2 research outputs found
Constructive Polynomial Partitioning for Algebraic Curves in with Applications
In 2015, Guth proved that for any set of -dimensional bounded complexity
varieties in and for any positive integer , there exists a
polynomial of degree at most whose zero set divides into
open connected sets, so that only a small fraction of the given varieties
intersect each of these sets. 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 ,
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 bounded-degree algebraic
curves (or even lines) in .
We present an efficient algorithmic construction for this setting. Given a
set of input algebraic curves and a positive integer , we efficiently
construct a decomposition of space into open "cells," each of
which meets curves from the input. The construction time is
. For the case of lines in -space we present an improved
implementation, whose running time is . The constant
of proportionality in both time bounds depends on and the maximum degree of
the polynomials defining the input curves.
As an application, we revisit the problem of eliminating depth cycles among
non-vertical lines in -space, recently studied by Aronov and Sharir (2018),
and show an algorithm that cuts such lines into
pieces that are depth-cycle free, for any . The algorithm runs in
time, which is a considerable improvement over the
previously known algorithms.Comment: 20 pages, 0 figures. v2: final version, to appear in SIAM J. Comput.
A preliminary version of this work was presented in Proc. 30th Annual
ACM-SIAM Sympos. Discrete Algorithms, 201
Eliminating Depth Cycles among Triangles in Three Dimensions
Given pairwise openly disjoint triangles in 3-space, their vertical depth
relation may contain cycles. We show that, for any , the
triangles can be cut into connected semi-algebraic
pieces, whose description complexity depends only on the choice of
, such that the depth relation among these pieces is now a proper
partial order. This bound is nearly tight in the worst case. We are not aware
of any previous study of this problem, in this full generality, with a
subquadratic bound on the number of pieces.
This work extends the recent study by two of the authors (Aronov,
Sharir~2018) on eliminating depth cycles among lines in 3-space. Our approach
is again algebraic, and makes use of a recent variant of the polynomial
partitioning technique, due to Guth, which leads to a recursive procedure for
cutting the triangles. In contrast to the case of lines, our analysis here is
considerably more involved, due to the two-dimensional nature of the objects
being cut, so additional tools, from topology and algebra, need to be brought
to bear.
Our result essentially settles a 35-year-old open problem in computational
geometry, motivated by hidden-surface removal in computer graphics.Comment: 28 pages, 4 figure