540 research outputs found
Graph Theory versus Minimum Rank for Index Coding
We obtain novel index coding schemes and show that they provably outperform
all previously known graph theoretic bounds proposed so far. Further, we
establish a rather strong negative result: all known graph theoretic bounds are
within a logarithmic factor from the chromatic number. This is in striking
contrast to minrank since prior work has shown that it can outperform the
chromatic number by a polynomial factor in some cases. The conclusion is that
all known graph theoretic bounds are not much stronger than the chromatic
number.Comment: 8 pages, 2 figures. Submitted to ISIT 201
The Balanced Unicast and Multicast Capacity Regions of Large Wireless Networks
We consider the question of determining the scaling of the -dimensional
balanced unicast and the -dimensional balanced multicast capacity
regions of a wireless network with nodes placed uniformly at random in a
square region of area and communicating over Gaussian fading channels. We
identify this scaling of both the balanced unicast and multicast capacity
regions in terms of , out of total possible, cuts. These cuts
only depend on the geometry of the locations of the source nodes and their
destination nodes and the traffic demands between them, and thus can be readily
evaluated. Our results are constructive and provide optimal (in the scaling
sense) communication schemes.Comment: 37 pages, 7 figures, to appear in IEEE Transactions on Information
Theor
Algebraic Network Coding Approach to Deterministic Wireless Relay Networks
The deterministic wireless relay network model, introduced by Avestimehr et
al., has been proposed for approximating Gaussian relay networks. This model,
known as the ADT network model, takes into account the broadcast nature of
wireless medium and interference. Avestimehr et al. showed that the Min-cut
Max-flow theorem holds in the ADT network.
In this paper, we show that the ADT network model can be described within the
algebraic network coding framework introduced by Koetter and Medard. We prove
that the ADT network problem can be captured by a single matrix, called the
"system matrix". We show that the min-cut of an ADT network is the rank of the
system matrix; thus, eliminating the need to optimize over exponential number
of cuts between two nodes to compute the min-cut of an ADT network.
We extend the capacity characterization for ADT networks to a more general
set of connections. Our algebraic approach not only provides the Min-cut
Max-flow theorem for a single unicast/multicast connection, but also extends to
non-multicast connections such as multiple multicast, disjoint multicast, and
two-level multicast. We also provide sufficiency conditions for achievability
in ADT networks for any general connection set. In addition, we show that the
random linear network coding, a randomized distributed algorithm for network
code construction, achieves capacity for the connections listed above.
Finally, we extend the ADT networks to those with random erasures and cycles
(thus, allowing bi-directional links). Note that ADT network was proposed for
approximating the wireless networks; however, ADT network is acyclic.
Furthermore, ADT network does not model the stochastic nature of the wireless
links. With our algebraic framework, we incorporate both cycles as well as
random failures into ADT network model.Comment: 9 pages, 12 figures, submitted to Allerton Conferenc
On the Impact of a Single Edge on the Network Coding Capacity
In this paper, we study the effect of a single link on the capacity of a
network of error-free bit pipes. More precisely, we study the change in network
capacity that results when we remove a single link of capacity . In a
recent result, we proved that if all the sources are directly available to a
single super-source node, then removing a link of capacity cannot
change the capacity region of the network by more than in each
dimension. In this paper, we extend this result to the case of multi-source,
multi-sink networks for some special network topologies.Comment: Originally presented at ITA 2011 in San Diego, CA. The arXiv version
contains an updated proof of Theorem
Optimal Reverse Carpooling Over Wireless Networks - A Distributed Optimization Approach
We focus on a particular form of network coding, reverse carpooling, in a
wireless network where the potentially coded transmitted messages are to be
decoded immediately upon reception. The network is fixed and known, and the
system performance is measured in terms of the number of wireless broadcasts
required to meet multiple unicast demands. Motivated by the structure of the
coding scheme, we formulate the problem as a linear program by introducing a
flow variable for each triple of connected nodes. This allows us to have a
formulation polynomial in the number of nodes. Using dual decomposition and
projected subgradient method, we present a decentralized algorithm to obtain
optimal routing schemes in presence of coding opportunities. We show that the
primal sub-problem can be expressed as a shortest path problem on an
\emph{edge-graph}, and the proposed algorithm requires each node to exchange
information only with its neighbors.Comment: submitted to CISS 201
Alignment based Network Coding for Two-Unicast-Z Networks
In this paper, we study the wireline two-unicast-Z communication network over
directed acyclic graphs. The two-unicast-Z network is a two-unicast network
where the destination intending to decode the second message has apriori side
information of the first message. We make three contributions in this paper:
1. We describe a new linear network coding algorithm for two-unicast-Z
networks over directed acyclic graphs. Our approach includes the idea of
interference alignment as one of its key ingredients. For graphs of a bounded
degree, our algorithm has linear complexity in terms of the number of vertices,
and polynomial complexity in terms of the number of edges.
2. We prove that our algorithm achieves the rate-pair (1, 1) whenever it is
feasible in the network. Our proof serves as an alternative, albeit restricted
to two-unicast-Z networks over directed acyclic graphs, to an earlier result of
Wang et al. which studied necessary and sufficient conditions for feasibility
of the rate pair (1, 1) in two-unicast networks.
3. We provide a new proof of the classical max-flow min-cut theorem for
directed acyclic graphs.Comment: The paper is an extended version of our earlier paper at ITW 201
Computation in Multicast Networks: Function Alignment and Converse Theorems
The classical problem in network coding theory considers communication over
multicast networks. Multiple transmitters send independent messages to multiple
receivers which decode the same set of messages. In this work, computation over
multicast networks is considered: each receiver decodes an identical function
of the original messages. For a countably infinite class of two-transmitter
two-receiver single-hop linear deterministic networks, the computing capacity
is characterized for a linear function (modulo-2 sum) of Bernoulli sources.
Inspired by the geometric concept of interference alignment in networks, a new
achievable coding scheme called function alignment is introduced. A new
converse theorem is established that is tighter than cut-set based and
genie-aided bounds. Computation (vs. communication) over multicast networks
requires additional analysis to account for multiple receivers sharing a
network's computational resources. We also develop a network decomposition
theorem which identifies elementary parallel subnetworks that can constitute an
original network without loss of optimality. The decomposition theorem provides
a conceptually-simpler algebraic proof of achievability that generalizes to
-transmitter -receiver networks.Comment: to appear in the IEEE Transactions on Information Theor
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