Shortest Paths in Planar Graphs with Real Lengths in $O(n\log^2n/\log\log n)$ Time

TL;DR

This paper presents an improved algorithm for computing single-source shortest paths in planar graphs with real lengths, reducing the time complexity to $O(n ext{log}^2 n / ext{log} ext{log} n)$, enhancing efficiency over previous methods.

Contribution

The authors develop a faster algorithm for shortest path computation in planar graphs with real lengths, improving the time complexity from $O(n ext{log}^2 n)$ to $O(n ext{log}^2 n / ext{log} ext{log} n)$.

Findings

Achieved $O(n ext{log}^2 n / ext{log} ext{log} n)$ time complexity for shortest path computation.

Improved upon the previous $O(n ext{log}^2 n)$ bound by Klein et al.

Maintained linear space complexity of O(n).

Abstract

Given an $n$-vertex planar directed graph with real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in $O(n\log^2n/\log\log n)$ time with O(n) space. This is an improvement of a recent time bound of $O(n\log^2n)$ by Klein et al.