Abstract
Given a set P of points in the Euclidean plane and two triangulations of P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O⁎(k⋅ck) (c≤2×1411), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By analyzing the underlying properties of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O⁎(k⋅32k).
Highlights
Given a set P of n points in the Euclidean plane, a triangulation of P is a maximal planar subdivision whose vertex set is P [10]
Triangulations play an important role in computational geometry, which are applied in areas such as computer-aided geometric design and numerical analysis [11, 13, 21]
In this paper we presented an fixed-parameter tractable (FPT) algorithm running in time O∗(k·32k) for the Parameterized Flip Distance problem, improving the previous O∗(k · ck)-time (c ≤ 2 × 1411) FPT algorithm by Kanj and Xia [15]
Summary
Given a set P of n points in the Euclidean plane, a triangulation of P is a maximal planar subdivision whose vertex set is P [10]. 65:2 An Improved FPT Algorithm for the Flip Distance Problem corresponding triangulations can be transformed into each other through one flip operation. The Parameterized Flip Distance problem is: given two triangulations of a set of points in the plane and an integer k, deciding if the flip distance between these two triangulations is equal to k. For the Parameterized Flip Distance problem on triangulations of a convex polygon, Lucas [19] gave a kernel of size 2k and an O∗(kk)-time algorithm. Kanj and Xia [15] studied the Parameterized Flip Distance problem on triangulations of a set of points in the plane, and presented an O∗(k · ck)-time algorithm (c ≤ 2 · 1411), which applies to triangulations of general polygonal regions (even with holes or points inside it). We get an improved O∗(k · 32k)-time FPT algorithm, which applies to triangulations of general polygonal regions (even with holes or points inside it)
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