Tight Time Bounds for the Minimum Local Convex Partition Problem

Let v be a vertex with n edges incident to it, such that the n edges partition an infinitesimally small circle C around v into convex pieces. The minimum local convex partition (MLCP) problem asks for two or three out of the n edges that still partition C into convex pieces and that are of minimum total length. We present an optimal algorithm solving the problem in linear time if the edges incident to v are sorted clockwise by angle. For unsorted edges our algorithm runs in O(n log n) time. For unsorted edges we also give a linear time approximation algorithm and a lower time bound.