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.

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doi.org/10.1016/j.jda.2005.06.008
Journal of Discrete Algorithms
School of Computer Science

Bose, P, Cabello, S. (Sergio), Cheong, O. (Otfried), Gudmundsson, J. (Joachim), van Kreveld, M. (Marc), & Speckmann, B. (Bettina). (2006). Area-preserving approximations of polygonal paths. Journal of Discrete Algorithms, 4(4), 554–566. doi:10.1016/j.jda.2005.06.008