An $O(3.82^k)$ Time FPT Algorithm for Convex Flip Distance

TL;DR

This paper introduces a fixed-parameter tractable algorithm with a runtime of O(3.82^k) for computing the minimum number of flips needed to transform one triangulation of a convex polygon into another, improving previous methods.

Contribution

The paper presents the first FPT algorithm with a runtime of O(3.82^k) for the convex flip distance problem, advancing computational efficiency.

Findings

Algorithm runs in O(3.82^k) time

Uses polynomial space

Significantly improves previous algorithms

Abstract

Let $P$ be a convex polygon in the plane, and let $T$ be a triangulation of $P$. An edge $e$ in $T$ is called a diagonal if it is shared by two triangles in $T$. A flip of a diagonal $e$ is the operation of removing $e$ and adding the opposite diagonal of the resulting quadrilateral to obtain a new triangulation of $P$ from $T$. The flip distance between two triangulations of $P$ is the minimum number of flips needed to transform one triangulation into the other. The Convex Flip Distance problem asks if the flip distance between two given triangulations of $P$ is at most $k$, for some given parameter $k$. We present an FPT algorithm for the Convex Flip Distance problem that runs in time $O(3.82^k)$ and uses polynomial space, where $k$ is the number of flips. This algorithm significantly improves the previous best FPT algorithms for the problem.