Non-Crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

TL;DR

This paper presents a linear-time algorithm for computing non-crossing shortest paths connecting terminal pairs on the external face of an undirected planar graph, enabling efficient distance calculations and introducing a new incremental shortest path subgraph concept.

Contribution

The paper introduces a linear-time algorithm for non-crossing shortest paths in planar graphs and a novel incremental shortest path subgraph concept.

Findings

Efficient linear-time computation of non-crossing shortest paths.

Ability to compute distances between terminal pairs within the same time bound.

Introduction of a new incremental shortest path subgraph concept.

Abstract

Given a set of well-formed terminal pairs on the external face of an undirected planar graph with unit edge weights, we give a linear-time algorithm for computing the union of non-crossing shortest paths joining each terminal pair, where well-formed means that such a set of non-crossing paths exists. This allows us to compute distances between each terminal pair, within the same time bound. We also give a novel concept of incremental shortest path subgraph of a planar graph, i.e., a partition of the planar embedding in subregions that preserve distances, that can be of interest itself.