Uniformly monotone partitioning of polygons

We study the problem of partitioning a simple polygon P with n vertices (including R reflex vertices) and no holes into a minimum number of uniformly monotone subpolygons using diagonals that lie inside P , each connecting two distinct vertices of P . A set of polygons is uniformly monotone if there exists a line such that all of the polygons are monotone with respect to the line. We present an O ( nR log n + R 5 )-time algorithm for the problem. When Steiner points can be placed on the boundary of P and each Steiner point is considered as a vertex of P , we present an O ( n + R 5 )-time algorithm for the problem. We present an O ( n + R 4 )-time algorithm when Steiner points can be placed anywhere in P . Our algorithms improve upon the previously best ones. We also present simple and efficient 2-approximation algorithms.