42 research outputs found
On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs
We show that the maximum cardinality of an anti-chain composed of
intersections of a given set of n points in the plane with half-planes is close
to quadratic in n. We approach this problem by establishing the equivalence
with the problem of the maximum monotone path in an arrangement of n lines. For
a related problem on antichains in families of convex pseudo-discs we can
establish the precise asymptotic bound: it is quadratic in n. The sets in such
a family are characterized as intersections of a given set of n points with
convex sets, such that the difference between the convex hulls of any two sets
is nonempty and connected.Comment: 10 pages, 3 figures. revised version correctly attributes the idea of
Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in
the paper and the title. Accepted for publication in Israel Journal of
Mathematic
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Fission of Halving Edges Graphs
In this paper we discuss an operation on halving edges graph that we call
fission. Fission replaces each point in a given configuration with a small
cluster of k points. The operation interacts nicely with halving edges, so we
examine its properties in detail.Comment: 17 pages, 10 figure
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201