16 research outputs found
4-Holes in point sets
We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft
The visible perimeter of an arrangement of disks
Given a collection of n opaque unit disks in the plane, we want to find a
stacking order for them that maximizes their visible perimeter---the total
length of all pieces of their boundaries visible from above. We prove that if
the centers of the disks form a dense point set, i.e., the ratio of their
maximum to their minimum distance is O(n^1/2), then there is a stacking order
for which the visible perimeter is Omega(n^2/3). We also show that this bound
cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform
grid. On the other hand, if the set of centers is dense and the maximum
distance between them is small, then the visible perimeter is O(n^3/4) with
respect to any stacking order. This latter bound cannot be improved either.
Finally, we address the case where no more than c disks can have a point in
common. These results partially answer some questions of Cabello, Haverkort,
van Kreveld, and Speckmann.Comment: 12 pages, 5 figure
Two trees are better than one
We consider partitions of a point set into two parts, and the lengths of the
minimum spanning trees of the original set and of the two parts. If
denotes the length of a minimum spanning tree of , we show that every set
of points admits a bipartition for which the
ratio is strictly larger than ; and that is the
largest number with this property. Furthermore, we provide a very fast
algorithm that computes such a bipartition in time and one that computes
the corresponding ratio in time. In certain settings, a ratio
larger than can be expected and sometimes guaranteed. For example, if
is a set of random points uniformly distributed in (), then for any \eps>0, the above ratio in a maximizing partition is
at least \sqrt2 -\eps with probability tending to . As another example, if
is a set of points with spread at most , for some
constant , then the aforementioned ratio in a maximizing partition is
. All our results and techniques are extendable to
higher dimensions.Comment: 2 figures, 11 page
On -Gons and -Holes in Point Sets
We consider a variation of the classical Erd\H{o}s-Szekeres problems on the
existence and number of convex -gons and -holes (empty -gons) in a set
of points in the plane. Allowing the -gons to be non-convex, we show
bounds and structural results on maximizing and minimizing their numbers. Most
noteworthy, for any and sufficiently large , we give a quadratic lower
bound for the number of -holes, and show that this number is maximized by
sets in convex position
Every Large Point Set contains Many Collinear Points or an Empty Pentagon
We prove the following generalised empty pentagon theorem: for every integer
, every sufficiently large set of points in the plane contains
collinear points or an empty pentagon. As an application, we settle the
next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and
Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]
Two-sided convexity testing with certificates
We revisit the problem of property testing for convex position for point sets
in . Our results draw from previous ideas of Czumaj, Sohler, and
Ziegler (ESA 2000). First, the algorithm is redesigned and its analysis is
revised for correctness. Second, its functionality is expanded by
(i)~exhibiting both negative and positive certificates along with the convexity
determination, and (ii)~significantly extending the input range for moderate
and higher dimensions. The behavior of the randomized tester is as follows:
(i)~if is in convex position, it accepts; (ii)~if is far from convex
position, with probability at least , it rejects and outputs a
-point witness of non-convexity as a negative certificate; (iiii)~if
is close to convex position, with probability at least , it accepts and
outputs an approximation of the largest subset in convex position. The
algorithm examines a sublinear number of points and runs in subquadratic time
for every dimension (and is faster in low dimensions).Comment: 14 pages, 2 figure
Convex polytopes in restricted point sets in
For a finite point set , denote by
the ratio of the largest to the smallest distances between pairs of points in
. Let be the largest integer such that any -point
set in general position, satisfying , contains an -point convex independent subset. We
determine the asymptotics of as by showing
the existence of positive constants and such that for .Comment: 20 pages, 1 figur
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm