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.

翻译：设$P$为平面上的一个凸多边形，$T$为$P$的一个三角剖分。若$T$中的一条边$e$被$T$中的两个三角形共享，则称$e$为一条对角线。翻转一条对角线$e$的操作是指：删除$e$，并添加所得四边形的另一条对角线，从而从$T$得到$P$的一个新三角剖分。$P$的两个三角剖分之间的翻转距离是指将一个三角剖分转换为另一个所需的最少翻转次数。凸翻转距离问题询问：对于给定参数$k$，$P$的两个给定三角剖分之间的翻转距离是否至多为$k$。我们提出一个针对凸翻转距离问题的FPT算法，其运行时间为$O(3.82^k)$，并使用多项式空间，其中$k$为翻转次数。该算法显著改进了该问题此前最佳的FPT算法。