Flip Distance Is in FPT Time O(n plus k . c(k))
Leibniz International Proceedings in Informatics(2015)
摘要
Let T be a triangulation of a set P of n points in the plane, and let e be an edge shared by two triangles in 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 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 FLIP DISTANCE problem asks if the flip distance between two given triangulations of P is k, for some given k is an element of N. It is a fundamental and a challenging problem. In this paper we present an algorithm for the FLIP DISTANCE problem that runs in time O(n+ k . c(k)), for a constant c <= 2.14(11), which implies that the problem is fixed-parameter tractable. The NP-hardness reduction for the FLIP DISTANCE problem given by Lubiw and Pathak can be used to show that, unless the exponential-time hypothesis (ETH) fails, the FLIP DISTANCE problem cannot be solved in time O*(2 degrees((k))). Therefore, one cannot expect an asymptotic improvement in the exponent of the running time of our algorithm.
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关键词
triangulations,flip distance,parameterized algorithms
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