291 research outputs found
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
Connecting Multiple-unicast and Network Error Correction: Reduction and Unachievability
We show that solving a multiple-unicast network coding problem can be reduced
to solving a single-unicast network error correction problem, where an
adversary may jam at most a single edge in the network. Specifically, we
present an efficient reduction that maps a multiple-unicast network coding
instance to a network error correction instance while preserving feasibility.
The reduction holds for both the zero probability of error model and the
vanishing probability of error model. Previous reductions are restricted to the
zero-error case. As an application of the reduction, we present a constructive
example showing that the single-unicast network error correction capacity may
not be achievable, a result of separate interest.Comment: ISIT 2015. arXiv admin note: text overlap with arXiv:1410.190
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
Precoding-Based Network Alignment For Three Unicast Sessions
We consider the problem of network coding across three unicast sessions over
a directed acyclic graph, where each sender and the receiver is connected to
the network via a single edge of unit capacity. We consider a network model in
which the middle of the network only performs random linear network coding, and
restrict our approaches to precoding-based linear schemes, where the senders
use precoding matrices to encode source symbols. We adapt a precoding-based
interference alignment technique, originally developed for the wireless
interference channel, to construct a precoding-based linear scheme, which we
refer to as as a {\em precoding-based network alignment scheme (PBNA)}. A
primary difference between this setting and the wireless interference channel
is that the network topology can introduce dependencies between elements of the
transfer matrix, which we refer to as coupling relations, and can potentially
affect the achievable rate of PBNA. We identify all possible such coupling
relations, and interpret these coupling relations in terms of network topology
and present polynomial-time algorithms to check the presence of these coupling
relations. Finally, we show that, depending on the coupling relations present
in the network, the optimal symmetric rate achieved by precoding-based linear
scheme can take only three possible values, all of which can be achieved by
PBNA.Comment: arXiv admin note: text overlap with arXiv:1202.340
A Linear Network Code Construction for General Integer Connections Based on the Constraint Satisfaction Problem
The problem of finding network codes for general connections is inherently
difficult in capacity constrained networks. Resource minimization for general
connections with network coding is further complicated. Existing methods for
identifying solutions mainly rely on highly restricted classes of network
codes, and are almost all centralized. In this paper, we introduce linear
network mixing coefficients for code constructions of general connections that
generalize random linear network coding (RLNC) for multicast connections. For
such code constructions, we pose the problem of cost minimization for the
subgraph involved in the coding solution and relate this minimization to a
path-based Constraint Satisfaction Problem (CSP) and an edge-based CSP. While
CSPs are NP-complete in general, we present a path-based probabilistic
distributed algorithm and an edge-based probabilistic distributed algorithm
with almost sure convergence in finite time by applying Communication Free
Learning (CFL). Our approach allows fairly general coding across flows,
guarantees no greater cost than routing, and shows a possible distributed
implementation. Numerical results illustrate the performance improvement of our
approach over existing methods.Comment: submitted to TON (conference version published at IEEE GLOBECOM 2015
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