15 research outputs found
On the pseudolinear crossing number
A drawing of a graph is {\em pseudolinear} if there is a pseudoline
arrangement such that each pseudoline contains exactly one edge of the drawing.
The {\em pseudolinear crossing number} of a graph is the minimum number of
pairwise crossings of edges in a pseudolinear drawing of . We establish
several facts on the pseudolinear crossing number, including its computational
complexity and its relationship to the usual crossing number and to the
rectilinear crossing number. This investigation was motivated by open questions
and issues raised by Marcus Schaefer in his comprehensive survey of the many
variants of the crossing number of a graph.Comment: 12 page
The Erd\H{o}s-Szekeres problem for non-crossing convex sets
We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th
about arrangements of planar convex bodies and a conjecture of Goodman and
Pollack about point sets in topological affine planes. As a corollary of this
equivalence we improve the upper bound of Pach and T\'{o}th on the
Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent
upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres
theorem for non-crossing convex bodies. Our methods also imply improvements on
the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and
non-crossing) convex bodies, as well as a generalization of the partitioned
Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing
convex bodies
Subquadratic Encodings for Point Configurations
For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension
Realization spaces of arrangements of convex bodies
We introduce combinatorial types of arrangements of convex bodies, extending
order types of point sets to arrangements of convex bodies, and study their
realization spaces. Our main results witness a trade-off between the
combinatorial complexity of the bodies and the topological complexity of their
realization space. First, we show that every combinatorial type is realizable
and its realization space is contractible under mild assumptions. Second, we
prove a universality theorem that says the restriction of the realization space
to arrangements polygons with a bounded number of vertices can have the
homotopy type of any primary semialgebraic set
An Improved Lower Bound on the Minimum Number of Triangulations
Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest:
(1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations.
(2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull.
(3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Regular systems of paths and families of convex sets in convex position
In this paper we show that every sufficiently large family of convex bodies
in the plane has a large subfamily in convex position provided that the number
of common tangents of each pair of bodies is bounded and every subfamily of
size five is in convex position. (If each pair of bodies have at most two
common tangents it is enough to assume that every triple is in convex position,
and likewise, if each pair of bodies have at most four common tangents it is
enough to assume that every quadruple is in convex position.) This confirms a
conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes
Toth. Our results on families of convex bodies are consequences of more general
Ramsey-type results about the crossing patterns of systems of graphs of
continuous functions . On our way towards proving the
Pach-Toth conjecture we obtain a combinatorial characterization of such systems
of graphs in which all subsystems of equal size induce equivalent crossing
patterns. These highly organized structures are what we call regular systems of
paths and they are natural generalizations of the notions of cups and caps from
the famous theorem of Erdos and Szekeres. The characterization of regular
systems is combinatorial and introduces some auxiliary structures which may be
of independent interest