A Linear Time Algorithm for Finding Minimum Flip Sequences between Plane Spanning Paths in Convex Point Sets

TL;DR

This paper introduces a linear time algorithm to compute the minimum number of flips needed to transform one plane spanning path into another on convex point sets, contrasting with previous hardness results.

Contribution

It presents the first linear time algorithm for flip distance in convex point sets and explores its adaptation to related flip problems.

Findings

The flip distance can be computed in linear time for convex point sets.

The happy edge property does not hold in this setting.

The algorithm extends to compatible, local, and polygon spanning path flips.

Abstract

We provide a linear time algorithm to determine the flip distance between two plane spanning paths on a point set in convex position. At the same time, we show that the happy edge property does not hold in this setting. This has to be seen in contrast to several results for reconfiguration problems where the absence of the happy edge property implies algorithmic hardness of the flip distance problem. Further, we show that our algorithm can be adapted for (1) compatible flips (2) local flips and (3) flips for plane spanning paths in simple polygons.