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Ham-Sandwich Cuts for Abstract Order Types


The linear-time ham-sandwich cut algorithm of Lo, Matoušek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen vertical lines. We consider the setting where we are not given the coordinates of the point set, but only the orientation of each point triple (the order type) and give a deterministic linear-time algorithm for the mentioned sub-algorithm. This yields a linear-time ham-sandwich cut algorithm even in our restricted setting. We also show that our methods are applicable to abstract order types.

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    Lo et al. [30] refer to [32], where [27, Lemma 4.5] (Lemma 12 herein) is used, and also refer to [33] in this context, where a general algorithm for constructing \(\varepsilon \)-approximations is given.


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Author information

Correspondence to Alexander Pilz.

Additional information

This work has been supported by the ESF EUROCORES programme EuroGIGA-ComPoSe. A.P. is supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. A preliminary version of this paper appeared in the proceedings of ISAAC 2014 [17]. Part of this work was presented in the PhD thesis [40] of the second author.

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Felsner, S., Pilz, A. Ham-Sandwich Cuts for Abstract Order Types. Algorithmica 80, 234–257 (2018).

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  • Ham-sandwich cut
  • Abstract order type
  • Pseudo-line arrangement