Computing the Flip Distance Between Triangulations

Let $$\mathcal{{T}}$$T be a triangulation of a set $$\mathcal{{P}}$$P of n points in the plane, and let e be an edge shared by two triangles in $$\mathcal{{T}}$$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 $$\mathcal{{P}}$$P from $$\mathcal{{T}}$$T. The flip distance between two triangulations of $$\mathcal{{P}}$$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 $$\mathcal{{P}}$$P is at most k, for some given $$k \in \mathbb {N}$$kźN. It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time $$\mathcal {O}(n + k \cdot c^{k})$$O(n+k·ck), for a constant $$c \le 2 \cdot 14^{11}$$c≤2·1411, 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.