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863
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.
Information exchange in wireless networks with network coding and physicallayer broadcast,”
 in Proceedings of the 39th Annual Conference on information Sciences and Systems (CISS ’05),
, 2005
"... AbstractWe show that mutual exchange of independent information between two nodes in a wireless network can be efficiently performed by exploiting network coding and the physicallayer broadcast property offered by the wireless medium. The proposed approach improves upon conventional solutions that ..."
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Cited by 196 (5 self)
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AbstractWe show that mutual exchange of independent information between two nodes in a wireless network can be efficiently performed by exploiting network coding and the physicallayer broadcast property offered by the wireless medium. The proposed approach improves upon conventional solutions that separate the processing of the two unicast sessions, corresponding to information transfer along one direction and the opposite direction. We propose a distributed scheme that obviates the need for synchronization and is robust to random packet loss and delay, and so on. The scheme is simple and incurs minor overhead. I. INTRODUCTION In this paper, we investigate the mutual exchange of independent information between two nodes in a wireless network. Let us name the two nodes in consideration a and b, respectively. Consider a packetbased communication network with all packets of equal size. The basic problem is very simple: a wants to transmit a sequence of packets {X 1 (n)} to b and b wants to transmit a sequence of packets {X 2 (n)} to a. Assume the two sequences of information packets, {X 1 (n)} and {X 2 (n)}, are from two independent information sources. Information exchange finds many useful applications. These include voice conversations, video conferencing between two participants, and instant messaging. In fact, the scope of information exchange goes much further beyond the generic twoway endtoend communications listed above. Note that a and b do not have to be the true communication endpoints for the packets {X 1 (n)} and {X 2 (n)}. For example, in a wireless ad hoc network where every node can act as a router, information exchange occurs as long as there are some packets {X 1 (n)} to be routed through a to b and some other packets {X 2 (n)} to be routed through b to a. This is illustrated in An information exchange session between a and b is essentially two unicast sessions, one from a to b and the other from b to a. Since the two unicast sessions carry independent information, it may appear that the two sessions can be treated separately, by devoting a first route for packets {X 1 (n)} to flow from a to b and a second route for packets {X 2 (n)} to flow from b to a. In this paper, we show that a joint
Resilient Network Coding in the Presence of Byzantine Adversaries
"... Network coding substantially increases network throughput. But since it involves mixing of information inside the network, a single corrupted packet generated by a malicious node can end up contaminating all the information reaching a destination, preventing decoding. This paper introduces distribu ..."
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Cited by 169 (32 self)
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Network coding substantially increases network throughput. But since it involves mixing of information inside the network, a single corrupted packet generated by a malicious node can end up contaminating all the information reaching a destination, preventing decoding. This paper introduces distributed polynomialtime rateoptimal network codes that work in the presence of Byzantine nodes. We present algorithms that target adversaries with different attacking capabilities. When the adversary can eavesdrop on all links and jam zO links, our first algorithm achieves a rate of C − 2zO, where C is the network capacity. In contrast, when the adversary has limited eavesdropping capabilities, we provide algorithms that achieve the higher rate of C − zO. Our algorithms attain the optimal rate given the strength of the adversary. They are informationtheoretically secure. They operate in a distributed manner, assume no knowledge of the topology, and can be designed and implemented in polynomialtime. Furthermore, only the source and destination need to be modified; nonmalicious nodes inside the network are oblivious to the presence of adversaries and implement a classical distributed network code. Finally, our algorithms work over wired and wireless networks.
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.
Computeandforward: Harnessing interference through structured codes
 IEEE TRANS. INF. THEORY
, 2009
"... ..."
Capacity of Wireless Erasure Networks
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2006
"... In this paper, a special class of wireless networks, called wireless erasure networks, is considered. In these networks, each node is connected to a set of nodes by possibly correlated erasure channels. The network model incorporates the broadcast nature of the wireless environment by requiring eac ..."
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Cited by 149 (12 self)
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In this paper, a special class of wireless networks, called wireless erasure networks, is considered. In these networks, each node is connected to a set of nodes by possibly correlated erasure channels. The network model incorporates the broadcast nature of the wireless environment by requiring each node to send the same signal on all outgoing channels. However, we assume there is no interference in reception. Such models are therefore appropriate for wireless networks where all information transmission is packetized and where some mechanism for interference avoidance is already built in. This paper looks at multicast problems over these networks. The capacity under the assumption that erasure locations on all the links of the network are provided to the destinations is obtained. It turns out that the capacity region has a nice maxflow mincut interpretation. The definition of cutcapacity in these networks incorporates the broadcast property of the wireless medium. It is further shown that linear coding at nodes in the network suffices to achieve the capacity region. Finally, the performance of different coding schemes in these networks when no side information is available to the destinations is analyzed.
Computation over MultipleAccess Channels
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2007
"... The problem of reliably reconstructing a function of sources over a multipleaccess channel is considered. It is shown that there is no sourcechannel separation theorem even when the individual sources are independent. Joint sourcechannel strategies are developed that are optimal when the structure ..."
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Cited by 139 (24 self)
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The problem of reliably reconstructing a function of sources over a multipleaccess channel is considered. It is shown that there is no sourcechannel separation theorem even when the individual sources are independent. Joint sourcechannel strategies are developed that are optimal when the structure of the channel probability transition matrix and the function are appropriately matched. Even when the channel and function are mismatched, these computation codes often outperform separationbased strategies. Achievable distortions are given for the distributed refinement of the sum of Gaussian sources over a Gaussian multipleaccess channel with a joint sourcechannel lattice code. Finally, computation codes are used to determine the multicast capacity of finite field multipleaccess networks, thus linking them to network coding.
Algebraic gossip: A network coding approach to optimal multiple rumor mongering
 IEEE Transactions on Information Theory
, 2004
"... We study the problem of simultaneously disseminating multiple messages in a large network in a decentralized and distributed manner. We consider a network with n nodes and k (k = O(n)) messages spread throughout the network to start with, but not all nodes have all the messages. Our communication mo ..."
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Cited by 134 (12 self)
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We study the problem of simultaneously disseminating multiple messages in a large network in a decentralized and distributed manner. We consider a network with n nodes and k (k = O(n)) messages spread throughout the network to start with, but not all nodes have all the messages. Our communication model is such that the nodes communicate in discretetime steps, and in every timestep, each node communicates with a random communication partner chosen uniformly from all the nodes (known as the random phone call model). The system is bandwidth limited and in each timestep, only one message can be transmitted. The goal is to disseminate rapidly all the messages among all the nodes. We study the time required for this dissemination to occur with high probability, and also in expectation. We present a protocol based on random linear coding (RLC) that disseminates all the messages among all the nodes in O(n) time, which is order optimal, if we ignore the small overhead associated with each transmission. The overhead does not depend on the size of the messages and is less than 1 % for k = 100 and messages of size 100 KB. We also consider a store and forward mechanism without coding, which is a natural extension of gossipbased dissemination with one message in the network. We show that, such an uncoded scheme can do no better than a sequential approach (instead of doing it simultaneously) of disseminating the messages which takes Θ(n ln(n)) time, since disseminating a single message in a gossip network takes Θ(ln(n)) time. 1
Network Coding for Efficient Communication in Extreme Networks
, 2005
"... Some forms of adhoc networks need to operate in extremely performancechallenged environments where endtoend connectivity is rare. Such environments can be found for example in very sparse mobile networks where nodes ”meet ” only occasionally and are able to exchange information, or in wireless s ..."
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Cited by 128 (3 self)
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Some forms of adhoc networks need to operate in extremely performancechallenged environments where endtoend connectivity is rare. Such environments can be found for example in very sparse mobile networks where nodes ”meet ” only occasionally and are able to exchange information, or in wireless sensor networks where nodes sleep most of the time to conserve energy. Forwarding mechanisms in such networks usually resort to some form of intelligent flooding, as for example in probabilistic routing. We propose a communication algorithm that significantly reduces the overhead of probabilistic routing algorithms, making it a suitable building block for a delaytolerant network architecture. Our forwarding scheme is based on network coding. Nodes do not simply forward packets they overhear but may send out information that is coded over the contents of several packets they received. We show by simulation that this algorithm achieves the reliability and robustness of flooding at a small fraction of the overhead.