Rectilinear approximation of a set of points in the plane

We derive algorithms for approximating a set S of n points in the plane by an x-monotone rectilinear polyline with k horizontal segments. The quality of the approximation is measured by the maximum distance from a point in S to the segment above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for k horizontal segments and min-#, where the goal is to minimize the number of segments for error ε. After O(n) preprocessing time, we solve the latter in O(min{klogn, n}) time per instance. We then solve the former in O(min{n2, nklog n}) time. We also describe an approximation algorithm for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k–1 segments. Both approximations run in O(nlog n) time.