Abstract
In 2015, Guth proved that, for any set of k-dimensional varieties in Rdand for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides Rdinto open connected “cells,” so that only a small fraction of the given varieties intersect each cell. Guth’s result generalized an earlier result of Guth and Katz for points. Guth’s proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R3. We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D3log3D) open cells, each of which meets at most O(n/D2) curves from the input. The construction time is O(n2), where the constant of proportionality depends on D and the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation, whose running time is O(n4/3polylog n). As an application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently studied by Aronov et al. (2017) and De Berg (2017). Our main result is an algorithm that cuts n triangles into O(n3/2+ε) pieces that are depth cycle free, for any ε > 0. The algorithm runs in O(n3/2+ε) time, which is nearly worst-case optimal.
Original language | English |
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Pages | 2636-2648 |
Number of pages | 13 |
State | Published - 2019 |
Event | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: 6 Jan 2019 → 9 Jan 2019 |
Conference
Conference | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 |
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Country/Territory | United States |
City | San Diego |
Period | 6/01/19 → 9/01/19 |
Bibliographical note
Publisher Copyright:Copyright © 2019 by SIAM.
Funding
Work on this paper by Boris Aronov has been supported by NSA MSP Grant H98230-10-1-0210, by NSF Grants CCF-08-30691, CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by BSF grant 2014/170. Work on this paper by Esther Ezra has been supported by NSF CAREER under grant CCF:AF 1553354 and by Grant 824/17 from the Israel Science Foundation. Work on this paper by Joshua Zahl was supported by a NSERC Discovery grant.
Funders | Funder number |
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National Science Foundation | CCF-08-30691, CCF-11-17336, CCF-12-18791, CCF-15-40656 |
National Security Agency | H98230-10-1-0210 |
Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences | 1553354 |