# Representing a Functional Curve by Curves with Fewer Peaks

## 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* + *m*log*m*) time, where *m* < *n* is the number of peaks of **f**. Our algorithms are based on new geometric observations and interesting techniques.

## Keywords

Linear Time Algorithm Skeleton Curve Dose Function Piecewise Linear Curve Geometric Observation## Preview

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