TY - JOUR

T1 - Area-preserving approximations of polygonal paths

AU - Bose, P.

AU - Cabello, S.

AU - Cheong, O.

AU - Gudmundsson, J.

AU - Kreveld, van, M.J.

AU - Speckmann, B.

PY - 2006

Y1 - 2006

N2 - Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P. Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q)+WB(Q)C. However, given P, an integer k, and a cost C, it is NP-hard to determine if a path Q with at most k edges exists such that max{WA(Q),WB(Q)}C. We describe an approximation algorithm for this setting. Finally, it is also NP-hard to decide whether a path Q exists such that |WA(Q)-WB(Q)|=0. Nevertheless, in this error measure we provide an algorithm for computing an optimal approximation up to an additive error.

AB - Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P. Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q)+WB(Q)C. However, given P, an integer k, and a cost C, it is NP-hard to determine if a path Q with at most k edges exists such that max{WA(Q),WB(Q)}C. We describe an approximation algorithm for this setting. Finally, it is also NP-hard to decide whether a path Q exists such that |WA(Q)-WB(Q)|=0. Nevertheless, in this error measure we provide an algorithm for computing an optimal approximation up to an additive error.

U2 - 10.1016/j.jda.2005.06.008

DO - 10.1016/j.jda.2005.06.008

M3 - Article

VL - 4

SP - 554

EP - 566

JO - Journal of Discrete Algorithms

JF - Journal of Discrete Algorithms

SN - 1570-8667

IS - 4

ER -