Computing the Flip Distance Between Triangulations

Abstract

Let $\mathcal{{T}}$ be a triangulation of a set $\mathcal{{P}}$ of n points in the plane, and let e be an edge shared by two triangles in $\mathcal{{T}}$ such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of $\mathcal{{P}}$ from $\mathcal{{T}}$. The flip distance between two triangulations of $\mathcal{{P}}$ is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of $\mathcal{{P}}$ is at most k, for some given $k \in \mathbb {N}$. It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time $\mathcal {O}(n + k \cdot c^{k})$, for a constant $c \le 2 \cdot 14^{11}$, which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.