270 research outputs found
On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks
We consider the multiple unicast problem with three source-terminal pairs
over directed acyclic networks with unit-capacity edges. The three
pairs wish to communicate at unit-rate via network coding. The connectivity
between the pairs is quantified by means of a connectivity level
vector, such that there exist edge-disjoint paths between
and . In this work we attempt to classify networks based on the
connectivity level. It can be observed that unit-rate transmission can be
supported by routing if , for all . In this work,
we consider, connectivity level vectors such that . We present either a constructive linear network coding scheme or an
instance of a network that cannot support the desired unit-rate requirement,
for all such connectivity level vectors except the vector (and its
permutations). The benefits of our schemes extend to networks with higher and
potentially different edge capacities. Specifically, our experimental results
indicate that for networks where the different source-terminal paths have a
significant overlap, our constructive unit-rate schemes can be packed along
with routing to provide higher throughput as compared to a pure routing
approach.Comment: To appear in the IEEE/ACM Transactions on Networkin
Communicating the sum of sources over a network
We consider the network communication scenario, over directed acyclic
networks with unit capacity edges in which a number of sources each
holding independent unit-entropy information wish to communicate the sum
to a set of terminals . We show that in the case in which
there are only two sources or only two terminals, communication is possible if
and only if each source terminal pair is connected by at least a
single path. For the more general communication problem in which there are
three sources and three terminals, we prove that a single path connecting the
source terminal pairs does not suffice to communicate . We then
present an efficient encoding scheme which enables the communication of
for the three sources, three terminals case, given that each source
terminal pair is connected by {\em two} edge disjoint paths.Comment: 12 pages, IEEE JSAC: Special Issue on In-network
Computation:Exploring the Fundamental Limits (to appear
Capacity of Sum-networks for Different Message Alphabets
A sum-network is a directed acyclic network in which all terminal nodes
demand the `sum' of the independent information observed at the source nodes.
Many characteristics of the well-studied multiple-unicast network communication
problem also hold for sum-networks due to a known reduction between instances
of these two problems. Our main result is that unlike a multiple unicast
network, the coding capacity of a sum-network is dependent on the message
alphabet. We demonstrate this using a construction procedure and show that the
choice of a message alphabet can reduce the coding capacity of a sum-network
from to close to
Network monitoring in multicast networks using network coding
In this paper we show how information contained in robust network codes can be used for passive inference of possible locations of link failures or losses in a network. For distributed randomized network coding, we bound the probability of being able to distinguish among a given set of failure events, and give some experimental results for one and two link failures in randomly generated networks. We also bound the required field size and complexity for designing a robust network code that distinguishes among a given set of failure events
On routing-optimal networks for multiple unicasts
In this paper, we consider the problem of multiple unicast sessions over a directed acyclic graph. It is well known that linear network coding is insufficient for achieving the capacity region, in the general case. However, there exist networks for which routing is sufficient to achieve the whole rate region, and we refer to them as routing-optimal networks. We identify a class of routing-optimal networks, which we refer to as information-distributive networks, defined by three topological features. Due to these features, for each rate vector achieved by network coding, there is always a routing scheme such that it achieves the same rate vector, and the traffic transmitted through the network is exactly the information transmitted over the cut-sets between the sources and the sinks in the corresponding network coding scheme. We present examples of information-distributive networks, including some examples from (1) index coding and (2) from a single unicast session with hard deadline constraint. © 2014 IEEE
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
On network coding for sum-networks
A directed acyclic network is considered where all the terminals need to
recover the sum of the symbols generated at all the sources. We call such a
network a sum-network. It is shown that there exists a solvably (and linear
solvably) equivalent sum-network for any multiple-unicast network, and thus for
any directed acyclic communication network. It is also shown that there exists
a linear solvably equivalent multiple-unicast network for every sum-network. It
is shown that for any set of polynomials having integer coefficients, there
exists a sum-network which is scalar linear solvable over a finite field F if
and only if the polynomials have a common root in F. For any finite or cofinite
set of prime numbers, a network is constructed which has a vector linear
solution of any length if and only if the characteristic of the alphabet field
is in the given set. The insufficiency of linear network coding and
unachievability of the network coding capacity are proved for sum-networks by
using similar known results for communication networks. Under fractional vector
linear network coding, a sum-network and its reverse network are shown to be
equivalent. However, under non-linear coding, it is shown that there exists a
solvable sum-network whose reverse network is not solvable.Comment: Accepted to IEEE Transactions on Information Theor
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