Results 1  10
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19
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 109 (27 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 93 (13 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 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 61 (16 self)
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We consider applying network coding in settings where there is a cost associated with network use.
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 44 (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.
Network Routing Capacity
, 2005
"... We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every nonnegative ratio ..."
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Cited by 34 (12 self)
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We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every nonnegative rational number is the routing capacity of some network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used.
Network Coding Fundamentals
 FOUNDATIONS AND TRENDS IN NETWORKING
, 2007
"... Network coding is an elegant and novel technique introduced at the turn of the millennium to improve network throughput and performance. It is expected to be a critical technology for networks of the future. This tutorial addresses the first most natural questions one would ask about this new techni ..."
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Cited by 28 (7 self)
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Network coding is an elegant and novel technique introduced at the turn of the millennium to improve network throughput and performance. It is expected to be a critical technology for networks of the future. This tutorial addresses the first most natural questions one would ask about this new technique: how network coding works and what are its benefits, how network codes are designed and how much it costs to deploy networks implementing such codes, and finally, whether there are methods to deal with cycles and delay that are present in all real networks. A companion issue deals primarily with applications of network coding.
Linear Network Codes and Systems of Polynomial Equations
 (SUBMITTED TO ISIT 2008)
, 2008
"... If β and γ are nonnegative integers and F is a field, then a polynomial collection {p1,..., pβ} ⊆ Z[α1,..., αγ] is said to be solvable over F if there exist ω1,..., ωγ ∈ F such that for all i = 1,..., β we have pi(ω1,..., ωγ) = 0. We say that a network and a polynomial collection are solvably equi ..."
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Cited by 15 (3 self)
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If β and γ are nonnegative integers and F is a field, then a polynomial collection {p1,..., pβ} ⊆ Z[α1,..., αγ] is said to be solvable over F if there exist ω1,..., ωγ ∈ F such that for all i = 1,..., β we have pi(ω1,..., ωγ) = 0. We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalarlinear solution over F if and only if the polynomial collection is solvable over F. Koetter and Médard’s work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalarlinear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set Ψ of prime numbers is a set of characteristics of a network if for every q ∈ Ψ, the network has a scalarlinear solution over some finite field with characteristic q and does not have a scalarlinear solution over any finite field whose characteristic lies outside of Ψ. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or cofinite. Two networks N and N ′ are lsequivalent if for any finite field F, N is scalarlinearly solvable over F if and only if N ′ is scalarlinearly solvable over F. We further show that every network is lsequivalent to a multipleunicast matroidal network.
Toward a random operation of networks
 IEEE Transactions on Information Theory
, 2004
"... We present a distributed randomized network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves optimal ca ..."
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Cited by 14 (4 self)
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We present a distributed randomized network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves optimal capacity with probability rapidly approaching 1 with the code length. We also demonstrate that randomized coding performs compression when necessary in a network, generalizing error exponents for linear SlepianWolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved performance and robustness. We illustrate some potential advantages of randomized network coding over routing in two examples of practical scenarios: distributed network operation and online algorithms for networks with dynamically varying connections. Our mathematical development of these results also provides a link between network coding and network flows/bipartite matching, leading to a new bound on required field size for centralized network coding on general multicast networks. 1
Linearity and solvability in multicast networks
 IEEE TRANS. INF. THEORY
, 2004
"... It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not ..."
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Cited by 12 (3 self)
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It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not every binary solvable multicast network has a binary scalar linear solution; iii) a multicast network that has a solution for a given alphabet might not have a solution for all larger alphabets.