Algorithms on minimizing the maximum sensor movement for barrier coverage of a linear domain
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let $\bf f$ be an input piecewise linear functional curve of size n. We consider several variations of the problems. (1) Uphill–downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents $\bf f$. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents $\bf f$. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents $\bf f$. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given ε0, determine whether there is a feasible UDPR solution for $\bf f$ with an approximation error ε; its min-ε version: Compute the minimum approximation error ε ∗ such that there is a feasible UDPR solution for $\bf f$ with error ε ∗. For the UR problem, we study its min-k version: Given ε0, find a feasible solution with the minimum number k ∗ of unimodal curves for $\bf f$ with an error ε; its min-ε version: given k0, compute the minimum error ε ∗ such that there is a feasible solution with at most k unimodal curves for $\bf f$ with error ε ∗. For the FPR problem, we study its min-k version: Given ε0, find one feasible curve with the minimum number k ∗ of peaks for $\bf f$ with an error ε; its min-ε version: given k≥0, compute the minimum error ε ∗ such that there is a feasible curve with at most k peaks for $\bf f$ with error ε ∗. Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-ε version) in optimal O(n) time, and the UR min-ε version in O(n+mlog m) time, where mn is the number of peaks of $\bf f$. Our algorithms are based on new geometric observations and interesting techniques.