7 research outputs found
Covering point sets with two disjoint disks or squares
Open archive-ElsevierWe study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB
with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized.
We give an algorithm to solve this problem in O(n8/3 log2 n) time, where n denotes the total number of points. We also show that
the analogous problem of finding two axis-aligned unit squares SR and SB instead of unit disks can be solved in O(nlog n) time,
which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation,
we give a solution using O(n3 log n) time
Tight Hardness Results for Maximum Weight Rectangles
Given weighted points (positive or negative) in dimensions, what is
the axis-aligned box which maximizes the total weight of the points it
contains?
The best known algorithm for this problem is based on a reduction to a
related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in
time . It was conjectured [Barbay et al., CCCG'13] that this runtime is
tight up to subpolynomial factors. We answer this conjecture affirmatively by
providing a matching conditional lower bound. We also provide conditional lower
bounds for the special case when points are arranged in a grid (a well studied
problem known as Maximum Subarray problem) as well as for other related
problems.
All our lower bounds are based on assumptions that the best known algorithms
for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique
problem in edge-weighted graphs are essentially optimal
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n â„ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Planar Case of the Maximum Box and Related Problems â
Given two finite sets of points X + and X â in R d,the maximum box problem is the problem of finding a box (hyperrectangle) B = {x: l †x †u} such that B â© X â = â
, and the cardinality of B â© X + is maximized. The maximum bichromatic discrepancy problem is to find a box B maximizing the difference between the number of the points of X + and X â inside the box, i.e. max ||B â© X + |â|B â© X â ||. In this paper, we discuss an exact algorithm for the maximum box problem on the plane. In addition, we provide factor 2 approximation algorithms for planar cases of both problems and give an extension to the numerical discrepancy problem.