Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Covering of high-dimensional cubes and quantization   As the main problem, we consider covering of a $d$-dimensional cube by $n$
balls with reasonably large $d$ (10 or more) and reasonably small $n$, like
$n=100$ or $n=1000$. We do not require the full coverage but only 90\% or 95\%
coverage. We establish that efficient covering schemes have several important
properties which are not seen in small dimensions and in asymptotical
considerations, for very large $n$. One of these properties can be termed `do
not try to cover the vertices' as the vertices of the cube and their close
neighbourhoods are very hard to cover and for large $d$ there are far too many
of them. We clearly demonstrate that, contrary to a common belief, placing
balls at points which form a low-discrepancy sequence in the cube, makes for a
very inefficient covering scheme. For a family of random coverings, we are able
to provide very accurate approximations to the coverage probability. We then
extend our results to the problems of coverage of a cube by smaller cubes and
quantization, the latter being also referred to as facility location. Along
with theoretical considerations and derivation of approximations, we discuss
results of a large-scale numerical investigation.

25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs