31 research outputs found
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
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
Network Coding for Computing: Cut-Set Bounds
The following \textit{network computing} problem is considered. Source nodes
in a directed acyclic network generate independent messages and a single
receiver node computes a target function of the messages. The objective is
to maximize the average number of times can be computed per network usage,
i.e., the ``computing capacity''. The \textit{network coding} problem for a
single-receiver network is a special case of the network computing problem in
which all of the source messages must be reproduced at the receiver. For
network coding with a single receiver, routing is known to achieve the capacity
by achieving the network \textit{min-cut} upper bound. We extend the definition
of min-cut to the network computing problem and show that the min-cut is still
an upper bound on the maximum achievable rate and is tight for computing (using
coding) any target function in multi-edge tree networks and for computing
linear target functions in any network. We also study the bound's tightness for
different classes of target functions. In particular, we give a lower bound on
the computing capacity in terms of the Steiner tree packing number and a
different bound for symmetric functions. We also show that for certain networks
and target functions, the computing capacity can be less than an arbitrarily
small fraction of the min-cut bound.Comment: Submitted to the IEEE Transactions on Information Theory (Special
Issue on Facets of Coding Theory: from Algorithms to Networks); Revised on
Aug 9, 201
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
An Achievable Region for the Double Unicast Problem Based on a Minimum Cut Analysis
We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, s1-t1 and s2-t2. The capacity region for this problem is not known; furthermore, the outer bounds on the region have a large number of inequalities which makes them hard to explicitly evaluate. In this work we consider a related problem. We assume that we only know certain minimum cut values for the network, e.g., mincut(Si, Tj), where Si ⊆ {s1, s2} and Tj ⊆ {t1, t2} for different subsets Si and Tj. Based on these values, we propose an achievable rate region for this problem using linear network codes. Towards this end, we begin by defining a multicast region where both sources are multicast to both the terminals. Following this we enlarge the region by appropriately encoding the information at the source nodes, such that terminal ti is only guaranteed to decode information from the intended source si, while decoding a linear function of the other source. The rate region depends upon the relationship of the different cut values in the network