1,208 research outputs found
On the zone of the boundary of a convex body
We consider an arrangement \A of hyperplanes in and the zone
in \A of the boundary of an arbitrary convex set in in such an
arrangement. We show that, whereas the combinatorial complexity of is
known only to be \cite{APS}, the outer part of the zone has
complexity (without the logarithmic factor). Whether this bound
also holds for the complexity of the inner part of the zone is still an open
question (even for )
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
The Complexity of Order Type Isomorphism
The order type of a point set in maps each -tuple of points to
its orientation (e.g., clockwise or counterclockwise in ). Two point sets
and have the same order type if there exists a mapping from to
for which every -tuple of and the
corresponding tuple in have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an algorithm for
this task, thereby improving upon the algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on
Discrete Algorithms (SODA14
Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra
AbstractIn 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi’s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3)
Fast Algorithms for Geometric Consensuses
Let P be a set of n points in ?^d in general position. A median hyperplane (roughly) splits the point set P in half. The yolk of P is the ball of smallest radius intersecting all median hyperplanes of P. The egg of P is the ball of smallest radius intersecting all hyperplanes which contain exactly d points of P.
We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time O(n^(d-1) log n). The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others
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