computation in weighted planar digraphs.

Handles arbitrary real edge weights, not just non-negative.

Provides an algorithm that can output the shortest cycle if it exists.

Abstract

The girth of a graph is the length of its shortest cycle. We give an algorithm that computes in O(n(log n)^3) time and O(n) space the (weighted) girth of an n-vertex planar digraph with arbitrary real edge weights. This is an improvement of a previous time bound of O(n^(3/2)), a bound which was only valid for non-negative edge-weights. Our algorithm can be modified to output a shortest cycle within the same time and space bounds if such a cycle exists.