SWAT 2010: Algorithm Theory - SWAT 2010 pp 200-211

# Representing a Functional Curve by Curves with Fewer Peaks

• Danny Z. Chen
• Chao Wang
• Haitao Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

## 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 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Danny Z. Chen
• 1
• Chao Wang
• 1
• Haitao Wang
• 1
1. 1.Department of Computer Science and EngineeringUniversity of Notre DameNotre DameUSA