7,197 research outputs found
Drawing the Horton Set in an Integer Grid of Minimum Size
In 1978 Erd\H os asked if every sufficiently large set of points in general
position in the plane contains the vertices of a convex -gon, with the
additional property that no other point of the set lies in its interior.
Shortly after, Horton provided a construction---which is now called the Horton
set---with no such -gon. In this paper we show that the Horton set of
points can be realized with integer coordinates of absolute value at most
. We also show that any set of points
with integer coordinates combinatorially equivalent (with the same order type)
to the Horton set, contains a point with a coordinate of absolute value at
least , where is a positive constant
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
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
Recommended from our members
Data structures for retrieval on integer grids
A family of data structures is presented for retrieval of the sum of values of points within a half-plane or polygon, given that the points are on integer coordinates in the plane. Fredman has shown that the problem has a lower bound of Ω(N^2/3) for intermixed updates and retrievals. Willard has shown an upper bound of O(N^2log6^4) for the case where the points are not restricted to integer coordinates.We have developed families of related data structures for retrievals of half-planes or polygons. One of the data structures permits intermixed updates and half-plane retrievals in O(N^2/3log N) time, where N is the size of the grid.We use a technique we call "Rotation" to permit a better match of a portion of the data structure to the particular problem. Rotations appear to be an effective method for trading-off storage redundancy against retrieval time for certain classes of problems
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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