Representing a Functional Curve by Curves with Fewer Peaks
نویسندگان
چکیده
We study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Let f be an input nonnegative piecewise linear functional curve of size n. We consider the following problems. (1) Uphill-downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing and one nonincreasing, such that their sum approximately represents f . (2) Unimodal representation (UR): Find a set of k nonnegative unimodal (single-peak) curves such that their sum approximately represents f . (3) Fewer-peak representation (FPR): Find a nonnegative piecewise linear curve with at most k peaks that approximately represents f . For each problem, we consider two versions. For UDPR, we study the feasibility version and the minversion. For each of the UR and FPR problems, we study the min-k version and the minversion. Little work has been done previously on these problems. We solve all problems (except the UR min) in optimal O(n) time, and the UR minversion in O(n+m logm) time, where m < n is the number of peaks of f . Our algorithms are based on new geometric observations and interesting techniques.
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