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Current medium access control mechanisms are based on collision avoidance and collided packets are discarded. The recent work on ZigZag decoding departs from this approach by recovering the original packets from multiple collisions. In this paper, we present an algebraic representation of collisions which allows us to view each collision as a linear combination of the original packets. The transmitted, colliding packets may themselves be a coded version of the original packets. We propose a new acknowledgment (ACK) mechanism for collisions based on the idea that if a set of packets collide, the receiver can afford to ACK exactly one of them and still decode all the packets eventually. We analytically compare delay and throughput performance of such collision recovery schemes with other collision avoidance approaches in the context of a single hop wireless erasure network. In the multiple receiver case, the broadcast constraint calls for combining collision recovery methods with network coding across packets at the sender. From the delay perspective, our scheme, without any coordination, outperforms not only a ALOHA-type random access mechanisms, but also centralized scheduling. For the case of streaming arrivals, we propose a priority-based ACK mechanism and show that its stability region coincides with the cut-set bound of the packet erasure network.

In previous work (`Network coding meets TCP') we proposed a new protocol that interfaces network coding with TCP by means of a coding layer between TCP and IP. Unlike the usual batch-based coding schemes, the protocol uses a TCP-compatible sliding window code in combination with new rules for acknowledging bytes to TCP that take into account the network coding operations in the lower layer. The protocol was presented in a theoretical framework and considered only in conjunction with TCP Vegas. In this paper we present a real-world implementation of this protocol that addresses several important practical aspects of incorporating network coding and decoding with TCP's window management mechanism. Further, we work with the more widespread and practical TCP Reno. Our implementation significantly advances the goal of designing a deployable, general, TCP-compatible protocol that provides the benefits of network coding., Comment: 9 pages; Submitted to INFOCOM 2010

Current approaches to the practical implementation of network coding are batch-based, and often do not use feedback, except possibly to signal completion of a file download. In this paper, the various benefits of using feedback in a network coded system are studied. It is shown that network coding can be performed in a completely online manner, without the need for batches or generations, and that such online operation does not affect the throughput. Although these ideas are presented in a single-hop packet erasure broadcast setting, they naturally extend to more general lossy networks which employ network coding in the presence of feedback. The impact of feedback on queue size at the sender and decoding delay at the receivers is studied. Strategies for adaptive coding based on feedback are presented, with the goal of minimizing the queue size and delay. The asymptotic behavior of these metrics is characterized, in the limit of the traffic load approaching capacity. Different notions of decoding delay are considered, including an order-sensitive notion which assumes that packets are useful only when delivered in order. Our work may be viewed as a natural extension of Automatic Repeat reQuest (ARQ) schemes to coded networks., Comment: 27 pages, 7 figures, submitted to IEEE Transactions on Information Theory

The problem of serving multicast flows in a crossbar switch is considered. Intra-flow linear network coding is shown to achieve a larger rate region than the case without coding. A traffic pattern is presented which is achievable with coding but requires a switch speedup when coding is not allowed. The rate region with coding can be characterized in a simple graph-theoretic manner, in terms of the stable set polytope of the "enhanced conflict graph". No such graph-theoretic characterization is known for the case of fanout splitting without coding. The minimum speedup needed to achieve 100% throughput with coding is shown to be upper bounded by the imperfection ratio of the enhanced conflict graph. When applied to KxN switches with unicasts and broadcasts only, this gives a bound of min{(2K-1)/K,2N/(N+1)} on the speedup. This shows that speedup, which is usually implemented in hardware, can often be substituted by network coding, which can be done in software. Computing an offline schedule (using prior knowledge of the flow rates) is reduced to fractional weighted graph coloring. A graph-theoretic online scheduling algorithm (using only queue occupancy information) is also proposed, that stabilizes the queues for all rates within the rate region., Comment: 26 pages, 27 figures, submitted to IEEE Transactions on Information Theory

We propose a mechanism that incorporates network coding into TCP with only minor changes to the protocol stack, thereby allowing incremental deployment. In our scheme, the source transmits random linear combinations of packets currently in the congestion window. At the heart of our scheme is a new interpretation of ACKs - the sink acknowledges every degree of freedom (i.e., a linear combination that reveals one unit of new information) even if it does not reveal an original packet immediately. Such ACKs enable a TCP-like sliding-window approach to network coding. Our scheme has the nice property that packet losses are essentially masked from the congestion control algorithm. Our algorithm therefore reacts to packet drops in a smooth manner, resulting in a novel and effective approach for congestion control over networks involving lossy links such as wireless links. Our experiments show that our algorithm achieves higher throughput compared to TCP in the presence of lossy wireless links. We also establish the soundness and fairness properties of our algorithm., Comment: 9 pages, 9 figures, submitted to IEEE INFOCOM 2009

For a packet erasure broadcast channel with three receivers, we propose a new coding algorithm that makes use of feedback to dynamically adapt the code. Our algorithm is throughput optimal, and we conjecture that it also achieves an asymptotically optimal average decoding delay at the receivers. We consider heavy traffic asymptotics, where the load factor \rho approaches 1 from below with either the arrival rate (\lambda) or the channel parameter (\mu) being fixed at a number less than 1. We verify through simulations that our algorithm achieves an asymptotically optimal decoding delay of O(1/(1-\rho))., Comment: This is the final version. The content has been changed to incorporate reviewer comments and recent results

We present a graph theoretic upper bound on speedup needed to achieve 100% throughput in a multicast switch using network coding. By bounding speedup, we show the equivalence between network coding and speedup in multicast switches - i.e. network coding, which is usually implemented using software, can in many cases substitute speedup, which is often achieved by adding extra switch fabrics. This bound is based on an approach to network coding problems called the "enhanced conflict graph". We show that the "imperfection ratio" of the enhanced conflict graph gives an upper bound on speedup. In particular, we apply this result to K-by-N switches with traffic patterns consisting of unicasts and broadcasts only to obtain an upper bound of min{(2K-1)/K, 2N/(N+1)}., Comment: 5 pages, 4 figures, IEEE ISIT 2007

A new coding and queue management algorithm is proposed for communication networks that employ linear network coding. The algorithm has the feature that the encoding process is truly online, as opposed to a block-by-block approach. The setup assumes a packet erasure broadcast channel with stochastic arrivals and full feedback, but the proposed scheme is potentially applicable to more general lossy networks with link-by-link feedback. The algorithm guarantees that the physical queue size at the sender tracks the backlog in degrees of freedom (also called the virtual queue size). The new notion of a node "seeing" a packet is introduced. In terms of this idea, our algorithm may be viewed as a natural extension of ARQ schemes to coded networks. Our approach, known as the drop-when-seen algorithm, is compared with a baseline queuing approach called drop-when-decoded. It is shown that the expected queue size for our approach is $O(\frac1{1-\rho})$ as opposed to $\Omega(\frac1{(1-\rho)^2})$ for the baseline approach, where $\rho$ is the load factor., Comment: Submitted to the 2008 IEEE International Symposium on Information Theory (ISIT 2008)

We consider the problem of serving multicast flows in a crossbar switch. We show that linear network coding across packets of a flow can sustain traffic patterns that cannot be served if network coding were not allowed. Thus, network coding leads to a larger rate region in a multicast crossbar switch. We demonstrate a traffic pattern which requires a switch speedup if coding is not allowed, whereas, with coding the speedup requirement is eliminated completely. In addition to throughput benefits, coding simplifies the characterization of the rate region. We give a graph-theoretic characterization of the rate region with fanout splitting and intra-flow coding, in terms of the stable set polytope of the 'enhanced conflict graph' of the traffic pattern. Such a formulation is not known in the case of fanout splitting without coding. We show that computing the offline schedule (i.e. using prior knowledge of the flow arrival rates) can be reduced to certain graph coloring problems. Finally, we propose online algorithms (i.e. using only the current queue occupancy information) for multicast scheduling based on our graph-theoretic formulation. In particular, we show that a maximum weighted stable set algorithm stabilizes the queues for all rates within the rate region., Comment: 9 pages, submitted to IEEE INFOCOM 2007