28 research outputs found
Edge-Cut Bounds on Network Coding Rates
Active networks are network architectures with processors that are capable of executing code carried by the packets passing through them. A critical network management concern is the optimization of such networks and tight bounds on their performance serve as useful design benchmarks. A new bound on communication rates is developed that applies to network coding, which is a promising active network application that has processors transmit packets that are general functions, for example a bit-wise XOR, of selected received packets. The bound generalizes an edge-cut bound on routing rates by progressively removing edges from the network graph and checking whether certain strengthened d -separation conditions are satisfied. The bound improves on the cut-set bound and its efficacy is demonstrated by showing that routing is rate-optimal for some commonly cited examples in the networking literature.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43451/1/10922_2005_Article_9019.pd
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
Beyond the Cut-Set Bound: Uncertainty Computations in Network Coding with Correlated Sources
Cut-set bounds on achievable rates for network communication protocols are
not in general tight. In this paper we introduce a new technique for proving
converses for the problem of transmission of correlated sources in networks,
that results in bounds that are tighter than the corresponding cut-set bounds.
We also define the concept of "uncertainty region" which might be of
independent interest. We provide a full characterization of this region for the
case of two correlated random variables. The bounding technique works as
follows: on one hand we show that if the communication problem is solvable, the
uncertainty of certain random variables in the network with respect to
imaginary parties that have partial knowledge of the sources must satisfy some
constraints that depend on the network architecture. On the other hand, the
same uncertainties have to satisfy constraints that only depend on the joint
distribution of the sources. Matching these two leads to restrictions on the
statistical joint distribution of the sources in communication problems that
are solvable over a given network architecture.Comment: 12 pages, A short version appears in ISIT 201
Structural Routability of n-Pairs Information Networks
Information does not generally behave like a conservative fluid flow in
communication networks with multiple sources and sinks. However, it is often
conceptually and practically useful to be able to associate separate data
streams with each source-sink pair, with only routing and no coding performed
at the network nodes. This raises the question of whether there is a nontrivial
class of network topologies for which achievability is always equivalent to
routability, for any combination of source signals and positive channel
capacities. This chapter considers possibly cyclic, directed, errorless
networks with n source-sink pairs and mutually independent source signals. The
concept of downward dominance is introduced and it is shown that, if the
network topology is downward dominated, then the achievability of a given
combination of source signals and channel capacities implies the existence of a
feasible multicommodity flow.Comment: The final publication is available at link.springer.com
http://link.springer.com/chapter/10.1007/978-3-319-02150-8_
Generalized Cut-Set Bounds for Broadcast Networks
A broadcast network is a classical network with all source messages
collocated at a single source node. For broadcast networks, the standard
cut-set bounds, which are known to be loose in general, are closely related to
union as a specific set operation to combine the basic cuts of the network.
This paper provides a new set of network coding bounds for general broadcast
networks. These bounds combine the basic cuts of the network via a variety of
set operations (not just the union) and are established via only the
submodularity of Shannon entropy. The tightness of these bounds are
demonstrated via applications to combination networks.Comment: 30 pages, 4 figures, submitted to the IEEE Transaction on Information
Theor
Capacity Bounds for Networks of Broadcast Channels
This paper derives new outer bounds on the capacities
of networks comprised of broadcast and point-to-point
channels. The results are tight in some cases, and methods
for bounding their error in general are discussed. The results
given demonstrate the simplicity of generalizing network coding
results to networks of noisy channels using the family of network
equivalence tools. The approach taken is not inherently a cut-set
approach and frequently yields tighter bounds than those
achieved by traditional cut-sets