6,306 research outputs found
On Ray Shooting for Triangles in 3-Space and Related Problems
We consider several problems that involve lines in three dimensions, and
present improved algorithms for solving them. The problems include (i) ray
shooting amid triangles in , (ii) reporting intersections between query
lines (segments, or rays) and input triangles, as well as approximately
counting the number of such intersections, (iii) computing the intersection of
two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections
in a set of lines in , and (v) output-sensitive construction of an
arrangement of triangles in three dimensions.
Our approach is based on the polynomial partitioning technique.
For example, our ray-shooting algorithm processes a set of triangles in
into a data structure for answering ray shooting queries amid the given
triangles, which uses storage and preprocessing, and
answers a query in time, for any . This
is a significant improvement over known results, obtained more than 25 years
ago, in which, with this amount of storage, the query time bound is roughly
. The algorithms for the other problems have similar performance
bounds, with similar improvements over previous results.
We also derive a nontrivial improved tradeoff between storage and query time.
Using it, we obtain algorithms that answer queries on objects in time, for any
, again an improvement over the earlier bounds.Comment: 33 pages, 7 figure
Distance-Sensitive Planar Point Location
Let be a connected planar polygonal subdivision with edges
that we want to preprocess for point-location queries, and where we are given
the probability that the query point lies in a polygon of
. We show how to preprocess such that the query time
for a point~ depends on~ and, in addition, on the distance
from to the boundary of~---the further away from the boundary, the
faster the query. More precisely, we show that a point-location query can be
answered in time , where
is the shortest Euclidean distance of the query point~ to the
boundary of . Our structure uses space and
preprocessing time. It is based on a decomposition of the regions of
into convex quadrilaterals and triangles with the following
property: for any point , the quadrilateral or triangle
containing~ has area . For the special case where
is a subdivision of the unit square and
, we present a simpler solution that achieves a
query time of . The latter solution can be extended to
convex subdivisions in three dimensions
Weak Visibility Queries of Line Segments in Simple Polygons
Given a simple polygon P in the plane, we present new algorithms and data
structures for computing the weak visibility polygon from any query line
segment in P. We build a data structure in O(n) time and O(n) space that can
compute the visibility polygon for any query line segment s in O(k log n) time,
where k is the size of the visibility polygon of s and n is the number of
vertices of P. Alternatively, we build a data structure in O(n^3) time and
O(n^3) space that can compute the visibility polygon for any query line segment
in O(k + log n) time.Comment: 16 pages, 9 figures. A preliminary version of this paper appeared in
ISAAC 2012 and we have improved results in this full versio
A technique for adding range restrictions to generalized searching problems
In a generalized searching problem, a set of colored geometric objects has to be stored in a data structure, such that for any given query object , the distinct colors of the objects of intersected by can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp.\ fat triangles) with a fat triangle (resp.\ point). For both problems, a data structure is obtained having size and query time . Here, denotes the number of colors reported by the query, and is an arbitrarily small positive constant
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