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Constructive Polynomial Partitioning for Lines in R^3: Revisiting Depth Cycle Elimination Ezra, Esther
Description
A recent extension of Guth (2015) to the basic polynomial partitioning technique of Guth and Katz (2015) shows the existence of a partitioning polynomial, for a given set of $k$-dimensional varieties in ${\reals}^d$, which subdivides space into open cells each of which meeting only a small fraction of the total number of varieties. For most instances, it is unknown how to obtain an explicit representation of such a partitioning polynomial (and how to construct it efficiently). This, in particular, applies to the (simple) case of lines in 3-space. In this work we present an efficient algorithmic (but somewhat suboptimal) construction for this setting, under the assumption that the lines are non-vertical and pairwise disjoint. We then revisit the problem of eliminating depth cycles among $n$ non-vertical pairwise disjoint triangles in $3$-space, recently been studied by Aronov etal. (2017) and de Berg (2017). Our main result is an algorithmic $O(n^{5/3+\eps})$ bound, for any $\eps > 0$, on the number of pieces one needs to cut the triangles such that the depth relation they induce does not contain cycles. Joint work with Boris Aronov.
Item Metadata
Title |
Constructive Polynomial Partitioning for Lines in R^3: Revisiting Depth Cycle Elimination
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-02-08T13:33
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Description |
A recent extension of Guth (2015) to the basic polynomial partitioning technique of Guth and Katz (2015) shows the existence of a partitioning polynomial, for a given set of $k$-dimensional varieties in ${\reals}^d$, which subdivides space into open cells each of which meeting only a small fraction of the total number of varieties. For most instances, it is unknown how to obtain an explicit representation of such a partitioning polynomial (and how to construct it efficiently). This, in particular, applies to the (simple) case of lines in 3-space. In this work we present an efficient algorithmic (but somewhat suboptimal) construction for this setting, under the assumption that the lines are non-vertical and pairwise disjoint. We then revisit the problem of eliminating depth cycles among $n$ non-vertical pairwise disjoint triangles in $3$-space, recently been studied by Aronov etal. (2017) and de Berg (2017). Our main result is an algorithmic $O(n^{5/3+\eps})$ bound, for any $\eps > 0$, on the number of pieces one needs to cut the triangles such that the depth relation they induce does not contain cycles.
Joint work with Boris Aronov.
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Extent |
37 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Georgia Tech
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Series | |
Date Available |
2018-08-08
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0369741
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International