A new algorithm for fitting a rectilinear x-monotone curve to a set of points in the plane

Abstract

Let S={p i=(x i,y i), i=1,2,…,n} , x 1< x 2<⋯< x n , be a set of n points in the plane and k be a positive integer. We give an O( n 2)-time algorithm for finding a rectilinear x-monotone curve R to fit S such that the number of links of R is not larger than k and the error of R with respect to S is minimized. In this paper, we will deal with the L ∞ measure of error [Comput. Graphics Image Process. 19 (1982) 248; Comput. Vision Graphics Image Process. 53 (1991) 132], i.e., the error of R with respect to S is defined to be the maximum vertical distance from the points of S to R. We take a plane sweep strategy with two sweeping lines. The previous best running time of this problem is O( n 2log n) [Eur. J. Oper. Res. 130 (2001) 214].