5,810 research outputs found
Detecting all regular polygons in a point set
In this paper, we analyze the time complexity of finding regular polygons in
a set of n points. We combine two different approaches to find regular
polygons, depending on their number of edges. Our result depends on the
parameter alpha, which has been used to bound the maximum number of isosceles
triangles that can be formed by n points. This bound has been expressed as
O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068.
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through
the set of points, while larger polygons are found by random sampling. We can
find all regular polygons with high probability in O(n^{2+alpha+epsilon})
expected time for every positive epsilon. This compares well to the
O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.Comment: 11 pages, 4 figure
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
From symmetry breaking to Poisson Point Process in 2D Voronoi Tessellations: the generic nature of hexagons
We bridge the properties of the regular triangular, square, and hexagonal honeycomb
Voronoi tessellations of the plane to the Poisson-Voronoi case, thus analyzing in
a common framework symmetry breaking processes and the approach to uniform random
distributions of tessellation-generating points. We resort to ensemble simulations of tessellations
generated by points whose regular positions are perturbed through a Gaussian noise,
whose variance is given by the parameter α2 times the square of the inverse of the average
density of points. We analyze the number of sides, the area, and the perimeter of the
Voronoi cells. For all valuesα >0, hexagons constitute the most common class of cells, and
2-parameter gamma distributions provide an efficient description of the statistical properties
of the analyzed geometrical characteristics. The introduction of noise destroys the triangular
and square tessellations, which are structurally unstable, as their topological properties are
discontinuous in α = 0. On the contrary, the honeycomb hexagonal tessellation is topologically
stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise
withα <0.12. For all tessellations and for small values of α, we observe a linear dependence
on α of the ensemble mean of the standard deviation of the area and perimeter of the cells.
Already for a moderate amount of Gaussian noise (α >0.5), memory of the specific initial
unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations
are indistinguishable. When α >2, results converge to those of Poisson-Voronoi
tessellations. The geometrical properties of n-sided cells change with α until the Poisson-
Voronoi limit is reached for α > 2; in this limit the Desch law for perimeters is shown to
be not valid and a square root dependence on n is established. This law allows for an easy
link to the Lewis law for areas and agrees with exact asymptotic results. Finally, for α >1,
the ensemble mean of the cells area and perimeter restricted to the hexagonal cells agree
remarkably well with the full ensemble mean; this reinforces the idea that hexagons, beyond
their ubiquitous numerical prominence, can be interpreted as typical polygons in 2D
Voronoi tessellations
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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