Representing a functional curve by curves with fewer peaks

Authors:
Danny Z. Chen;Chao Wang;Haitao Wang
Affiliations:
Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN;Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN;Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN
Venue:
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Year:
2010

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Abstract

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 min-ε version. For each of the UR and FPR problems, we study the min-k version and the min-ε version. Little work has been done previously on these problems. We solve all problems (except the UR min-ε) in optimal O(n) time, and the UR min-ε version in O(n+mlogm) time, where mn is the number of peaks of f. Our algorithms are based on new geometric observations and interesting techniques.