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

Let ${\cal P}$ be a convex polygon in the plane, and let ${\cal T}$ be a triangulation of ${\cal P}$. An edge $e$ in ${\cal T}$ is called a diagonal if it is shared by two triangles in ${\cal T}$. A {\em 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 ${\cal P}$ from ${\cal T}$. The {\em flip distance} between two triangulations of ${\cal P}$ is the minimum number of flips needed to transform one triangulation into the other. The {\sc Convex Flip Distance} problem asks if the flip distance between two given triangulations of ${\cal P}$ is at most $k$, for some given parameter $k$. We present an FPT algorithm for the {\sc 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.