2,285 research outputs found
Incidences between points and lines in three dimensions
We give a fairly elementary and simple proof that shows that the number of
incidences between points and lines in , so that no
plane contains more than lines, is (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between and ).
This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions
Dominance Product and High-Dimensional Closest Pair under
Given a set of points in , the Closest Pair problem is
to find a pair of distinct points in at minimum distance. When is
constant, there are efficient algorithms that solve this problem, and fast
approximate solutions for general . However, obtaining an exact solution in
very high dimensions seems to be much less understood. We consider the
high-dimensional Closest Pair problem, where for some , and the underlying metric is .
We improve and simplify previous results for Closest Pair, showing
that it can be solved by a deterministic strongly-polynomial algorithm that
runs in time, and by a randomized algorithm that runs in
expected time, where is the time bound for computing the
{\em dominance product} for points in . That is a matrix ,
such that ; this is the
number of coordinates at which dominates . For integer coordinates
from some interval , we obtain an algorithm that runs in
time, where
is the exponent of multiplying an matrix by an
matrix.
We also give slightly better bounds for , by using more recent
rectangular matrix multiplication bounds. Computing the dominance product
itself is an important task, since it is applied in many algorithms as a major
black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths),
and variants of APSP (all pairs shortest paths)
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
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