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Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions

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Abstract

Given n pairwise disjoint non-vertical lines in 3-space, their vertical depth (i.e., above/below) relation may contain cycles. We show that the lines can be cut into \(O(n^{3/2}{{\mathrm{polylog}}}\, n)\) pieces, such that the depth relation among these pieces is a proper partial order. This bound is nearly tight in the worst case. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. Our technique can be extended to eliminating all cycles in the depth relation among segments and among constant-degree algebraic arcs. Our results almost completely settle a 35-year-old open problem in computational geometry motivated by hidden-surface removal in computer graphics. We also discuss several algorithms for constructing a small set of cuts so as to eliminate all depth-relation cycles among the lines.

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Notes

  1. That is, the points, at which we cut the lines, are removed and do not belong to the resulting pieces.

  2. The correctness of this naïve procedure is immediate when the lines are in general position and requires a short justification when they are not. The argument is offered in Sect. 2.

  3. Note that, by definition, these “limit” cuts, at the endpoints of \(e_i\) and \(e_{i+1}\), do eliminate the cycle.

  4. For the algorithmic part of the analysis, discussed in Sect. 3 below, it is preferable to work with constant degree D, which is why, up to this point, we have not committed to a particular choice of \(D=D(n)\); see the remark below and Sect. 3 for more details.

  5. The assumption of general position is not really necessary, but, to simplify the presentation, we refrain from spelling out all the details needed to handle degeneracies, both here and below, and so state the results for general position only.

  6. This extension of the analysis of [12], which had originally been done explicitly only for the case of segments, to more general well-behaved arcs, appears to be considered folklore (see, e.g., [1, 18]); refer to the discussion of this issue in the subsequent paper [22].

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Acknowledgements

We thank an anonymous referee for many helpful comments on the paper.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, 11201, USA

    Boris Aronov

  2. Blavatnik School of Computer Science, Tel Aviv University, 69978, Tel-Aviv, Israel

    Micha Sharir

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  1. Boris Aronov
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  2. Micha Sharir
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Correspondence to Boris Aronov.

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Editor in Charge: János Pach

Work on this paper by B.A. has been partially supported by NSF Grants CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by BSF Grant 2014/170. Work by M.S. has been supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper has appeared in Proceedings of the 48th Annual ACM Symposium on Theory of Computing, 2016, 1–8 [6].

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Aronov, B., Sharir, M. Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions. Discrete Comput Geom 59, 725–741 (2018). https://doi.org/10.1007/s00454-017-9920-9

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