19,899 research outputs found
Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model
We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle â â C, the number of intersection points between the segments of A and those of B that lie in â. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n2/log2 n) logO(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+Δ), for any Δ > 0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a âhandicapâ that turns out to be beneficial for speeding up our algorithm.SCOPUS: cp.pinfo:eu-repo/semantics/publishe
Triangle areas in line arrangements
A widely investigated subject in combinatorial geometry, originated from
Erd\H{o}s, is the following. Given a point set of cardinality in the
plane, how can we describe the distribution of the determined distances? This
has been generalized in many directions. In this paper we propose the following
variants. Consider planar arrangements of lines. Determine the maximum
number of triangles of unit area, maximum area or minimum area, determined by
these lines. Determine the minimum size of a subset of these lines so that
all triples determine distinct area triangles.
We prove that the order of magnitude for the maximum occurrence of unit areas
lies between and . This result is strongly connected
to both additive combinatorial results and Szemer\'edi--Trotter type incidence
theorems. Next we show a tight bound for the maximum number of minimum area
triangles. Finally we present lower and upper bounds for the maximum area and
distinct area problems by combining algebraic, geometric and combinatorial
techniques.Comment: Title is shortened. Some typos and small errors were correcte
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
We formulate and give partial answers to several combinatorial problems on
volumes of simplices determined by points in 3-space, and in general in
dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by
points in \RR^3 is at most , and there are point sets
for which this number is . We also present an time
algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby
extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for
every k,d\in \NN, , the maximum number of -dimensional
simplices of minimum (nonzero) volume spanned by points in \RR^d is
. (ii) The number of unit-volume tetrahedra determined by
points in \RR^3 is , and there are point sets for which this
number is . (iii) For every d\in \NN, the minimum
number of distinct volumes of all full-dimensional simplices determined by
points in \RR^d, not all on a hyperplane, is .Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200
Cubic Partial Cubes from Simplicial Arrangements
We show how to construct a cubic partial cube from any simplicial arrangement
of lines or pseudolines in the projective plane. As a consequence, we find nine
new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
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