### Abstract

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.

Original language | English |
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Pages (from-to) | 554-566 |

Journal | Journal of Discrete Algorithms |

Volume | 4 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2006 |

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## Cite this

Bose, P., Cabello, S., Cheong, O., Gudmundsson, J., Kreveld, van, M. J., & Speckmann, B. (2006). Area-preserving approximations of polygonal paths.

*Journal of Discrete Algorithms*,*4*(4), 554-566. https://doi.org/10.1016/j.jda.2005.06.008