Computing polygonal path simplification under area measures

Abstract

In this paper, we consider the restricted version of the well-known 2D line simplification problem under area measures and for restricted version. We first propose a unified definition for both of sum-area and difference-area measures that can be used on a general path of n vertices. The optimal simplification runs in O(n3) under both of these measures. Under sum-area measure and for a realistic input path, we propose an approximation algorithm of On2ϵ time complexity to find a simplification of the input path, where ϵ is the absolute error of this algorithm compared to the optimal solution. Furthermore, for difference-area measure, we present an algorithm that finds the optimal simplification in O(n2) time. The best previous results work only on x-monotone paths while both of our algorithms work on general paths. To the best of our knowledge, the results presented here are the first sub-cubic simplification algorithms on these measures for general paths.