2,538 research outputs found
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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
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