13 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
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
Zero-error Function Computation on a Directed Acyclic Network
We study the rate region of variable-length source-network codes that are
used to compute a function of messages observed over a network. The particular
network considered here is the simplest instance of a directed acyclic graph
(DAG) that is not a tree. Existing work on zero-error function computation in
DAG networks provides bounds on the \textit{computation capacity}, which is a
measure of the amount of communication required per edge in the worst case.
This work focuses on the average case: an achievable rate tuple describes the
expected amount of communication required on each edge, where the expectation
is over the probability mass function of the source messages.
We describe a systematic procedure to obtain outer bounds to the rate region
for computing an arbitrary demand function at the terminal. Our bounding
technique works by lower bounding the entropy of the descriptions observed by
the terminal conditioned on the function value and by utilizing the
Schur-concave property of the entropy function. We evaluate these bounds for
certain example demand functions.Comment: 18 pages, 2 figures, submitted to IEEE Transactions on Information
Theor
Sum-networks from incidence structures: construction and capacity analysis
A sum-network is an instance of a network coding problem over a directed acyclic network in which each terminal node wants to compute the sum over a finite field of the information observed at all 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. In this work, we describe an algorithm to construct families of sum-network instances using incidence structures. The computation capacity of several of these sum-network families is characterized. We demonstrate that unlike the multiple unicast problem, the computation capacity of sum-networks depends on the characteristic of the finite field over which the sum is computed. This dependence is very strong; we show examples of sum-networks that have a rate-1 solution over one characteristic but a rate close to zero over a different characteristic. Additionally, a sum-network can have an arbitrary different number of computation capacities for different alphabets. This is contrast to the multiple unicast problem where it is known that the capacity is independent of the network coding alphabet
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 unitrate 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 paper, 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 paper, 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 unitrate requirement, for all such connectivity-level vectors except the vector [1 2 4] (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.This is a manuscript of an article from IEEE/ACM Transactions on Networking 22 (2014): 285, doi: 10.1109/TNET.2013.2270438. Posted with permission.</p