657 research outputs found
A new characterization of the center of a polytope
The main contribution of this work is the introduction of a new function which has the analytic center of a polytope as its maximizer. At the function's optimal point, it assumes a value equal to m, the total number of constraints used to define the polytope. For this reason we call it the m-function of the polytope. We also prove that given a p-dimensional face of a nondegenerate polytope the m-function for that polytope assumes the value m-(n-p) at the analytic center of the face. In particular the m-function assumes the value m at the analytic center of the polytope.16318520
Smooth Distance Approximation
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities.
In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation.
We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in ?^d from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute ?-approximation to this query in time O(log (1/?)) using O(1/?^{d/2}) storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian
Smooth Distance Approximation
Traditional problems in computational geometry involve aspects that are both
discrete and continuous. One such example is nearest-neighbor searching, where
the input is discrete, but the result depends on distances, which vary
continuously. In many real-world applications of geometric data structures, it
is assumed that query results are continuous, free of jump discontinuities.
This is at odds with many modern data structures in computational geometry,
which employ approximations to achieve efficiency, but these approximations
often suffer from discontinuities.
In this paper, we present a general method for transforming an approximate
but discontinuous data structure into one that produces a smooth approximation,
while matching the asymptotic space efficiencies of the original. We achieve
this by adapting an approach called the partition-of-unity method, which
smoothly blends multiple local approximations into a single smooth global
approximation.
We illustrate the use of this technique in a specific application of
approximating the distance to the boundary of a convex polytope in
from any point in its interior. We begin by developing a novel
data structure that efficiently computes an absolute
-approximation to this query in time
using storage space. Then, we proceed to apply the
proposed partition-of-unity blending to guarantee the smoothness of the
approximate distance field, establishing optimal asymptotic bounds on the norms
of its gradient and Hessian.Comment: To appear in the European Symposium on Algorithms (ESA) 202
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