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106
Polynomial time algorithms for multicast network code construction
 IEEE TRANS. ON INFO. THY
, 2005
"... The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediat ..."
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Cited by 316 (29 self)
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The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to reencode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.
On Randomized Network Coding
 In Proceedings of 41st Annual Allerton Conference on Communication, Control, and Computing
, 2003
"... We consider a randomized network coding approach for multicasting from several sources over a network, in which nodes independently and randomly select linear mappings from inputs onto output links over some field. This approach was first described in [3], which gave, for acyclic delayfree netwo ..."
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Cited by 200 (35 self)
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We consider a randomized network coding approach for multicasting from several sources over a network, in which nodes independently and randomly select linear mappings from inputs onto output links over some field. This approach was first described in [3], which gave, for acyclic delayfree networks, a bound on error probability, in terms of the number of receivers and random coding output links, that decreases exponentially with code length. The proof was based on a result in [2] relating algebraic network coding to network flows. In this paper, we generalize these results to networks with cycles and delay. We also show, for any given acyclic network, a tighter bound in terms of the probability of connection feasibility in a related network problem with unreliable links. From this we obtain a success probability bound for randomized network coding in linkredundant networks with unreliable links, in terms of link failure probability and amount of redundancy.
MinimumCost Multicast over Coded Packet Networks
 IEEE TRANS. ON INF. THE
, 2006
"... We consider the problem of establishing minimumcost multicast connections over coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as b ..."
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Cited by 164 (28 self)
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We consider the problem of establishing minimumcost multicast connections over coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomialtime solvable optimization problem, ... and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimumcost multicast. By contrast, establishing minimumcost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections. For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved.
Insufficiency of linear coding in network information flow
 IEEE TRANSACTIONS ON INFORMATION THEORY (REVISED JANUARY
, 2005
"... It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finitefield alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solutio ..."
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Cited by 162 (14 self)
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It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finitefield alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finitefield alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10 % greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filterbank coding, or linear time sharing, the network has no linear solution.
Network Information Flow with Correlated Sources
 TO APPEAR IN THE IEEE TRANSACTIONS ON INFORMATION THEORY
, 2005
"... Consider the following network communication setup, originating in a sensor networking application we refer to as the “sensor reachback ” problem. We have a directed graph G = (V, E), where V = {v0v1...vn} and E ⊆ V × V. If (vi, vj) ∈ E, then node i can send messages to node j over a discrete memor ..."
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Cited by 93 (7 self)
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Consider the following network communication setup, originating in a sensor networking application we refer to as the “sensor reachback ” problem. We have a directed graph G = (V, E), where V = {v0v1...vn} and E ⊆ V × V. If (vi, vj) ∈ E, then node i can send messages to node j over a discrete memoryless channel (Xij, pij(yx), Yij), of capacity Cij. The channels are independent. Each node vi gets to observe a source of information Ui (i = 0...M), with joint distribution p(U0U1...UM). Our goal is to solve an incast problem in G: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the M + 1 nodes (v0 by convention) must have received enough information to reproduce the entire field of observations (U0U1...UM), with arbitrarily small probability of error. In this paper, we prove that such perfect reconstruction is possible if and only if H(USUS c) < i∈S,j∈S c for all S ⊆ {0...M}, S = ∅, 0 ∈ S c. Our main finding is that in this setup a general source/channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a
Network Coding with a Cost Criterion
 in Proc. 2004 International Symposium on Information Theory and its Applications (ISITA 2004
, 2004
"... We consider applying network coding in settings where there is a cost associated with network use. ..."
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Cited by 85 (18 self)
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We consider applying network coding in settings where there is a cost associated with network use.
Information flow decomposition for network coding
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2006
"... We propose a method to identify structural properties of multicast network configurations, by decomposing networks into regions through which the same information flows. This decomposition allows us to show that very different networks are equivalent from a coding point of view, and offers a means t ..."
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Cited by 68 (10 self)
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We propose a method to identify structural properties of multicast network configurations, by decomposing networks into regions through which the same information flows. This decomposition allows us to show that very different networks are equivalent from a coding point of view, and offers a means to identify such equivalence classes. It also allows us to divide the network coding problem into two almost independent tasks: one of graph theory and the other of classical channel coding theory. This approach to network coding enables us to derive the smallest code alphabet size sufficient to code any network configuration with two sources as a function of the number of receivers in the network. But perhaps the most significant strength of our approach concerns future network coding practice. Namely, we propose deterministic algorithms to specify the coding operations at network nodes without the knowledge of the overall network topology. Such decentralized designs facilitate the construction of codes that can easily accommodate future changes in the network, e.g., addition of receivers and loss of links.
On Coding for NonMulticast Networks
 IN PROC. 41ST ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL AND COMPUTING
, 2003
"... We consider the issue of coding for nonmulticast networks. For multicast networks, it is known that linear operations over a field no larger than the number of receivers are sufficient to achieve all feasible connections. In the case of nonmulticast networks, necessary and sufficient conditions ..."
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Cited by 68 (13 self)
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We consider the issue of coding for nonmulticast networks. For multicast networks, it is known that linear operations over a field no larger than the number of receivers are sufficient to achieve all feasible connections. In the case of nonmulticast networks, necessary and sufficient conditions are known, if we restrict ourselves to linear codes over a finite field [1]. However, no linearity sufficiency results exist for nonmulticast networks. Indeed, [2] shows that linearity over a field is not sufficient in general. We present a coding theorem that provides necessary and sufficient conditions, in terms of receiver entropies, for an arbitrary set of connections to be achievable on any network. We conjecture that linearity is sufficient to satisfy the coding theorem, when linear operations are performed over vectors rather than scalars in a field. We illustrate the intuition of this conjecture with an example. This work is part of an ongoing cooperation with R. Koetter.
Network Coding for Correlated Sources
 in CISS
, 2004
"... We consider the ability of a distributed randomized network coding approach to multicast, to one or more receivers, correlated sources over a network where compression may be required. We give, for two arbitrarily correlated sources in a general network, upper bounds on the probability of decoding e ..."
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Cited by 64 (9 self)
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We consider the ability of a distributed randomized network coding approach to multicast, to one or more receivers, correlated sources over a network where compression may be required. We give, for two arbitrarily correlated sources in a general network, upper bounds on the probability of decoding error at each receiver, in terms of network parameters. In the special case of a SlepianWolf source network consisting of a link from each source to the receiver, our error exponents reduce to known error exponents for linear SlepianWolf coding.