10,174 research outputs found
Largest Empty Circle Centered on a Query Line
The Largest Empty Circle problem seeks the largest circle centered within the
convex hull of a set of points in and devoid of points
from . In this paper, we introduce a query version of this well-studied
problem. In our query version, we are required to preprocess so that when
given a query line , we can quickly compute the largest empty circle
centered at some point on and within the convex hull of .
We present solutions for two special cases and the general case; all our
queries run in time. We restrict the query line to be horizontal in
the first special case, which we preprocess in time and
space, where is the slow growing inverse of the Ackermann's
function. When the query line is restricted to pass through a fixed point, the
second special case, our preprocessing takes time and space. We use insights from the two special cases to solve the
general version of the problem with preprocessing time and space in and respectively.Comment: 18 pages, 13 figure
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability
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