40,694 research outputs found
On Approximating the Sum-Rate for Multiple-Unicasts
We study upper bounds on the sum-rate of multiple-unicasts. We approximate
the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts
network coding problem with independent sources. Our approximation
algorithm runs in polynomial time and yields an upper bound on the joint source
entropy rate, which is within an factor from the GNS cut. It
further yields a vector-linear network code that achieves joint source entropy
rate within an factor from the GNS cut, but \emph{not} with
independent sources: the code induces a correlation pattern among the sources.
Our second contribution is establishing a separation result for vector-linear
network codes: for any given field there exist networks for which
the optimum sum-rate supported by vector-linear codes over for
independent sources can be multiplicatively separated by a factor of
, for any constant , from the optimum joint entropy
rate supported by a code that allows correlation between sources. Finally, we
establish a similar separation result for the asymmetric optimum vector-linear
sum-rates achieved over two distinct fields and
for independent sources, revealing that the choice of field
can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium
on Information Theory) 2015; some typos correcte
Optimal Staged Self-Assembly of General Shapes
We analyze the number of tile types , bins , and stages necessary to
assemble squares and scaled shapes in the staged tile assembly
model. For squares, we prove stages suffice and
are necessary for almost all .
For shapes with Kolmogorov complexity , we prove
stages suffice and are necessary to
assemble a scaled version of , for almost all . We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201
Note on the upper bound of the rainbow index of a graph
A path in an edge-colored graph , where adjacent edges may be colored the
same, is a rainbow path if every two edges of it receive distinct colors. The
rainbow connection number of a connected graph , denoted by , is the
minimum number of colors that are needed to color the edges of such that
there exists a rainbow path connecting every two vertices of . Similarly, a
tree in is a rainbow~tree if no two edges of it receive the same color. The
minimum number of colors that are needed in an edge-coloring of such that
there is a rainbow tree connecting for each -subset of is
called the -rainbow index of , denoted by , where is an
integer such that . Chakraborty et al. got the following result:
For every , a connected graph with minimum degree at least
has bounded rainbow connection, where the bound depends only on
. Krivelevich and Yuster proved that if has vertices and the
minimum degree then . This bound was later
improved to by Chandran et al. Since , a
natural problem arises: for a general determining the true behavior of
as a function of the minimum degree . In this paper, we
give upper bounds of in terms of the minimum degree in
different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected
-step dominating sets, connected -dominating sets and -dominating
sets of .Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by
other author
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