14 research outputs found
Matching random colored points with rectangles
Let S ¿ [0, 1]2 be a set of n points, randomly and uniformly selected. Let R ¿ B be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random number M(n) of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n) = 0.83 n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Postprint (published version
10-Gabriel graphs are Hamiltonian
Given a set of points in the plane, the -Gabriel graph of is the
geometric graph with vertex set , where are connected by an
edge if and only if the closed disk having segment as diameter
contains at most points of . We consider the
following question: What is the minimum value of such that the -Gabriel
graph of every point set contains a Hamiltonian cycle? For this value, we
give an upper bound of 10 and a lower bound of 2. The best previously known
values were 15 and 1, respectively
Matching random colored points with rectangles
Let S[0,1]2 be a set of n points, randomly and uniformly selected. Let RB be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n)=0.83n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Research supported by projects MTM2015-63791-R MINECO/FEDER and Gen.
Cat. DGR 2017SGR1640Postprint (author's final draft
Matching points with disks with a common intersection
We consider matchings with diametral disks between two sets of points R and
B. More precisely, for each pair of matched points p in R and q in B, we
consider the disk through p and q with the smallest diameter. We prove that for
any R and B such that |R|=|B|, there exists a perfect matching such that the
diametral disks of the matched point pairs have a common intersection. In fact,
our result is stronger, and shows that a maximum weight perfect matching has
this property
Recommended from our members
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry