This paper reduces multiple-unicast network coding to single-unicast secure network coding and single-unicast network error correction. Specifically, we present reductions that map an arbitrary multiple-unicast network coding instance to a unicast secure network coding instance in which at most one link is eavesdropped, or a unicast network error correction instance in which at most one link is erroneous, such that a rate tuple is achievable in the multiple-unicast network coding instance if and only if a corresponding rate is achievable in the unicast secure network coding instance, or in the unicast network error correction instance. Conversely, we show that an arbitrary unicast secure network coding instance in which at most one link is eavesdropped can be reduced back to a multiple-unicast network coding instance. In addition, we show that the capacity of a unicast network error correction instance in general is not (exactly) achievable.

The real-time streams play an increasingly important role in both traditional and emerging networks. Whereas, transmitting such streams requires vast channel resources, posing a huge challenge to the networks. Motivated by this problem, this paper considers a real-time stream system, where real-time messages with different importance should be transmitted through a burst erasure channel, and be decoded by the receiver within a fixed delay. We introduce the concepts of Symmetric Real-time (SR) stream and Asymmetric Real-time (AR) stream according to the stream characteristics. Different from the classical Shannon capacity, the channel transmission capacities are derived for SR and AR streams respectively, which can be used to guide the design of efficient real-time stream coding approaches. The capacity for SR streams is shown asymptotically achievable with a Proportional Time-invariant Intra-session Code (PTIC). Moreover, a Maximum Time-invariant Intra-session Code (MTIC) is presented for AR streams, which can also asymptotically achieve the capacity. Finally, the performances of PTIC and MTIC are verified by simulation experiments.

We consider a simple multiple access network in which a destination node receives information from multiple sources via a set of relay nodes. Each relay node has access to a subset of the sources, and is connected to the destination by a unit capacity link. Arbitrary errors may be introduced by up to z of the relay nodes. We propose an efficient distributed error correction coding scheme, where the relay nodes encode independently such that the overall codewords received at the destination are codewords from a single Reed-Solomon code. We show that it achieves the full capacity region for up to three sources.

We consider the problem of secure communication over a network in the presence of wiretappers. We give a new cut-set bound on secrecy capacity which takes into account the contribution of both forward and backward edges crossing the cut, and the connectivity between their endpoints in the rest of the network. We show the bound is tight on a class of networks, which demonstrates that it is not possible to find a tighter bound by considering only cut-set edges and their connectivity.

In this paper, we propose a new cooperation model for discrete memoryless multiple access channels. Unlike in prior cooperation models (e.g., conferencing encoders), where the transmitters cooperate directly, in this model the transmitters cooperate through a larger network. We show that under this indirect cooperation model, there exist channels for which the increase in sum-capacity resulting from cooperation is significantly larger than the rate shared by the transmitters to establish the cooperation. This result contrasts both with results on the benefit of cooperation under prior models and results in the network coding literature, where attempts to find examples in which similar small network modifications yield large capacity benefits have to date been unsuccessful.

In this paper, we present an achievable rate region for double-unicast networks by assuming that the intermediate nodes perform random linear network coding, and the source and sink nodes optimize their strategies to maximize the achievable region. Such a setup can be modeled as a deterministic interference channel, whose capacity region is known. For the particular class of linear deterministic interference channels of our interest, in which the outputs and interference are linear deterministic functions of the inputs, we show that the known capacity region can be achieved by linear strategies. As a result, for a given set of network coding coefficients chosen by the intermediate nodes, the proposed linear precoding and decoding for the source and sink nodes will give the maximum achievable rate region for double-unicast networks. We further derive a suboptimal but easy-to-compute rate region that is independent of the network coding coefficients used at the intermediate nodes, and is instead specified by the min-cuts of the network. It is found that even this suboptimal region is strictly larger than the existing achievable rate regions in the literature.

We consider the delay of network coding compared to routing with retransmissions in packet erasure networks with probabilistic erasures. We investigate the sublinear term in the block delay required for unicasting n packets and show that there is an unbounded gap between network coding and routing. In particular, we show that delay benefit of network coding scales at least as √n. Our analysis of the delay function for the routing strategy involves a major technical challenge of computing the expectation of the maximum of two negative binomial random variables. Previous characterizations of this expectation are approximate; we derive an exact characterization and analyze its scaling behavior, which may be of independent interest. We also use a martingale bounded differences argument to show that the actual coding delay is concentrated around its expectation.

Given the growing popularity of peer-to-peer file distribution in commercial applications, it is important to understand the challenges of using p2p file-sharing protocols for file distribution, and how extreme conditions such as flash crowds affect the efficiency of file distribution. In this light, there is a need to understand the impact of the utilization of available bandwidth on the performance of peer-assisted file distribution systems. With a simple measurement study on PlanetLab, we identified distinct phases in peer bandwidth utilization over the download duration. Based on the evolution of the utilization of available peer bandwidth over time, we formulated an analytical model for flash crowds in homogeneous and heterogeneous bandwidth swarms. The model estimates the instantaneous download rate and the average file download time with 10 percent error for swarms up to 160 peers. Our model can be used to predict the scalability of the system when the number of peers increases, and to provision for flash crowds by estimating the server bandwidth to achieve a minimum quality of service. Lastly, we demonstrate how our model is applied to new p2p protocols to understand their design and performance problems.

This paper studies rateless network error correction codes for reliable multicast in the presence of adversarial errors. We present rateless coding schemes for two adversarial models, where the source sends more redundancy over time, until decoding succeeds. The first model assumes there is a secret channel between the source and the destination that the adversaries cannot overhear. The rate of the channel is negligible compared to the main network. In the second model the source and destination share random secrets independent of the input information. The amount of secret information required is negligible compared to the amount of information sent. Both schemes are capacity optimal, distributed, polynomial-time and end-to-end in that other than the source and destination nodes, other intermediate nodes carry out classical random linear network coding.

We study two variations of the distributed storage allocation problem. The goal is to allocate a given storage budget in a distributed storage system for maximum reliability. It was recently discovered that this problem is related to an old conjecture in extremal combinatorics, on the maximum number of edges in a hypergraph subject to a constraint on its maximum matching number. The conjecture was recently verified in some regimes. In this paper we assume that the conjecture is true and establish new results for the optimal allocation for a variety of parameter values. We also derive new performance bounds that are independent of the conjecture, and compare them to the best previously known bounds.

We consider a real-time streaming system where messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link; the receiver must subsequently decode each message within a given delay from its creation time. We study a bursty erasure model in which all erasure patterns containing erasure bursts of a limited length are admissible. For certain classes of parameter values, we provide code constructions that asymptotically achieve the maximum message size among all codes that allow decoding under all admissible erasure patterns. We also study an i.i.d. erasure model in which each transmitted packet is erased independently with the same probability; the objective is to maximize the decoding probability for a given message size. We derive an upper bound on the decoding probability for any time-invariant code, and show that the gap between this bound and the performance of a family of time-invariant intrasession codes is small in the high reliability regime.

We consider multi-resolution streaming in fully-connected peer-to-peer networks, where transmission rates are constrained by arbitrarily specified upload capacities of the source and peers. We fully characterize the capacity region of rate vectors achievable with arbitrary coding, where an achievable rate vector describes a vector of throughputs of the different resolutions that can be supported by the network. We then prove that all rate vectors in the capacity region can be achieved using pure routing strategies. This shows that coding has no capacity advantage over routing in this scenario.

The secrecy capacity of a network, for a given collection of permissible wiretap sets, is the maximum rate of communication such that observing links in any permissible wiretap set reveal no information about the message. This paper considers secure network coding with nonuniform or restricted wiretap sets, for example, networks with unequal link capacities where a wiretapper can wiretap any subset ofklinks, or networks where only a subset of links can be wiretapped. Existing results show that for the case of uniform wiretap sets (networks with equal capacity links/packets where anykcan be wiretapped), the secrecy capacity is given by a cut-set bound if random keys are injected at the source (and decoded at the sink), whether or not the communicating users have information about the choice of wiretap set. In contrast, we show that for the nonuniform case, this secrecy rate is achievable for the case of known but not unknown wiretap set. We give achievable linear optimization-based strategies where random keys are canceled at intermediate nonsink nodes or injected at intermediate nonsource nodes. Finally, we show that determining the secrecy capacity is an NP-hard problem.

We consider a real-time streaming system where messages are created sequentially at the source, and are encoded for transmission over a packet erasure channel. Each message must subsequently be decoded at the receiver within a given delay from its creation time. We consider code design and maximum message rates when all messages must be decodable by their respective deadlines under a specified set of erasure patterns (erasure model). Specifically, we provide a code construction that achieves the optimal rate for an asymptotic number of messages, under erasure models containing a limited number of erasures per coding window, per sliding window, and containing erasure bursts of a limited length.

We consider lossy data compression in capacity-constrained networks with correlated sources. We derive, using dual decomposition, a distributed algorithm that maximizes an aggregate utility measure defined in terms of the distortion levels of the sources. No coordination among sources is required; each source adjusts its distortion level according to distortion prices fed back by the sinks. The algorithm is developed for the case of squared error distortion and high resolution coding where the rate-distortion region is known, and can be easily extended to consider achievable regions that can be expressed in a related form. Our distributed optimization framework applies to unicast and multicast with and without network coding. Numerical examples show relatively fast convergence, allowing the algorithm to be used in time-varying networks.

We consider communication protocols for N-way all-cast relay networks, which comprise N source terminals such that each source terminal demands messages from all other source terminals with the help of a relay. The derived protocols are characterized by the fact that physical layer network coding is employed at the relay, where each source has side information about the signals it has sent. Amplify-and-forward (AF) and decode-and-forward (DF) protocols are applied to the N-way relay network setting, where the achievable rate regions for those protocols are derived and compared with outer capacity bounds. We propose several practical space-time coding schemes for AF and DF, and introduce two new protocols denoted as denoise-and-forward (DNF) and estimate-and-forward (EF). Further, for AF and DF the fundamental diversity-multiplexing trade-off is characterized.

Recent advances in network coding have shown great potential for efficient information multicasting in communication networks, in terms of both network throughput and network management. In this paper, the problem of flow control at end-systems for network-coding-based multicast flows is addressed. Optimization-based models are formulated for network resource allocation, based on which two sets of decentralized controllers at sources and links/nodes for congestion control are developed for wired networks with given coding subgraphs and without given coding subgraphs, respectively. With random network coding, both sets of controllers can be implemented in a distributed manner, and work at the transport layer to adjust source rates and at network layer to carry out network coding. The convergence of the proposed controllers to the desired equilibrium operating points is proved, and numerical examples are provided to complement the theoretical analysis. The extension to wireless networks is also briefly discussed.

We examine the problem of allocating a given total storage budget in a distributed storage system for maximum reliability. A source has a single data object that is to be coded and stored over a set of storage nodes; it is allowed to store any amount of coded data in each node, as long as the total amount of storage used does not exceed the given budget. A data collector subsequently attempts to recover the original data object by accessing only the data stored in a random subset of the nodes. By using an appropriate code, successful recovery can be achieved whenever the total amount of data accessed is at least the size of the original data object. The goal is to find an optimal storage allocation that maximizes the probability of successful recovery. This optimization problem is challenging in general because of its combinatorial nature, despite its simple formulation. We study several variations of the problem, assuming different allocation models and access models. The optimal allocation and the optimal symmetric allocation (in which all nonempty nodes store the same amount of data) are determined for a variety of cases. Our results indicate that the optimal allocations often have nonintuitive structure and are difficult to specify. We also show that depending on the circumstances, coding may or may not be beneficial for reliable storage.

The paper studies the construction of error-estimation-codes (EEC), which estimate the bit-error-rate (BER) of packet transmissions. The concept of EEC was first proposed by Chen et.al in SIGCOMM 2010. In the same work group-sampling-error-estimation-codes (GSEEC) were constructed. Assuming the packet length is n, GSEEC requires communication overhead O(log(n)) and coding complexity O(n) to reliably estimate BER. In this paper, random walk based error-estimation-codes (RAKEE) are proposed, which are able to achieve constant communication overhead O(1) and linear coding complexity O(n). In the end of the paper, numerical experiments is shown to support our theoretical analysis.

Random linear network codes can be designed and implemented in a distributed manner, with low computational complexity. However, these codes are classically implemented over finite fields whose size depends on some global network parameters (size of the network, the number of sinks) that may be unknown prior to code design. Also, the entire network code may have to be redesigned when a new node joins. In this work, we present the first universal and robust distributed linear network coding schemes. Our schemes are universal since they are independent of all network parameters. They are robust since in case nodes join or leave, the remaining nodes do not need to change their coding operations and the receivers can still decode. They are distributed since nodes need only have topological information about the part of the network upstream of them, which can be naturally streamed as part of the communication protocol. We present both probabilistic and deterministic schemes that are all asymptotically rate-optimal in the coding block-length, and have guarantees of correctness. Our probabilistic designs are computationally efficient, with order-optimal complexity. Our deterministic designs guarantee zero error decoding, albeit via codes with high computational complexity in general. Our coding schemes are based on network codes over “scalable fields”. Instead of choosing coding coefficients from one field at every node as in, each node uses linear coding operations over an “effective field-size” which depends on the node's distance from the source node. The analysis of our schemes requires technical tools that may be of independent interest. In particular, we generalize the Schwartz-Zippel lemma by proving a nonuniform version, wherein variables are chosen from sets of possibly different sizes. We also provide a novel robust distributed algorithm to assign unique IDs to network nodes.

We consider the problem of finding network coding capacities of networks of independent point-to-point channels in the presence of a Byzantine adversary. We assume that the adversary knows all messages, and noise values and the code used to communicate across the network. The adversary controls an unknown subset of edges and can replace the channel output vectors from those edges. We show that finding the capacity for the above network is equivalent to finding the capacity of a network that is obtained by replacing each finite input alphabet point-to-point channel by a noiseless link of the noisy channel capacity. Our result shows the asymptotic optimality of separation between channel coding for each link followed by network coding for the resulting network under the corresponding model of adversarial attack.

We examine the problem of creating an encoded distributed storage representation of a data object for a network of mobile storage nodes so as to achieve the optimal recovery delay. A source node creates a single data object and disseminates an encoded representation of it to other nodes for storage, subject to a given total storage budget. A data collector node subsequently attempts to recover the original data object by contacting other nodes and accessing the data stored in them. By using an appropriate code, successful recovery is achieved when the total amount of data accessed is at least the size of the original data object. The goal is to find an allocation of the given budget over the nodes that optimizes the recovery delay incurred by the data collector; two objectives are considered: (i) maximization of the probability of successful recovery by a given deadline, and (ii) minimization of the expected recovery delay. We solve the problem completely for the second objective in the case of symmetric allocations (in which all nonempty nodes store the same amount of data), and show that the optimal symmetric allocation for the two objectives can be quite different. A simple data dissemination and storage protocol for a mobile delay-tolerant network is evaluated under various scenarios via simulations. Our results show that the choice of storage allocation can have a significant impact on the recovery delay performance, and that coding may or may not be beneficial depending on the circumstances.

An equivalence result is established between the Shannon capacity and the stable capacity of communication networks. Given a discrete-time network with memoryless, time-invariant, discrete-output channels, it is proved that the Shannon capacity equals the stable capacity. The results treat general demands (e.g., multiple unicast demands) and apply even when neither the Shannon capacity nor the stable capacity is known for the given demands. The result also generalize from discrete-alphabet channels to Gaussian channels.

This work considers the multiple-access multicast error-correction scenario over a packetized network withzmalicious edge adversaries. The network has min-cutmand packets of lengthl, and each sink demands all information from the set of sourcesS. The capacity region is characterized for both a “side-channel” model (where sources and sinks share some random bits that are secret from the adversary) and an “omniscient” adversarial model (where no limitations on the adversary's knowledge are assumed). In the “side-channel” adversarial model, the use of a secret channel allows higher rates to be achieved compared to the “omniscient” adversarial model, and a polynomial-complexity capacity-achieving code is provided. For the “omniscient” adversarial model, two capacity-achieving constructions are given: the first is based on random subspace code design and has complexity exponential inlm, while the second uses a novel multiple-field-extension technique and has O(lm|S|) complexity, which is polynomial in the network size. Our code constructions are “end-to-end” in that all nodes except the sources and sinks are oblivious to the adversaries and may simply implement predesigned linear network codes (random or otherwise). Also, the sources act independently without knowledge of the data from other sources.

This paper studies the capacity of single-source single-sink noiseless networks under adversarial or arbitrary errors on no more thanzedges. Unlike prior papers, which assume equal capacities on all links, arbitrary link capacities are considered. Results include new upper bounds, network error-correction coding strategies, and examples of network families where our bounds are tight. An example is provided of a network where the capacity is 50% greater than the best rate that can be achieved with linear coding. While coding at the source and sink suffices in networks with equal link capacities, in networks with unequal link capacities, it is shown that intermediate nodes may have to do coding, nonlinear error detection, or error correction in order to achieve the network error-correction capacity.

We consider the problem of optimally allocating a given total storage budget in a distributed storage system. A source has a data object which it can code and store over a set of storage nodes; it is allowed to store any amount of coded data in each node, as long as the total amount of storage used does not exceed the given budget. A data collector subsequently attempts to recover the original data object by accessing each of the nodes independently with some constant probability. By using an appropriate code, successful recovery occurs when the total amount of data in the accessed nodes is at least the size of the original data object. The goal is to find an optimal storage allocation that maximizes the probability of successful recovery. This optimization problem is challenging because of its discrete nature and nonconvexity, despite its simple formulation. Symmetric allocations (in which all nonempty nodes store the same amount of data), though intuitive, may be suboptimal; the problem is nontrivial even if we optimize over only symmetric allocations. Our main result shows that the symmetric allocation that spreads the budget maximally over all nodes is asymptotically optimal in a regime of interest. Specifically, we derive an upper bound for the suboptimality of this allocation and show that the performance gap vanishes asymptotically in the specified regime. Further, we explicitly find the optimal symmetric allocation for a variety of cases. Our results can be applied to distributed storage systems and other problems dealing with reliability under uncertainty, including delay tolerant networks (DTNs) and content delivery networks (CDNs).

We consider the problem of optimally allocating a given total storage budget in a distributed storage system. A source has a data object which it can code and store over a set of storage nodes; it is allowed to store any amount of data in each storage node, subject to a given total storage budget constraint. A data collector subsequently attempts to recover the original data object by accessing a random fixed-size subset of these storage nodes. Successful recovery of the data object occurs when the total amount of coded data in this subset of storage nodes is at least the size of the original data object. The goal is to determine the amount of data to store in each storage node so that the probability of successful recovery is maximized. We solve this problem in the high recovery probability regime. Our results can be applied to a variety of distributed storage systems, including delay tolerant networks (DTNs), content delivery networks (CDNs), and sensor networks.

We investigate the problem of maintaining an encoded distributed storage system when some nodes contain adversarial errors. Using the error-correction capabilities that are built into the existing redundancy of the system, we propose a simple linear hashing scheme to detect errors in the storage nodes. Our main result is that for storing a data object of total size M using an (n, k) MDS code over a finite field Fq, up to t1 = ⌊(n - k)/2⌋ errors can be detected, with probability of failure smaller than 1/M, by communicating only O(n(n-k) logM) bits to a trusted verifier. Our result constructs small projections of the data that preserve the errors with high probability and builds on a pseudorandom generator that fools linear functions. The transmission rate achieved by our scheme is asymptotically equal to the min-cut capacity between the source and any receiver.

We address secure network coding over networks with unequal link capacities in the presence of a wiretapper who has only access to a restricted number of k links in the network. Previous results show that for the case of equal link capacities and unrestricted wiretapping sets, the secrecy capacity is given by the cut-set bound, whether or not the location of the wiretapped links is known. The cut-set bound can be achieved by injecting k random keys at the source which are decoded at the sink along with the message. In contrast, for the case where the wiretapping set is restricted, or where link capacities are not equal, we show that the cut-set bound is not achievable in general. Finally, it is shown that determining the secrecy capacity is a NP-hard problem.

We consider the delay of network coding compared to routing for a family of simple networks with parallel links. We investigate the sub-linear term in the block delay required for unicasting n packets and show that there is an unbounded gap between network coding and routing. In particular, we show that delay benefit of network coding is scaling at least as fast as √n. The main technical contribution involves showing that the delay function for the routing retransmission strategy is unbounded. This problem is equivalent to computing the expected maximum of two negative binomial random variables. This problem has also been addressed previously and we derive the first exact characterization which might be of independent interest.

The network communication scenario where one or more receivers request all the information transmitted by different sources is considered. We introduce the first polynomial-time (in network size) network codes that achieve any point inside the rate-region for the problem of multiple-source multicast in the presence of malicious errors, for any fixed number of sources. Our codes are fully distributed and different sources require no knowledge of the data transmitted by their peers. Our codes are “end-to-end”, that is, all nodes apart from the sources and the receivers are oblivious to the adversaries present in the network and simply implement random linear network coding.

In this paper, we consider cross-layer optimization in wireless networks with wireless broadcast advantage, focusing on the problem of distributed scheduling of broadcast links. The wireless broadcast advantage is most useful in multicast scenarios. As such, we include network coding in our design to exploit the throughput gain brought in by network coding for multicasting. We derive a subgradient algorithm for joint rate control, network coding and scheduling, which however requires centralized link scheduling. Under the primary interference model, link scheduling problem is equivalent to a maximum weighted hypergraph matching problem that is NP-complete. To solve the scheduling problem distributedly, locally greedy and randomized approximation algorithms are proposed and shown to have bounded worst-case performance. With random network coding, we obtain a fully distributed cross-layer design. Numerical results show promising throughput gain using the proposed algorithms, and surprisingly, in some cases even with less complexity than cross-layer design without broadcast advantage.

This paper presents a new progressive distributed estimation scheme (DES) along with the power scheduling among sensors under AWGN channels. The progressive DES consists of a transmission bit allocation scheme and a quasi best linear unbiased estimate (BLUE) of the unknown parameter at each sensor. This scheme is shown to outperform the traditional progressive DES. Moreover, the power scheduling among sensors, which minimizes the total transmission power subject to the desired MSE performance tolerance, is formulated as a convex problem and the optimal solution is derived analytically.

In this paper, we develop a new class of medium access control protocol, which allows each user to transmit at different data rates chosen randomly from an appropriately determined set of rates. By using successive interference cancellation, multiple packets can be received simultaneously. In slotted Aloha type Gaussian networks, we show that the achievable total throughput of the proposed protocol is at least a constant fraction of the mac sum rate when the number of transmission rates at each node is equal to the number of users in the network. We also study the case when only a limited number of transmission rates is available at each node. Extension to rate splitting is discussed. Simulation results show that the proposed protocol can achieve a significant throughput gain over the conventional Aloha.

We analyze the achievable rate in interference free wireless networks with physical layer fading channels and orthogonal multiple access. As a starting point, the point-to-point channel is considered. We find the optimal physical and network layer rate trade-off which maximizes the achievable overall rate for both a fixed rate transmission scheme and an improved scheme based on multiple virtual users and superposition coding. These initial results are extended to the network setting, where, based on a cut-set formulation, the achievable rate at each node and its upper bound are derived. We propose a distributed optimization algorithm which allows to jointly determine the maximum achievable rate, the optimal physical layer rates on each network link, and an opportunistic back-pressure-type routing strategy on the network layer. This inherently justifies the layered architecture in existing wireless networks. Finally, we show that the proposed layered optimization approach can achieve almost all of the ergodic network capacity in high SNR.

We analyze a simple network where a source and a receiver are connected by a line of erasure channels of different reliabilities. Recent prior work has shown that random linear network coding can achieve the min-cut capacity and therefore the asymptotic rate is determined by the worst link of the line network. In this paper we investigate the delay for transmitting a batch of packets, which is a function of all the erasure probabilities and the number of packets in the batch. We show a monotonicity result on the delay function and derive simple expressions which characterize the expected delay behavior of line networks. Further, we use a martingale bounded differences argument to show that the actual delay is tightly concentrated around its expectation.

We consider the problem of correcting errors and erasures with network coding. Unlike existing works which consider performance limits for worst-case locations of given numbers of errors and erasures, we consider the performance of given (not necessarily optimal) coding and forwarding strategies for given (not necessarily worst-case) models of error and erasure locations. Our approach characterizes decoding success in terms of the rank of certain matrices corresponding to useful and erroneous information received at the sink nodes. We use this approach to analyze random coding and forwarding strategies on a family of simple networks with random error and erasure locations, and show that the relative performance of the strategies depends on the erasure and error probabilities.

In this paper we derive capacity regions for network error correction with both known and unknown topologies (coherent and non-coherent network coding) under a multiple-source multicast transmission scenario. For the multiple-source non-multicast scenario, given any achievable network code for the error-free case, we construct a code with a reduced rate region for the case with errors.

In this paper we study the capacity of the half-duplex wireless butterfly network, in which a relay node facilitates the communication between two interfering transmitter-receiver pairs. We use the deterministic approach to make progress towards approximating the capacity region of this network. We use the insights obtained from the analysis of the corresponding deterministic problem to derive a new upper bound on the capacity of this network. We also propose a transmission strategy and show that for symmetric channel gains the gap between its achievable rate region and the upper bound is at most equation bits/sec/Hz per user.

We propose relaying strategies for uncoded two-way relay channels, where two terminals transmit simultaneously to each other with the help of a relay. In particular, we consider a memoryless system, where the signal transmitted by the relay is obtained by applying an instantaneous relay function to the previously received signal. For binary antipodal signaling, a class of so called absolute (abs)-based schemes is proposed in which the processing at the relay is solely based on the absolute value of the received signal. We analyze and optimize the symbol-error performance of existing and new abs-based and non-abs-based strategies under an average power constraint, including abs-based and non-abs-based versions of amplify and forward (AF), detect and forward (DF), and estimate and forward (EF). Additionally, we optimize the relay function via functional analysis such that the average probability of error is minimized at the high signal-to-noise ratio (SNR) regime. The optimized relay function is shown to be a Lambert W function parameterized on the noise power and the transmission energy. The optimized function behaves like abs-AF at low SNR and like abs-DF at high SNR, respectively; EF behaves similarly to the optimized function over the whole SNR range. We find the conditions under which each class of strategies is preferred. Finally, we show that all these results can also be generalized to higher order constellations.

The problem of multiple multicast sessions with intra-session network coding in time-varying networks is considered. The network-layer capacity region of input rates that can be stably supported is established. Dynamic algorithms for multicast routing, network coding, power allocation, session scheduling, and rate allocation across correlated sources, which achieve stability for rates within the capacity region, are presented. This work builds on the back-pressure approach introduced by Tassiulas, extending it to network coding and correlated sources. In the proposed algorithms, decisions on routing, network coding, and scheduling between different sessions at a node are made locally at each node based on virtual queues for different sinks. For correlated sources, the sinks locally determine and control transmission rates across the sources. The proposed approach yields a completely distributed algorithm for wired networks. In the wireless case, power control among different transmitters is centralized while routing, network coding, and scheduling between different sessions at a given node are distributed.

We address communication protocols for N-way relay networks with M antennas at the relay and a single antenna at the N source terminals. In particular, amplify-and-forward (AF), decode-and-forward (DF), and compress-and-forward (CF) strategies are extended to these networks, and in addition, two new relaying protocols, denoise-and-forward and estimate-and-forward, are proposed. In the first part of the paper, the performance of these schemes is analyzed in terms of the achievable rate region. Also, the optimal diversity-multiplexing tradeoff is derived for both AF and DF. The second part of the paper is devoted to practical space-time transmission strategies. Linear dispersion codes are used, which are optimized by maximizing the sum rate. For AF a diversity order of close to M can be achieved by using a specific space-time code construction.

We consider distributed space-time coding for two-way wireless relay networks, where communication between two terminals is assisted by relay nodes. We compare existing and new protocols that operate over 2, 3 or 4 times slots. Particularly, a new class of relaying protocols, termed as partial decode-and- forward (PDF), is proposed for the 2 time slots transmission. We show that the proposed amplify and forward (AF) protocols achieve the diversity order of min{N,T} ( 1- (loglogP)/(logP))> where N is the number of relays, P is the total power of the network, and T is the number of symbols transmitted during each time slot. When linear dispersion (LD) codes with random unitary matrices are used, the proposed PDF protocols resemble random linear network coding, where the former operates on unitary group and the latter works on finite field.

We consider energy efficient network coding design in wireless networks with multiple unicast sessions. Our approach decomposes multiple unicast sessions into a superposition of multicast and unicast sessions, with coding occurring only within each session. We give an optimization approach that is more general than the existing poison-remedy optimization formulation. For the case of wireless, we consider XOR coding and give an achievable rate region for a primary interference model. To simplify network operation, we give an oblivious backpressure algorithm which does not optimize overhearing of transmissions, and a practical protocol called COPR based on the oblivious backpressure algorithm. Simulation experiments show that COPR largely reduces network power consumption over existing algorithms.

An information-theoretic approach for detecting Byzantine or adversarial modifications in networks employing random linear network coding is described. Each exogenous source packet is augmented with a flexible number of hash symbols that are obtained as a polynomial function of the data symbols. This approach depends only on the adversary not knowing the random coding coefficients of all other packets received by the sink nodes when designing its adversarial packets. We show how the detection probability varies with the overhead (ratio of hash to data symbols), coding field size, and the amount of information unknown to the adversary about the random code.

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 polynomial-time rate-optimal 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 links, our first algorithm achieves a rate of , where is the network capacity. In contrast, when the adversary has limited eavesdropping capabilities, we provide algorithms that achieve the higher rate of . Our algorithms attain the optimal rate given the strength of the adversary. They are information-theoretically secure. They operate in a distributed manner, assume no knowledge of the topology, and can be designed and implemented in polynomial time. 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.

We study a model for peer-to-peer file sharing. The goal is to distribute a file from a server to multiple peers. We assume the upload capacity of each peer is the only bottleneck. We examine the finish times of peers under different transmission strategies. Pareto optimality, min-max finish time, and optimal average finish time of the model are studied. We believe the results provide fundamental insights into practical peer-to-peer systems such as BitTorrent.

We show a statistical pruning approach for maximum likelihood (ML) detection of multiple-input multiple-output (MIMO) systems. We present a general pruning strategy for sphere decoder (SD), which can also be applied to any tree search algorithms. Our pruning rules are effective especially for the case when SD has high complexity. Three specific pruning rules are given and discussed. From analyzing the union bound on the symbol error probability, we show that the diversity order of the deterministic pruning is only one by fixing the pruning probability. By choosing different pruning probability distribution functions, the statistical pruning can achieve arbitrary diversity orders and SNR gains. Our statistical pruning strategy thus achieves a flexible trade-off between complexity and performance.

Recent advances in network coding have shown great potential for efficient information multicasting in communication networks, in terms of both network throughput and network management. In this paper, we address the problem of rate control at end-systems for network coding based multicast flows. We develop two adaptive rate control algorithms for the networks with given coding subgraphs and without given coding subgraphs, respectively. With random network coding, both algorithms can be implemented in a distributed manner, and work at transport layer to adjust source rates and at network layer to carry out network coding. We prove that the proposed algorithms converge to the globally optimal solutions for intra-session network coding. Some related issues are discussed, and numerical examples are provided to complement our theoretical analysis.

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 introducesthefirstdistributedpolynomial-timerate-optimalnetwork 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 zOlinks , our first algorithm achieves a rate ofC- 2zO, where C is the network capacity. In contrast, when the adversary has limited snooping capabilities, we provide algorithms that achieve the higher rate ofC- zO. Our algorithms attain the optimal rate given the strength of the adversary. They are information-theoretically secure. They operate in a distributed manner, assume no knowledge of the topology, and can be designed and implemented in polynomial-time. Furthermore, only the source and destination need to be modified; non-malicious 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.

We consider lossy data compression in capacity-constrained ned networks with correlated sources. We develop, using dual decomposition, a distributed algorithm that maximizes an aggregate utility measure defined in terms of the distortion levels of the sources. No coordination among sources is required; each source adjusts its distortion level according to distortion prices fed back by the sinks. The algorithm is developed for the case of squared error distortion and high resolution coding where the rate distortion region is known, and is easily extended to consider achievable regions that can be expressed in a related form. Our distributed optimization framework applies to unicast and multicast with and without network coding. Numerical example shows relatively fast convergence, allowing the algorithm to be used in time-varying networks.

In this paper, we consider finding a minimum cost multicast subgraph with network coding, where the rate to inject packets on each link is constrained to be integral. In the usual minimum cost network coding formulation, the optimal solution cannot always be integral. Fractional rates can be well approximated by choosing the time unit large enough, but this increases the encoding and decoding complexity as well as delay at the terminals. We formulate this problem as an integer program, which is NP-hard. A greedy algorithm and an algorithm based on linear programming rounding are proposed, which have approximation ratios k and 2k respectively, where k is the number of sinks. Moreover, both algorithms can be decentralized. We show by simulation that our algorithms' average performance substantially exceeds their bounds on random graphs.

The lossless rate region for the coded side information problem is "solved," but its solution is expressed in terms of an auxiliary random variable. As a result, finding the rate region for any fixed example requires an optimization over a family of allowed auxiliary random variables. While intuitive constructions are easy to come by and optimal solutions are known under some special conditions, proving the optimal solution is surprisingly difficult even for examples as basic as a binary source with binary side information. We derive the optimal auxiliary random variables and corresponding achievable rate regions for a family of problems where both the source and side information are binary. Our solution involves first tightening known bounds on the alphabet size of the auxiliary random variable and then optimizing the auxiliary random variable subject to this constraint. The technique used to tighten the bound on the alphabet size applies to a variety of problems beyond the one studied here.

For wireless fading networks where M source nodes wish to multicast their information to N destination nodes via a common relay, it has been shown that network coding at the relay can reduce the number of required transmissions, if the destination nodes overhear M - 1 source nodes transmitting to the relay. Nested coding is a recently proposed alternative where, unlike network coding, physical and network layers are jointly designed. We find that for a given network throughput ideal nested coding in many situations can lead to a significant increase in transmission reliability compared to network coding. We also consider the case of feedback and retransmissions when received packets fail to be decoded. For the case of two source nodes it is shown that by using nested codes the expected number of relay transmissions is reduced. However, for a larger number of source nodes the gain depends on the employed nested coding strategy at the relay.

We propose a jointly opportunistic source coding and opportunistic routing (OSCOR) protocol for correlated data gathering in wireless sensor networks. OSCOR improves data gathering efficiency by exploiting opportunistic data compression and cooperative diversity associated with wireless broadcast advantage. The design of OSCOR involves several challenging issues across different network protocol layers. At the MAC layer, sensor nodes need to coordinate wireless transmission and packet forwarding to exploit multiuser diversity in packet reception. At the network layer, in order to achieve high diversity and compression gains, routing must be based on a metric that is dependent on not only link-quality but also compression opportunities. At the application layer, sensor nodes need a distributed source coding algorithm that has low coordination overhead and does not require the source distributions to be known. OSCOR provides practical solutions to these challenges incorporating a slightly modified 802.11 MAC, a distributed source coding scheme based on network coding and Lempel-Ziv coding, and a node compression ratio dependent metric combined with a modified Dijkstra's algorithm for path selection. We evaluate the performance of OSCOR through simulations, and show that OSCOR can potentially reduce power consumption by over 30% compared with an existing greedy scheme, routing driven compression, in a 4 × 4 grid network.

A heuristic tree search algorithm is developed for the maximum likelihood detection and decoding problem in a general communication system. We propose several "cheap" heuristic functions using constrained linear detectors and the minimum mean square errors (MMSE) detector. Even though the MMSE heuristic function does not guarantee the optimal solution, it has a negligible performance loss and provides a good complexity-performance tradeoff. For linear block coded systems, heuristic tree search is modified for soft decision decoding. High rate codes are decoded via the minimum state trellis, and low rate codes via the minimum complexity tree. Preprocessing is also discussed to further speed up the algorithms.

We develop a polynomial-time detector for maximum likelihood (ML) detection over multiple-input multiple-output (MIMO) channels. Our proposed polynomial moment relaxation (PMR) detection gives a unified framework for MIMO detection with relaxation including semi-definite relaxation as a special case. We give three approaches to replace a finite alphabet constraint with a polynomial constraint. Since both the objective function and the constraints are polynomials, we use a moment relaxation approach by applying the dual theories of moments and positive polynomials solvable by semi-definite programming. With different relaxation orders, our PMR achieve a flexible trade-off between complexity and performance.

We develop an efficient linear programming detector (LPD) for multiple-input multiple-output (MIMO) systems. Instead of using the usual l2norm, our proposed LPD uses the l1norm as the detection metric, resulting in a mixed-integer linear program (MILP). Two branch-and-bound algorithms are proposed to solve the MILP. The solution of the MILP achieves the same full diversity order as the maximum likelihood detector. The MILP is further relaxed to a linear program (LP), which can be readily solved using the standard simplex method. We show that in some cases the solution of the LP is guaranteed to be that of the MILP. The LPD is also extended to the joint detection and decoding of linear block coded MIMO systems. Our LPD can be immediately implemented using mature circuits design for the simplex algorithm

Network coding has been shown to be useful for throughput and reliability in various network topologies, under a fixed-rate, point-to-multipoint wireless network model. We study the effect of introducing a wireless network model where link capacity depends on the network geometry and the signal to interference and noise ratio. In particular, we compare strategies with and without network coding on a multicast network with and without fading, and on single-user multiple path networks with fading. For the multicast network without fading, we find that the network geometry affects which scheme attains higher throughput. For the case with fading, we compare the throughput-outage probability curves achieved by network coding and repetition schemes. For the multiple path networks, we further consider the case where multiple simultaneous transmissions of identical information signals can be combined at a receiver. We find that the relative performance of the schemes we consider depends on the network geometry, the ratio of signal to noise power, whether multiple simultaneous transmissions can be combined, and the operating point on the throughput-outage probability curve

We describe a new tiling approach for network code design. The proposed method applies dynamic programming to find the best strategy among a restricted collection of network codes. We demonstrate the proposed strategy as a method for efficiently accommodating multiple unicasts in a wireless coding environment on a triangular lattice and discuss its generalization.

We present a distributed random linear 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 capacity with probability exponentially approaching 1 with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error exponents for linear Slepian-Wolf 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 success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two examples of practical scenarios: distributed network operation and networks with dynamically varying connections. Our derivation of these results also yields a new bound on required field size for centralized network coding on general multicast networks

We consider the problem of establishing minimum-cost 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 polynomial-time 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 minimum-cost multicast. By contrast, establishing minimum-cost 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.

In this paper we show how information contained in robust network codes can be used for passive inference of possible locations of link failures or losses in a network. For distributed randomized network coding, we bound the probability of being able to distinguish among a given set of failure events, and give some experimental results for one and two link failures in randomly generated networks. We also bound the required field size and complexity for designing a robust network code that distinguishes among a given set of failure events

We design codes to transmit information over a network, some subset of which is controlled by a malicious adversary. The computationally unbounded, hidden adversary knows the message to be transmitted, and can observe and change information over the part of the network being controlled. The network nodes do not share resources such as shared randomness or a private key. We first consider a unicast problem in a network with |epsiv parallel, unit-capacity, directed edges. The rate-region has two parts. If the adversary controls a fraction p < 0.5 of the |epsiv edges, the maximal throughput equals (1 - p) |epsiv|. We describe low-complexity codes that achieve this rate-region. We then extend these results to investigate more general multicast problems in directed, acyclic networks

We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes.

Distributed randomized network coding, a robust approach to multicasting in distributed network settings, can be extended to provide Byzantine modification detection without the use of cryptographic functions is presented in this paper.

We present an information theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a fundamental way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We give bounds on the network management information needed for link failure recovery in various network connection problems, in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. This is the first paper to our knowledge that looks at network management for general connections.

In large multiagent games, partial observability, coordination, and credit assignment persistently plague attempts to design good learning algo- rithms. We provide a simple and efficient algorithm that in part uses a linear system to model the world from a single agent’s limited per- spective, and takes advantage of Kalman filtering to allow an agent to construct a good training signal and learn an effective policy.