Optimal Point Set Partitioning using Rigid Motion Star Placement

We consider the problem of determining the placement of a star R on a set P of n points in the plane such that a given objective function is maximized. A star R is a set of m rays {r1, . . . , rm} in R, emanating from a point p such that the angle between two consecutive rays is 2π m . A cone defined by two consecutive rays is c-occupied if it contains at least c points of P. Our main result is an O(nm) expected time (and O(nm log nm) deterministic time) algorithm to find the rigid motion placement of R that maximizes the number of c-occupied cones. We then show how this technique can be extended to solve several other optimization problems.