43 research outputs found
On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions
Let be a collection of points in the plane, each moving along some
straight line at unit speed. We obtain an almost tight upper bound of
, for any , on the maximum number of discrete
changes that the Delaunay triangulation of experiences
during this motion. Our analysis is cast in a purely topological setting, where
we only assume that (i) any four points can be co-circular at most three times,
and (ii) no triple of points can be collinear more than twice; these
assumptions hold for unit speed motions.Comment: 138 pages+ Appendix of 7 pages. A preliminary version has appeared in
Proceedings of the 54th Annual Symposium on Foundations of Computer Science
(FOCS 2013). The paper extends the result of http://arxiv.org/abs/1304.3671
to more general motions. The presentation is self-contained with main ideas
delivered in Sections 1--
An Improved Bound for Weak Epsilon-Nets in the Plane
We show that for any finite set of points in the plane and
there exist
points in , for arbitrary small , that pierce every
convex set with . This is the first improvement
of the bound of that was
obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general
point sets in the plane.Comment: A preliminary version to appear in the proceedings of FOCS 201
On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves
A long standing conjecture of Richter and Thomassen states that the total
number of intersection points between any simple closed Jordan curves in
the plane, so that any pair of them intersect and no three curves pass through
the same point, is at least .
We confirm the above conjecture in several important cases, including the
case (1) when all curves are convex, and (2) when the family of curves can be
partitioned into two equal classes such that each curve from the first class is
touching every curve from the second class. (Two curves are said to be touching
if they have precisely one point in common, at which they do not properly
cross.)
An important ingredient of our proofs is the following statement: Let be
a family of the graphs of continuous real functions defined on
, no three of which pass through the same point. If there are
pairs of touching curves in , then the number of crossing points is
.Comment: To appear in SODA 201
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let be a set of points and a convex -gon in .
We analyze in detail the topological (or discrete) changes in the structure of
the Voronoi diagram and the Delaunay triangulation of , under the convex
distance function defined by , as the points of move along prespecified
continuous trajectories. Assuming that each point of moves along an
algebraic trajectory of bounded degree, we establish an upper bound of
on the number of topological changes experienced by the
diagrams throughout the motion; here is the maximum length of an
-Davenport-Schinzel sequence, and is a constant depending on the
algebraic degree of the motion of the points. Finally, we describe an algorithm
for efficiently maintaining the above structures, using the kinetic data
structure (KDS) framework