TY - GEN

T1 - Almost tight bounds for eliminating depth cycles in three dimensions

AU - Aronov, Boris

AU - Sharir, Micha

N1 - Funding Information:
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.

PY - 2016/6/19

Y1 - 2016/6/19

N2 - Given n non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2 polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a consequence, we deduce that the number of pairwise nonoverlapping cycles, namely, cycles whose xy-projections do not overlap, is O(n3/2 polylog n); this bound too is almost tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir, or only cycles in grid-like patterns, by Chazelle et al.), and the bounds were considerably weaker-much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our approach extends to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles as well. Our results almost completely settle a long-standing (35 years old) open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

AB - Given n non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2 polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a consequence, we deduce that the number of pairwise nonoverlapping cycles, namely, cycles whose xy-projections do not overlap, is O(n3/2 polylog n); this bound too is almost tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir, or only cycles in grid-like patterns, by Chazelle et al.), and the bounds were considerably weaker-much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our approach extends to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles as well. Our results almost completely settle a long-standing (35 years old) open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

KW - Computational geometry

KW - Depth order

KW - Lines in space

KW - Painter's algorithm

KW - Polynomial partitioning

UR - http://www.scopus.com/inward/record.url?scp=84979224126&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979224126&partnerID=8YFLogxK

U2 - 10.1145/2897518.2897539

DO - 10.1145/2897518.2897539

M3 - Conference contribution

AN - SCOPUS:84979224126

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1

EP - 8

BT - STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing

A2 - Mansour, Yishay

A2 - Wichs, Daniel

PB - Association for Computing Machinery

T2 - 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016

Y2 - 19 June 2016 through 21 June 2016

ER -