Existing studies on federated learning (FL) are mostly focused on system orchestration forstatic snapshotsof the network and makingstatic control decisions(e.g., spectrum allocation). However, real-world wireless networks are susceptible totemporal variationsof wireless channel capacity and users’ datasets. In this paper, we study the impacts of the dynamics: 1) wireless channels and 2) users’ datasets on the FL execution. The former is captured by introducing a set of discrete time events while the latter is characterized by a novelordinary differential equationand the metric ofdynamic model drift, formulated via apartial differential inequality, drawing concrete analytical connections between the dynamics of users’ datasets and FL accuracy. We then proposedynamiccooperative FLwith dedicatedMAC schedulers (DCLM), exploiting the unique features of open radio access network (O-RAN) to execute FL.DCLMentails: 1) a hierarchical device-to-device (D2D)-assisted model training; 2) dynamic control decisions through dedicated O-RAN MAC schedulers; and 3) asymmetric user selection. We provide extensive theoretical analysis to study the convergence ofDCLMand then aim to optimize its degrees of freedom (e.g., user selection and spectrum allocation) through a non-convex optimization problem. We develop a systematic and generic approach to obtain the solution for this problem. We finally show the efficiency ofDCLMvia numerical simulations and provide a series of future directions.

In an arbitrarily varying channel (AVC), the channel has a state which is under the control of an adversarial jammer and the corresponding capacities are often functions of the “power” constraints on the transmitter and jammer. In this paper we propose a model in which the constraints must hold almost surely over contiguous subsequences of the codeword and state, which we call a sliding window constraint. We study oblivious jammers and codes with stochastic encoding under maximum probability of error. We show that this extra limitation on the jammer is beneficial for the transmitter: in some cases, the capacity for unique decoding with a sliding window constraint is equal to the capacity for list decoding in the standard model without sliding windows, roughly implying that the addition of window constraints reduces list decoding to unique decoding. The list decoding capacity in the standard model can be strictly larger than the unique decoding capacity.

A mechanism called switched feedback is introduced; under switched feedback, each channel output goes forward to the receiver(s) or back to the transmitter(s) but never both. By studying the capacity of the Multiple-Access Channel (MAC) with switched feedback, this work investigates the benefits of feedback, seeking to maximize that benefit under reliable and unreliable feedback scenarios. The study is used to explore the tradeoffs between cooperation and transmission in the context of communication systems. Results include upper and lower bounds on the capacity region of the MAC with switched feedback.

Let$n_{q}(M, d)$be the minimum length of a$q$-ary code of size$M$and minimum distance$d$. Bounding$n_{q}(M, d)$is a fundamental problem that lies at the heart of coding theory. This work considers a generalization$n_{q}^{\bullet}(M, d)$of$n_{q}(M, d)$corresponding to codes in which codewords have protected and unprotected entries; where (analogs of) distance and of length are measured with respect to protected entries only. Such codes, here referred to as box codes, have seen prior studies in the context of bipartite graph covering. Upper and lower bounds on$n_{q}^{\bullet \bullet}(M, d)$are presented.

This work provides a framework for the design of codes for zero-error information theory problems via superposition coding and suitable expurgation processes. For the Hamming metric, the proposed code construction meets the Gilbert-Varshamov bound via a different construction/analysis than the folklore uniformly random ensemble usually considered in the literature.

We consider reward systems defined as iterative decision-making processes, where a player selects an action from the unit interval, and the environment responds by choosing a reward function from a known set of functions. The goal of the player is to accumulate rewards that exceed a given threshold in minimal time, and the performance is measured via regret with respect to an optimal player who knows the entire sequence of reward functions in advance. The central challenge lies in the dynamical nature of the reward system: each time step may involve a different reward function, requiring the player’s policy to adapt over time and making the regret an infinite-letter optimization problem. Our main result is an explicit expression for the optimal regret in the case of two linear reward functions that have opposing slopes. Moreover, we show that the optimal regret is achieved by a piecewise-constant action sequence, where both the transition times and action values exhibit special structural properties. These properties seem fundamental and may extend to classes of nonlinear reward functions. Finally, we highlight the implications of our solution in the context of communication, particularly, in characterizing the capacity of arbitrarily varying channels (AVCs) under competitive performance criteria.

For a multi-source multi-terminal noiseless network, the multicastkey-disseminationproblem, here called thekey-castproblem, involves the task of multicasting a (secret) keyKfrom the network sources to its terminals. Unlike traditional communication, where messages must be delivered from source to destination(s) unchanged, key-cast is more flexible since key-cast need not require source reconstruction at destination nodes. Instead, the distributed key can be a mixture of sources from which the sources themselves may be unrecoverable. Key-cast (also known as secret key agreement) in memoryless networks has seen significant studies over the past decades. This work initiates the study of key-cast in noiseless networks, i.e., network coding, and addresses the similarities and differences between traditional forms of communication that require source reconstruction and the less constrained form of communication in the key-cast task. The study varies according to three criteria. The setting can be secure, in which case the shared key is not revealed to an eavesdropper (whose capabilities we constrain), or non-secure, in which case the key need not be hidden. The setting can have one or multiple source nodes capable of generating independent randomness to be used in the key generation process. And the setting can have one or multiple terminal sets, where the latter case is referred to as multiple key-cast. In multiple key-cast, one requires that terminals in each terminal set decode a different shared key. In the given settings, this work derives combinatorial conditions for key-cast and multiple key-cast and designs corresponding communication schemes. In addition, the study compares the key-cast rate with and without the restriction of source reconstruction that is needed in traditional forms of communication. Key-cast achieves a strict advantage in rate when source reconstruction is relaxed.

We propose a concatenated code construction for a class of discrete-alphabet oblivious arbitrarily varying channels (AVCs) with cost constraints. The code has time and space complexity polynomial in the blocklength$n$. It uses a Reed-Solomon outer code, logarithmic blocklength random inner codes, and stochastic encoding by permuting the codeword before transmission. When the channel satisfies a condition called strong DS-nonsymmetrizability (a modified version of nonsymmetrizability originally due to Dobrushin and Stambler), we show that the code achieves a rate that for a variety of oblivious AVCs (such as classically studied error/erasure channels) match the known capacities.

Motivated by the design of low-complexity low-power coding solutions for the Gaussian relay channel, this work presents an upper bound on the minimum energy-per-bit achiev-able on the Gaussian relay channel using rank-1 linear relaying. Our study addresses high-dimensional relay codes and presents bounds that outperform prior known bounds using 2-dimensional schemes. A novelty of our analysis ties the optimization problem at hand to the solution of a certain differential equation which, in turn, leads to a low energy-per-bit achievable scheme.

Motivated by the study of multi-source multiterminal key-dissemination, here called “key-cast;’ the work at hand presents a combinatorial characterization of when positiverate secure multicast network coding in the presence of eaves-dropping nodes is possible. In key-cast, introduced by the authors in [ITW2022], network nodes hold independent random bits, and one seeks a communication scheme that allows all terminal nodes to share a secret key$K$. We here address positive (albeit, arbitrarily small) rate key-cast under the security requirement that no single non-terminal network node can gain information about the shared key$K$; this scenario is useful in cryptographic settings. The work at hand studies key-dissemination protocols based on secure network coding and presents a combinatorial characterization of networks that support positive-rate multicast resilient to eavesdroppers that control individual network nodes. The secure-multicast capacity solution in the same setting is a known open problem.

Arbitrary varying channels (AVC) are used to model communication settings in which a channel state may vary arbitrarily over time. Their primary objective is to circumvent statistical assumptions on channel variation. Traditional studies on AVCs optimize rate subject to the worst-case state sequence. While this approach is resilient to channel variations, it may result in low rates for state sequences that are associated with relatively good channels. This paper addresses the analysis of AVCs through the lens of competitive analysis, where solution quality is measured with respect to the optimal solution had the state sequence been known in advance. Our main result demonstrates that codes constructed by a single input distribution do not achieve optimal competitive performance over AVCs. This stands in contrast to the single-letter capacity formulae for AVCs, and it indicates, in our setting, that even though the encoder cannot predict the subsequent channel states, it benefits from varying its input distribution as time proceeds.

This paper focuses on the study of the minimum possible alphabet size of codes in a generalized setting where the coding scheme is required to handle a pre-specified set of erasure or error patterns, naturally represented by a hypergraph. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size of codes in this generalized setting and the combinatorial properties of the hypergraph that represents the pre-specified collection of erasure or error patterns. We also establish connections between error and erasure correcting codes in our generalized setting. Finally, we consider a variation of the problem that allows a small probability of decoding error and relate it to an approximate version of hypergraph coloring.

We consider communication over channels whose statistics are not known in full, but can be parameterized as a finite family of memoryless channels. A typical approach to address channel uncertainty is to design codes for the worst channel in the family, resulting in the well-known compound channel capacity. Although this approach is robust, it may suffer a significant loss of performance if the capacity-achieving distribution of the worst channel attains low rates over other channels. In this work, we cope with channel uncertainty through the lens ofcompetitive analysis. The main idea is to optimize a relative metric that compares the performance of the designed code and a clairvoyant code that has access to the true channel. To allow communication rates that adapt to the channel at use, we consider rateless codes with a fixed number of message bits and random decoding times. We propose two competitive metrics: the competitive ratio between the expected rates of the two codes, and a regret defined as the difference between the expected rates. The competitive ratio, for instance, provides a percentage guarantee on the expected rate of the designed code when compared to the rate of the clairvoyant code that knows the channel at hand. Our main results are single-letter expressions for the optimalcompetitive-ratioandregret, expressed as a max-min or minmax optimization. Several examples illustrate the benefits of the competitive analysis approach to code design compared to the compound channel.

The multiple access channel (MAC) capacity with feedback is considered under feedback models designed to tease out which factors contribute to the MAC feedback capacity benefit. Comparing the capacity of a MAC with "perfect" feedback, which causally delivers to the transmitters the true channel output, to that of a MAC with "independent" feedback, which causally delivers to the transmitters an independent instance of that same channel output, allows separation of effects like cooperation from alternative feedback benefits such as knowledge of the channel instance. Proving that the Cover-Leung (CL) achievability bound, which is known to be loose for some channels, is achievable also under (shared or distinct) independent feedback at the transmitters shows that the CL bound does not require transmitter knowledge of the channel instance. Proving that each transmitter’s maximal rate under independent feedback exceeds that under perfect feedback highlights the potential power of an independent look at the channel output.

We consider communication over channels whose statistics are not known in full, but can be parameterized as a finite family of memoryless channels. A typical approach to address channel uncertainty is to design codes for the worst channel in the family, resulting in the well-known compound channel capacity. Although this approach is robust, it may suffer a loss of performance if the capacity-achieving distribution of the worst channel attains low rates over other channels. In this work, we cope with channel uncertainty through the lens of competitive analysis. The idea is to optimize a relative metric that compares the performance of the designed code and a clairvoyant code that has access to the true channel. To allow communication rates that can adapt to the channel at use, we consider rateless codes with a fixed number of information bits and random decoding times. We propose two competitive metrics: the competitive ratio between the decoding times of the two codes, and a regret defined as the difference between the expected rates. Our main results are single-letter expressions for the competitive-ratio and the regret, expressed as a max-min or min-max optimization. Several examples illustrate our results and the benefits of the competitive analysis approach to code design.

We study communication models for channels with erasures in which the erasure pattern can be controlled by an adversary with partial knowledge of the transmitted codeword. In particular, we design block codes for channels with binary inputs with an adversary who can erase a fraction p of the transmitted bits. We consider causal adversaries, who must choose to erase an input bit using knowledge of that bit and previously transmitted bits, and myopic adversaries, who can choose an erasure pattern based on observing the transmitted codeword through a binary erasure channel with random erasures. For both settings we design efficient (polynomial time) encoding and decoding algorithms that use randomization at the encoder only. Our constructions achieve capacity for the causal and "sufficiently myopic" models. For the "insufficiently myopic" adversary, the capacity is unknown, but existing converses show the capacity is zero for a range of parameters. For all parameters outside of that range, our construction achieves positive rates.

Group testing is one of the fundamental problems in coding theory and combinatorics in which one is to identify a subset of contaminated items from a given ground set. There has been renewed interest in group testing recently due to its applications in diagnostic virology, including pool testing for the novel coronavirus. The majority of existing works on group testing focus on the uniform setting in which any subset of size d from a ground set V of size n is potentially contaminated.In this work, we consider a generalized version of group testing with an arbitrary set-system of potentially contaminated sets. The generalized problem is characterized by a hypergraph H = (V, E), where V represents the ground set and edges e ∈ E represent potentially contaminated sets. The problem of generalized group testing is motivated by practical settings in which not all subsets of a given size d may be potentially contaminated, rather, due to social dynamics, geographical limitations, or other considerations, there exist subsets that can be readily ruled out. For example, in the context of pool testing, the edge set E may consist of families, work teams, or students in a classroom, i.e., subsets likely to be mutually contaminated. The goal in studying the generalized setting is to leverage the additional knowledge characterized by H = (V, E) to reduce the number of tests.The paper considers both adaptive and non-adaptive group testing and makes the following contributions. First, for the non-adaptive setting, we show that finding an optimal solution for the generalized version of group testing is NP-hard. For this setting, we present a solution that requires O(d log |E|) tests, where d is the maximum size of a set e ∈ E. Our solutions generalize those given for the traditional setting and are shown to be of order-optimal size O(log |E|) for hypergraphs with edges that have “large” symmetric differences. For the adaptive setting, when edges in E are of size exactly d, we present a solution of size O(log |E| + d log2d) that comes close to the lower bound of $\Omega (\log |E| + d)$

This work addresses the cooperation facilitator (CF) model, in which network nodes coordinate through a rate limited communication device. For multiple-access channel (MAC) encoders, the CF model is known to show significant rate benefits, even when the rate of cooperation is negligible. Specifically, the benefit in MAC sum-rate, as a function of the cooperation rate CCF, sometimes has an infinite slope at CCF= 0 when the CF enables transmitter dependence where none was possible otherwise. This work asks whether cooperation through a CF can yield similar infinite-slope benefits when dependence among MAC transmitters has no benefit or when it can be established without the help of the CF. Specifically, this work studies the CF model when applied to relay nodes of a single-source, single-terminal, diamond network comprising a broadcast channel followed by a MAC. In the relay channel with orthogonal receiver components, careful generalization of the partial-decode-forward/compress-forward lower bound to the CF model yields sufficient conditions for an infinite-slope benefit. Additional results include derivation of a family of diamond networks for which the infinite-slope rate-benefit derives directly from the properties of the corresponding MAC studied in isolation.

We characterize the capacity for the discrete-time arbitrarily varying channel with discrete inputs, outputs, and states when (a) the encoder and decoder do not share common randomness, (b) the input and state are subject to cost constraints, (c) the transition matrix of the channel is deterministic given the state, and (d) at each time step the adversary can only observe the current and past channel inputs when choosing the state at that time. The achievable strategy involves stochastic encoding together with list decoding and a disambiguation step. The converse uses a two-phase "babble-and-push" strategy where the adversary chooses the state randomly in the first phase, list decodes the output, and then chooses state inputs to symmetrize the channel in the second phase. These results generalize prior work on specific channels models (additive, erasure) to general discrete alphabets and models.

For a multi-source multi-terminal noiseless network, the key-dissemination problem involves the task of multicasting a secret key K from the network sources to its terminals. As in secure multicast network-coding, in the key-dissemination problem the source nodes have access to independent randomness and, as the network is noiseless, the resulting key K is a function of the sources’ information. However, different from traditional forms of multicast, in key-dissemination the key K need not consist of source messages, but rather may be any function of the information generated at the sources, as long as it is shared by all terminals. Allowing the shared key K to be a mixture of source information grants a flexibility to the communication process which gives rise to the potential of increased key-rates when compared to traditional secure multicast. The multicast key-capacity is the supremum of achievable key-rates, subject to the security requirement that the shared key is not revealed to an eavesdropper with predefined eavesdropping capabilities. The key-dissemination problem (termed also, secret key-agreement) has seen significant studies over the past decades in memoryless network structures. In this work, we initiate the study of key-dissemination in the context of noiseless networks, i.e., network coding. In this context, we study similarities and differences between traditional secure-multicast and the more lenient task of key-dissemination.

We study the relation betweenlatencyandalphabet sizein the context of Multicast Network Coding. Given a graph$G = (V, E)$representing a communication network, a subset$S \subseteq V$of sources, each of which initially holds a set of information messages, and a set$T \subseteq V$of terminals; we consider the problem in which one wishes to design a communication scheme that eventually allows all terminals to obtain all the messages held by the sources. In this study we assume that communication is performed in rounds, where in each round each network node may transmit a single (possibly encoded) information packet on any of its outgoing edges. The objective is to minimize the communication latency, i.e., number of communication rounds needed until all terminals have all the messages of the source nodes. For sufficiently large alphabet sizes (i.e., large block length, packet sizes), it is known that traditional linear multicast network coding techniques (such as random linear network coding) minimize latency. In this work we seek to study the task of minimizing latency in the setting of limited alphabet sizes (i.e., finite block length), and alternatively, the task of minimizing the alphabet size in the setting of bounded latency. We focus on the establishing the computation complexity of the problem and present several intractability results. In particular, through reductive arguments, we prove that it is NP-hard to (i) approximate (and in particular to determine) the minimum alphabet size given a latency constraint; (ii) approximate (and in particular to determine) the minimum latency of communication schemes in the setting of limited alphabet sizes.

This paper focuses on error-correcting codes that can handle a predefined set of specific error patterns. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size for this generalized setting and the combinatorial properties of a hypergraph that represents the prespecified collection of error patterns. We also show a connection between error and erasure correcting codes in this specialized setting. This allows us to establish bounds on the minimum alphabet size and show an advantage of non-linear codes over linear codes in a generalized setting. We also consider a variation of the problem which allows a small probability of decoding error and relate it to an approximate version of the hypergraph coloring problem.

The work at hand presents a finite-blocklength analysis of the multiple access channel (MAC) sum-rate under the cooperation facilitator (CF) model. The CF model, in which independent encoders coordinate through an intermediary node, is known to show significant rate benefits, even when the rate of cooperation is limited. We continue this line of study for cooperation rates which are sub-linear in the blocklength n. Roughly speaking, our results show that if the facilitator transmits log$K$bits, then there is a sum-rate benefit of order √log K/n compared to the best-known achievable rate. This result extends across a wide range of K: even a single bit of cooperation is shown to provide a sum-rate benefit of order 1/√n.

The edge-removal problem asks whether the removal of a$\lambda$-capacity edge from a given network can decrease the communication rate between source-terminal pairs by more than$\lambda$. We prove that for undirected networks, removing a$\lambda$capacity edge decreases the rate by$O(\lambda$. Through previously known reductive arguments, here newly applied to undirected networks, our result implies that the zero-error capacity region of an undirected network equals its vanishing-error capacity region. Whether it is possible to prove similar results for directed networks remains an open question.

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve performance metrics such as the total rate delivered across the network. This work studies encoder cooperation in the setting of a discrete multiple access channel (MAC) with two encoders and a single decoder. A network node, here called the cooperation facilitator (CF), that is connected to both encoders via rate-limited links, enables the cooperation strategy. Previous work by the authors presents two classes of MACs: (i) one class where the average-error sum-capacity has an infinite derivative in the limit where CF output link capacities approach zero, and (ii) a second class of MACs where the maximal-error sum-capacity is not continuous at the point where the output link capacities of the CF equal zero. This work contrasts the power of the CF in the maximal- and average-error cases, showing that a constant number of bits communicated over the CF output link can yield a positive gain in the maximal-error sum-capacity, while a far greater number of bits, even a number that grows sublinearly in the blocklength, can never yield a non-negligible gain in the average-error sum-capacity.

Myopic arbitrarily varying channels (AVCs) are point-to-point communication models in which a channel state is controlled by a malicious adversary (a jammer) who receives side-information about the transmitted codeword via a side-channel (wiretapping) and wishes to maximize the probability of error. Compared to standard "oblivious" AVCs, myopic AVCs can potentially use the side information to launch a more effective attack, lowering the capacity of the channel. In this paper, we define a novel property, myopic symmetrizability, and prove it is a sufficient condition for the capacity of any myopic AVC to be zero. We also study the sufficiently myopic setting, in which, roughly speaking, the jammer's side information reveals less information on the codeword transmitted than eventually available at the receiver. In this scenario we show that myopic symmetrizability is also a necessary condition for the capacity to equal zero, by providing a novel code construction using non-i.i.d. codebooks. A key technical lemma, interesting in its own right, is an argument showing that for any positive-rate code (whether for myopic AVCs or not) one can identify a corresponding distribution PX,X'that is a convex combination of product distributions, and such that a constant fraction of pairs of codewords have an empirical distribution approximately equaling PX,X'.

A Maximum Distance Separable code over an alphabet F is defined via an encoding function C : Fk→ Fnthat allows to retrieve a message m ∈ Fkfrom the codeword C(m) even after erasing any n - k of its symbols. The minimum possible alphabet size of general (non-linear) MDS codes for given parameters n and k is unknown and forms one of the central open problems in coding theory. The paper initiates the study of the alphabet size of codes in a generalized setting where the coding scheme is required to handle a pre-specified subset of all possible erasure patterns, naturally represented by an n-vertex k-uniform hypergraph. We relate the minimum possible alphabet size of such codes to the strong chromatic number of the hypergraph and analyze the tightness of the obtained bounds for both the linear and non-linear settings. We further consider variations of the problem which allow a small probability of decoding error.

The edge-removal problem addresses the loss in capacity obtained by the removal of an edge of capacity λ from a given network. In the context of network coding, the edgere-moval problem has been solved for certain families of networks (such as instances with co-located sources). However, the problem is open for general instances. This work ties the edge-removal problem to two additional problems in the context of network communication: (a) the “zero-error” network coding problem, which asks whether the zero-error network coding capacity differs from the capacity when error probability is allowed to tend to zero as the blocklength grows; and (b) the “δ-dependence” problem, which measures the advantage of message dependence in network coding capacity.

It is common in the study of secure multicast network coding in the presence of an eavesdropper that has access to z network links, to assume that the source node is the only node that generates random keys. In this setting, the secure multicast rate is well understood. Computing the secure multicast rate, or even the secure unicast rate, in the more general setting in which all network nodes may generate (independent) random keys is known to be as difficult as computing the (non-secure) capacity of multiple-unicast network coding instances - a well known open problem. This work treats an intermediate model of secure unicast in which only one node can generate random keys, however that node need not be the source node. The secure communication rate for this setting is characterized again with an eavesdropper that has access to z network links.

The difference in capacity formulae between worst-case and average-case channel noise models has been part of information theory since the early days of the field. This paper continues a line of work studying intermediate models in which the channel behavior can depend partially on the transmitted codeword. In particular, we consider a model in which a binary erasure channel (with maximum fraction of erasures p) is controlled by an adversary who can observe the transmitted codeword through an independent and memoryless erasure channel (with erasure probability q). Upper and lower bounds on the capacity are given for two models: a noncausal model, in which the adversary can choose their erasures based on the entire (partially observed) codeword, and a causal model, in which at each time the adversary must choose its erasures based on the current and previously observed codeword bits. The achievable rate for the noncausal case is larger than the Gilbert-Varshamov bound and for some parameter ranges exceeds the linear programming (LP) bound; we also provide a non-trivial outer bound on the capacity. For the causal case, we show the capacity is 1-2p+q for p ≥ q (prior work shows the capacity to equal 1-p when p<;q). Our code construction in both scenarios are novel, requiring the encoder to carefully add “low-weight correlated noise” to its transmission.

The edge removal problem studies the loss in network coding rates that results when a network communication edge is removed from a given network. It is known, for example, that in networks restricted to linear coding schemes and networks restricted to Abelian group codes, removing an edge e* with capacity Re. reduces the achievable rate on each source by no more than Re*. In this work, we seek to uncover larger families of encoding functions for which the edge removal statement holds. We take a local perspective: instead of requiring that all network encoding functions satisfy certain restrictions (e.g., linearity), we limit only the function carried on the removed edge e*. Our central results give sufficient conditions on the function carried by edge e* in the code used to achieve a particular rate vector under which we can demonstrate the achievability of a related rate vector once e* is removed.

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries ($\FAQ$s) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Chattopadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an \em arbitrary communication topology. This model, which is closer to the well-studied $\congest$ model in distributed computing and generalizes Yao's two party communication complexity model, has so far only been studied for problems that are common in the two-party communication complexity literature. This is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two specific problems in this paper: Boolean Conjunctive Query ($\BCQ$) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is $O(1)$-degenerate and has $O(1)$-arity. In particular, the $O(1)$-degeneracy condition covers most well-studied queries that are efficiently computable in the centralized computation model like queries with constant treewidth. These tight bounds depend on a new notion of 'width' (namely \em internal-node-width ) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a sub-class of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over a finite field of two elements) using a novel min-entropy based argument.

In the q-ary online (or “causal”) channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1,..., xn) ∈ {0, 1,..., q-1}nsymbol-by-symbol via a channel limited to at most pn errors and p*n erasures. The channel is “online” in the sense that at the ith step of communication the channel decides whether to corrupt the ith symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols (x1, . . ., xi). This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge of the sent codeword x. In this paper, we study the capacity of q-ary online channels for a combined corruption model, in which the channel may impose at most pn errors and at most p*n erasures on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies, which present both upper and lower bounds on its capacity. In this paper, we give a full characterization of the capacity as a function of q, p, and p*.

We consider a communication problem in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial jammer, James, who is myopic. Specifically, for blocklength n, the codeword Xntransmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of Xnto corrupt and how to corrupt them) on the non-causal observation Znof Xnobtained through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channelpZ|Xfrom Alice to James, and an arbitrarily varying channel from Alice to Bob, pY|XSgoverned by a state Sndetermined by James. In standard adversarial channels, the states Snmay depend on the codeword Xn, but in our setting Sndepends non-causally only on James's view Zn. We present upper and lower bounds on the capacity of myopic channels. For a number of special cases of interest we show that our bounds are tight. We then extend our results to the setting of secure communication, in which we require that the transmitted message remains secret from James. For example, we show that if 1i) James may flip at most a p fraction of the bits communicated between Alice and Bob and 2) James views Xnthrough a binary symmetric channel with crossover probability q, then once James is “sufficiently myopic” (in this case, when pH(p)), then the optimal communication rate is that of an adversary who is “blind” (that is, an adversary that does not have any knowledge of Xnat all), which is 1- H(p) for standard communication, and H(q)- H(p) for secure communication. A similar phenomenon exists for more general models of communication.

We obtain a relationship between the rate and security for secure multi-cast linear network codes in the presence of a “wire-tap” adversary who can eavesdrop on a bounded number of network edges. Specifically, we show that a network code that can communicate information at rate R and is secure against an adversary eavesdropping on z network edges can be transformed to a code that allows communication at rate R-1 while being secure against a more potent adversary eavesdropping on z+1 network edges. Results of this nature are known under the assumption that only the network source node has the ability to generate random keys used to obfuscate the communicated messages. The novelty of this work lies in studying the setting in which each network node may generate independent random keys.

We consider the problem of communication over a channel with a causal jamming adversary subject to quadratic constraints. A sender Alice wishes to communicate a message to a receiver Bob by transmitting a real-valued length-n codeword x=(x1, ..., xn) through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, a jammer James chooses a real-valued length- n adversarial noise sequence s=(s1, ..., sn) in a causal manner: each st (1 ≤ t ≤ n) can only depend on (x1, ..., xt). Bob receives y, the sum (over \mathbbR) of Alice's transmission x and James' jamming vector s, and is required to reliably estimate Alice's message from this sum. In addition, Alice and James's transmission powers are restricted by quadratic constraints P > 0 and N > 0 such that Σt=1nxt2≤ nP and Σt=1nst2≤ nN. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values Cn([P/N]) of a series of optimizations. Upper and lower bounds on Cn([P/N]) are provided both analytically and numerically. Interestingly, unlike many communication problems, in this causal setting Alice's optimal codebook may not have a uniform power allocation - for certain SNR a codebook with a two-level uniform power allocation results in a strictly higher rate than a codebook with a uniform power allocation would.

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve network performance metrics such as sum-rate. This work studies encoder cooperation in the setting of a discrete multiple access channel with two encoders and a single decoder. A node in the network that is connected to both encoders via rate-limited links, referred to as the cooperation facilitator (CF), enables the cooperation strategy. Previously, the authors presented a class of multiple access channels where the average-error sum-capacity has an infinite derivative in the limit where CF output link capacities approach zero. The authors also demonstrated that for some channels, the maximal-error sum-capacity is not continuous at the point where the output link capacities of the CF equal zero. This work shows that the average-error sum-capacity is continuous when CF output link capacities converge to zero; that is, the infinite derivative of the average-error sum-capacity is not a result of its discontinuity as in the maximal-error case.

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.

Cooperation strategies allow communication devices to work together to improve network capacity. Consider a network consisting of k encoders, a multiple access channel (MAC), a decoder, and a node, referred to as a “cooperation facilitator” (CF), that is connected to each encoder via a pair of rate-limited links, with one link going from the encoder to the CF and the other link going back. Let the “cooperation rate” be the total outgoing rate of the CF. This paper demonstrates the existence of a class of MACs where the ratio of the sum-capacity gain to cooperation rate tends to infinity as the cooperation rate tends to zero. For any k ≥ 2, examples of channels in this class include the k-user binary adder MAC and the k-user Gaussian MAC.

In network cooperation strategies, nodes work together with the aim of increasing transmission rates or reliability. This paper demonstrates that enabling cooperation between the transmitters of a two-user multiple access channel via a cooperation facilitator that has access to both messages results in a network whose maximal- and average-error capacity regions are the same; this benefit ensues even when the information received by each transmitter is negligible. From this result, it follows that if a multiple access channel with no transmitter cooperation has different maximal- and average-error sum-capacities, then the maximal-error sum-capacity of the network consisting of this channel and a cooperation facilitator is not continuous with respect to the output edge capacities of the facilitator. Thus, there exist networks where adding negligible rate yields a non-negligible benefit.

We prove that there exist non-linear binary cyclic codes that attain the Gilbert-Varshamov bound.

Alice and Bob want to exchange information over an additive interference channel that also contains a malicious eavesdropper-jammer James who aims to disrupt this two-way communication. In the baseline model (motivated by wireless jamming scenarios), Alice and Bob transmit length-n q-ary encodings xAand xBrespectively of their own messages. James observes the interference pattern z = xA+ xB, and as a non-causal function of ζ and his knowledge of Alice and Bob's codebooks, chooses a jamming pattern s of power (Hamming weight) at most pn. Alice and Bob then both observe the interfered-jammed signal xA+ xB+ s, and aim to decode each others' messages despite the jamming pattern s. We demonstrate that in such a model, the fact of interference actually aids communication by allowing for communication to occur in each direction at a rate of 1 - Hq(p), i.e., the jammer can do no worse than act like “random noise”.1Interestingly, neither linear codes nor random codes (as “usually” defined) achieve this performance - we thus define and analyze a new class of codes we call linearish codes that do. We then extend our results to general q-ary additive-error channels with asymmetric jamming patterns (with potentially different powers) to Alice and Bob, and also demonstrate how to simultaneously ensure information-theoretic secrecy of both Alice and Bob's messages from James.

In many communication networks, the availability of channel state information at various nodes provides an opportunity for network nodes to work together, or “cooperate.” This work studies the benefit of cooperation in the multiple access channel with a cooperation facilitator, distributed state information at the encoders, and full state information available at the decoder. Under various causality constraints, sufficient conditions are obtained such that encoder cooperation through the facilitator results in a gain in sum-capacity that has infinite slope in the information rate shared with the encoders. This result extends the prior work of the authors on cooperation in networks where none of the nodes have access to state information.

A key result of classical information theory states that if the rate of a randomly generated codebook is less than the mutual information between the channel's input and output, then the probability that that codebook has negligible error goes to one as the blocklength goes to infinity. In an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this work derives necessary and sufficient conditions on the rate so that the probability that a randomly generated codebook operated under list decoding (for any fixed list size) has zero error probability goes to one as the blocklength goes to infinity. Furthermore, this work extends the classical birthday problem to an information-theoretic setting, which results in the definition of a “noisy” counterpart of Rényi entropy, analogous to how mutual information can be considered a noisy counterpart of Shannon entropy.

We consider a delay-constrained streaming model for zero-error communications and show that under this model, network coding and index coding problems are code equivalent. That is, any streaming network coding instance can be efficiently mapped to a corresponding acyclic streaming index coding instance such that an index code for the latter can be efficiently transformed into a network code for the former. This reduction holds even for network coding instances that contain cycles, thereby proving the first known reduction from cyclic to finite acyclic network coding networks.

We prove tight network topology dependent bounds on the round complexity of computing well studied k-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent such results on total communication that have received some attention. We also present some applications to distributed graph computation problems. Our main contribution is a proof technique that allows us to reduce the problem on a general graph topology to a relevant two-party communication complexity problem. However, unlike many previous works that also used the same high level strategy, we do not reason about a two-party communication problem that is induced by a cut in the graph. To ‘stitch’ back the various lower bounds from the two party communication problems, we use the notion of timed graph that has seen prior use in network coding. Our reductions use some tools from Steiner tree packing and multi-commodity flow problems that have a delay constraint.

We consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword x = (x1, ... , xn) is corrupted by a permutation π ∈ Snto yield the received wordy = (y1, . . . , yn), where yi= xπ(i). We initiate the study of worst case (or zero-error) communication over permutation channels that distort the information by applying permutations π, which are limited to displacing any symbol by at most r locations, i.e., permutations π with weight at most r in the ℓ∞-metric. We present direct and recursive constructions, as well as bounds on the rate of such channels for binary and general alphabets. Specific attention is given to the case of r = 1.

In the q-ary online (causal) channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, . . . , xn) ∈ {0, 1, . . . , q - 1}nsymbol-by-symbol via a channel limited to at most p*n errors (symbol changes) and p*n erasures. The channel is "online" (i.e., "causal") in the sense that at the ith step of communication the channel decides whether to corrupt the ith symbol or not only based on its view of the symbols (x1,. . . , xi). This is in contrast to the classical adversarial channel in which the corruption is chosen with full knowledge of the sent codeword x. In this work we extend the results obtained in [1]-[4] (in which the capacities of binary online bit-flip-only channels, and separately binary online erasure-only channels were characterized). We here extend those prior results in two important ways. First, we obtain the capacity of q-ary online channels for general q (rather than just q = 2). Second, we analyze combined error-erasure corruption models (rather than studying them separately). Characterization of this much broader class of symmetric online channels gives a fuller understanding of the effects of causality on jamming adversaries. The extensions in this paper require novel approaches for both optimal code designs, and matching information-theoretic converse arguments.

We consider the problem of communicating a message m in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to pn bits, out of n transmitted bits X = (x1, ..., xn). The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit xidepends on his observations (x1, ..., xi) was recently characterized [1], [2] to be 1 - 2p. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit xidepends only on (x1, ..., xi-1) (and is independent of the “current bit” xi) then the capacity increases to 1 - p when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword X is a deterministic function of the message m) then no rate asymptotically larger than 1 - 2p is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of 1- p against a one-bit-delayed Calvin.

In this work we characterize the capacity region for multi-source multi-terminal acyclic network coding with dependent information sources. We show that a full characterization of the capacity region can be derived in terms of entropy functions.

Cooperation strategies that allow communication devices to work together can improve network capacity. This paper generalizes the “cooperation facilitator” (CF) model from the 2-user to the k-user multiple access channel (MAC), extending capacity bounds, characterizing all k-user MACs for which the sum-capacity gain of encoder cooperation exceeds the capacity cost that enables it, and demonstrating an infinite benefit-cost ratio in the limit of small cost.

In network cooperation strategies, nodes work together with the aim of increasing transmission rates or reliability. This paper demonstrates that enabling cooperation between the transmitters of a two-user multiple access channel via a cooperation facilitator that has access to both messages, always results in a network whose maximal- and average-error sum-capacities are the same-even when the information shared with the encoders is negligible. Thus, for a multiple access channel whose maximal- and average-error sum-capacities differ, the maximal-error sum-capacity is not continuous with respect to the output edge capacities of the facilitator. This shows that for some networks, sharing even a negligible number of bits per channel use with the encoders can yield a non-negligible benefit.

In this work, we study the Yeung network coding entropic function outer bound and prove an equivalence relationship between its tightness and the edge removal problem. In addition, we derive an implicit characterization of the 0-error capacity region using restricted sets of entropic vectors.

We propose efficient coding schemes for two communication settings: 1) asymmetric channels and 2) channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes that resulted from the use of large Boolean functions. We propose an improved scheme that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder using a multicoding technique. We consider specific instances of the scheme for flash memories that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton's region, improving upon previous schemes for this setting.

A secret sharing scheme is a method to store information securely and reliably. Particularly, in a threshold secret sharing scheme, a secret is encoded into n shares, such that any set of at least t1shares suffice to decode the secret, and any set of at most t21shares reveal no information about the secret. Assuming that each party holds a share and a user wishes to decode the secret by receiving information from a set of parties; the question we study is how to minimize the amount of communication between the user and the parties. We show that the necessary amount of communication, termed “decoding bandwidth”, decreases as the number of parties that participate in decoding increases. We prove a tight lower bound on the decoding bandwidth, and construct secret sharing schemes achieving the bound. Particularly, we design a scheme that achieves the optimal decoding bandwidth when d parties participate in decoding, universally for all t1≤ d ≤ n. The scheme is based on a generalization of Shamir's secret sharing scheme and preserves its simplicity and efficiency. In addition, we consider the setting of secure distributed storage where the proposed communication efficient secret sharing schemes not only improve decoding bandwidth but further improve disk access complexity during decoding.

In this work we consider the communication setting in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial entity, Calvin, who is myopic. Roughly speaking, for blocklength n, the codeword Xntransmitted by Alice is corrupted by Calvin who must base his adversarial decisions, on which characters of Xnto corrupt and how to corrupt them, not on the entire view of the codeword Xnbut on Zn, the image of Xnthrough a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel p(z|x) from Alice to Calvin, and an arbitrarily varying channel from Alice to Bob, p(y|x, s) governed by a states Sndetermined by Calvin. In standard adversarial channels, the states Snmay depend on the codeword Xn, however in our setting Sndepends only on Calvin's view Zn. The myopic channel captures a broad range of channels and bridges between the standard models of memoryless and adversarial (zero error) channels. In this work we present upper and lower bounds on the capacity of myopic channels. For a number of special cases of interest we show that our bounds are tight. We extend our results to the setting of secure communication in which we require that the transmitted message remain secret from Calvin. For example, we show that if (i) Calvin may flip at most a p fraction of the bits communicated between Alice and Bob, and (ii) Calvin views Xnthrough a binary symmetric channel with parameter q, then once Calvin is “sufficiently myopic” (in this case, when q > p), then the optimal communication rate is that of an adversary who is “blind” (that is, an adversary that does not see Xnat all), which is 1-H(p) for standard communication, and H(q)-H(p) for secure communication. A similar phenomena exists for our general model of communication.

We show that solving a multiple-unicast network coding problem can be reduced to solving a single-unicast network error correction problem, where an adversary may jam at most a single edge in the network. Specifically, we present an efficient reduction that maps a multiple-unicast network coding instance to a network error correction instance while preserving feasibility. The reduction holds for both the zero probability of error model and the vanishing probability of error model. Previous reductions are restricted to the zero-error case. As an application of the reduction, we present a constructive example showing that the single-unicast network error correction capacity may not be achievable, a result of separate interest.

In this work we consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword x = (x1, ..., xn) is corrupted by a permutation π ∈ Snto yield the received word y = (y1, ..., yn) where yi= xπ(i). We initiate the study of worst case (or zero error) communication over permutation channels that distort the information by applying permutations π which are limited to displacing any symbol by at most r locations, i.e. permutations π with weight at most r in the ℓ∞-metric. We present direct and recursive constructions, as well as bounds on the rate of such channels for binary and general alphabets. Specific attention is given to the case of r = 1.

In cooperative communication, network nodes that would otherwise act independently instead coordinate their efforts with the aim of improving communication performance. To better understand cooperation, we consider communication over a multiple access channel using a “cooperation facilitator”, a node that receives rate-limited message descriptions from the transmitters and sends rate-limited message descriptions back. This model includes the conferencing encoders model and a prior model from the current authors as special cases. We characterize a class of multiple access channels for which there is no gain in sum-capacity under current or prior cooperation models. We then show that for all other multiple access channels, the gain in sum-capacity can be far greater than the capacity of the cooperation facilitator's output links. These channels violate the edge removal property. The Gaussian multiple access channel is an important special case for which we explicitly characterize the sum-rate cooperation gain.

In recent work, Kamath et al. show that network code design for any k-unicast network reduces to network code design for a related 2-unicast network. The proof assumes that codes achieve their desired rates precisely (rather than approaching them asymptotically) and that error probability equals zero. We study two questions posed in but left unanswered by the Kamath et al. paper. The first asks whether the reduction for 0-error code design can be extended to show an equivalence in 0-error network capacity, which includes rates approached asymptotically. The second asks whether the reduction can be generalized to show an equivalence in Shannon capacity, which requires that error probability approach (but not necessarily hit) zero. While we do not solve these questions, we show that finding the k-unicast capacity reduces to finding the 2-unicast capacity under this reduction if and only if the so called “edge removal statement” is true for all networks. This equivalence holds under both 0-error and asymptotic notions of reliability.

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1,...,xn) ∈ {0,1}n bit by bit via a channel limited to at most pn corruptions. The channel is "online" in the sense that at the ith step of communication the channel decides whether to corrupt the ith bit or not based on its view so far, i.e., its decision depends only on the transmitted bits (x1,...,xi). This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has full knowledge of the transmitted codeword x.

We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and nonlinear codes. Specifically, we present a reduction that maps a network coding instance to an index coding instance while preserving feasibility, i.e., the network coding instance has a feasible solution if and only if the corresponding index coding instance is feasible. In addition, we show that one can determine the capacity region of a given network coding instance with colocated sources by studying the capacity region of a corresponding index coding instance. Previous connections between network and index coding were restricted to the linear case.

We consider the coded cooperative data exchange problem for general graphs, both undirected and directed. In this problem, given a graph G = (V, E) representing clients in a broadcast network, each of which initially hold a (not necessarily disjoint) set of information packets; one wishes to design a communication scheme in which eventually all clients will hold all the packets of the network. Communication is performed in rounds, where in each round a single client broadcasts a single (possibly encoded) information packet to its neighbors in G. The objective is to design a broadcast scheme that satisfies all clients with the minimum number of broadcast rounds. The coded cooperative data exchange problem has seen significant research over the last few years; mostly when the graph G is the complete broadcast graph in which each client is adjacent to all other clients in the network, but also on general topologies, both in the fractional and integral setting. In this paper, we focus on the integral setting in general topologies G, both undirected and directed. For undirected graphs, we tie the coded cooperative data exchange problem on G to variants of the dominating set problem and in such show that solving the problem exactly or even approximately within a multiplicative factor of log |V| is intractable (i.e., NP-hard). We then turn to study efficient data exchange schemes for undirected topologies yielding a number of communication rounds comparable with our intractability result. Last, we tie the coded cooperative data exchange problem for directed topologies to the directed Steiner tree problem, yielding efficient data exchange approximation schemes. Our communication schemes do not involve encoding, and in such yield bounds on the coding advantage in the setting at hand.

Motivated by the study of deletion channels, this paper presents improved bounds on the number of subsequences obtained from a binary string X of length n under t deletions. It is known that the number of subsequences in this setting strongly depends on the number of runs in the string X; where a run is a maximal substring of the same character. Our improved bounds are obtained by a structural analysis of the family of r-run strings X, an analysis in which we identify the extremal strings with respect to the number of subsequences. Specifically, for every r, we present r-run strings with the minimum (respectively maximum) number of subsequences under any t deletions; we perform an exact analysis of the number of subsequences of these extremal strings; and show that this number can be calculated in polynomial time.

We consider the noisy write-once memory (WOM) model to capture the behavior of data-storage devices such as flash memories. The noisy WOM is an asymmetric channel model with non-causal state information at the encoder. We show that a nesting of non-linear polar codes achieves the corresponding Gelfand-Pinsker bound with polynomial complexity.

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.

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far1. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.

An equivalence between multiple multicast and multiple unicast network codes is proven in a 2007 paper by Dougherty and Zeger. A related equivalence between multiple multicast and multiple unicast capacity for general (possibly noisy) memoryless networks is proven in a 2013 paper by the current authors. While the construction used in the proof from the earlier paper maps any linear code for one network to a linear code for the other network, the construction from the later paper does not necessarily preserve linearity. As a result, the 2013 result does not prove an equivalence between the capacity achievable by linear codes in memoryless multiple multicast and memoryless multiple unicast networks. The linear capacity equivalence for memoryless multiple multicast and multiple unicast networks is proven in this work.

In recent years, work has begun to emerge demonstrating intriguing relationships between seemingly disparate information theoretic problems. For example, recent results establish powerful ties between solutions for networks of memoryless channels and networks of noiseless links (network coding networks), between network coding networks in which every internal node can code and a particular subset of network coding networks in which only a single internal node can code (index coding networks), and between multiple multicast demands on memoryless networks and multiple unicast demands on memoryless networks. While the results vary widely, together, they hint at the potential for a unifying theory. In this work, we consider one possible framework for such a theory. Inspired by ideas from the field of computational complexity theory, the proposed framework generalizes definitions and techniques for reduction, completeness, and approximation to the information theoretic domain. One possible outcome from such a theory is a taxonomy of information theoretic problems where problems in the same taxonomic class share similar properties in terms of their code designs, capacities, or other forms of solution. Another potential outcome is the identification of small classes of network information theoretic problems whose solutions, were they available, would solve all information theoretic problems in a much larger class. A third potential outcome is the development of techniques by which approximate solution for one family of network information theoretic problems can be obtained from precise or approximate solution of another family of networks.

We consider the problem of communicating information over a network secretly and reliably in the presence of a hidden adversary who can eavesdrop and inject malicious errors. We provide polynomial-time distributed network codes that are information-theoretically rate-optimal for this scenario, improving on the rates achievable in prior work by Ngai Our main contribution shows that as long as the sum of the number of links the adversary can jam (denoted by ZO) and the number of links he can eavesdrop on (denoted by ZI) is less than the network capacity (denoted by C) (i.e., ), our codes can communicate (with vanishingly small error probability) a single bit correctly and without leaking any information to the adversary. We then use this scheme as a module to design codes that allow communication at the source rate of C- ZO when there are no security requirements, and codes that allow communication at the source rate of C- ZO- ZI while keeping the communicated message provably secret from the adversary. Interior nodes are oblivious to the presence of adversaries and perform random linear network coding; only the source and destination need to be tweaked. We also prove that the rate-region obtained is information-theoretically optimal. In proving our results, we correct an error in prior work by a subset of the authors in this paper.

For a hypergraph H on the vertex set {1,...,n}, a distribution D = (D_1,...,D_n) over {0,1}^n is H-wise independent if every restriction of D to indices which form an edge in H is uniform. This generalizes the notion of k-wise independence obtained by taking H to be the complete n vertex k-uniform hypergraph. This generalization was studied by Schulman (STOC 1992), who presented constructions of H-wise independent distributions that are linear, i.e., the samples are strings of inner products (over F2) of a fixed set of vectors with a uniformly chosen random vector. Let l(H) denote the minimum possible size of a sample space of a uniform H-wise independent distribution. The l parameter is well understood for the special case of k-wise independence. In this work we study the notion of H-wise independence and the l parameter for general graphs and hypergraphs. For graphs, we show how the l parameter relates to standard graph parameters (e.g., clique number, chromatic number, Lovasz theta function, minrank). We derive algorithmic and hardness results for this parameter as well as an explicit construction of graphs G for which l(G) is exponentially smaller than the size of the sample space of any linear G-wise independent distribution. For hypergraphs, we study the problem of testing whether a given distribution is H-wise independent, generalizing results of Alon et al. (STOC 2007).

We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and non-linear codes. Specifically, we present an efficient reduction that maps a network coding instance to an index coding instance while preserving feasibility. Previous connections were restricted to the linear case.

Both rewriting and error correction are important technologies for non-volatile memories, especially flash memories. However, coding schemes that combine them have been limited. This paper presents a new coding scheme that combines rewriting and error correction for the write-once memory model. Its construction is based on polar codes, and it supports any number of rewrites and corrects a substantial number of errors. The code is analyzed for the binary symmetric channel, and experimental results verify its performance. The results can be extended to multi-level cells and more general noise models.

We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.

In recent work by Effros, El Rouayheb, and Langberg, an equivalence of code feasibility between network and index coding is derived. The authors ask whether the capacity region of a network coding problem can be obtained by solving the capacity region of an index coding problem. We answer in the affirmative for the linear coding case. While the question is still open for the general case, we show that it is related to the edge removal problem, which has been studied recently.

The typical paradigm in voting theory involves n voters and m candidates. Every voter ranks the candidates resulting in a permutation of the m candidates. A key problem is to derive the aggregate result of the voting. A popular method for vote aggregation is based on the Condorcet criterion. The Condorcet winner is the candidate who wins every other candidate by pairwise majority. However, the main disadvantage of this approach, known as the Condorcet paradox, is that such a winner does not necessarily exist since this criterion does not admit transitivity. This paradox is mathematically likely (if voters assign rankings uniformly at random, then with probability approaching one with the number of candidates, there will not be a Condorcet winner), however, in real life scenarios such as elections, it is not likely to encounter the Condorcet paradox. In this paper we attempt to improve our intuition regarding the gap between the mathematics and reality of voting systems. We study a special case where there is global intransitivity between all candidates. We introduce tools from information theory and derive an entropy-based characterization of global intransitivity. In addition, we tighten this characterization by assuming that votes tend to be similar; in particular they can be modeled as permutations that are confined to a sphere defined by the Kendalls τ distance.

The sequence-reconstruction problem was first proposed by Levenshtein in 2001. This problem studies the model where the same word is transmitted over multiple channels. If the transmitted word belongs to some code of minimum distance d and there are at most r errors in every channel, then the minimum number of channels that guarantees a successful decoder (under the assumption that all channel outputs are distinct) has to be greater than the largest intersection of two balls of radius r and with distance at least d between their centers. This paper studies the combinatorial problem of computing the largest intersection of two balls for two cases. In the first part we solve this problem in the Grassmann graph for all values of d and r. In the second part we derive similar results for permutations under Kendall's τ-metric for some special cases of d and r.

In this paper, we investigate the impact of a single edge on the capacity region of a network of error-free, point-to-point links. A family of networks and edges is said to exhibit the “edge removal property” if for any network and edge in the family, removing a δ-capacity edge changes the capacity region by at most δ in each dimension. We derive a sufficient condition on network coding functions to guarantee that the edge removal property holds when the network is operated using functions satisfying the condition. Also, we extend the family of network capacity bounds for which it is known that removing a single edge of capacity δ changes the capacity bound by at most f(δ) in each dimension. Specifically, we show that removing a single δ-capacity edge changes the Generalized Network Sharing outer bound by at most δ in each dimension and the Linear Programming outer bound by at most a constant times δ in each dimension.

Traditional studies of multi-source, multi-terminal interference channels typically allow a vanishing probability of error in communication. Motivated by the study of network coding, this work addresses the task of quantifying the loss in rate when insisting on zero error communication in the context of interference channels.

We consider the network communication scenario, over directed acyclic networks with unit capacity edges in which a number of sources si each holding independent unit-entropy information Xiwish to communicate the sum ΣXito a set of terminals tj. We show that in the case in which there are only two sources or only two terminals, communication is possible if and only if each source terminal pair si/tjis connected by at least a single path. For the more general communication problem in which there are three sources and three terminals, we prove that a single path connecting the source terminal pairs does not suffice to communicate ΣXi. We then present an efficient encoding scheme which enables the communication of ΣXifor the three sources, three terminals case, given that each source terminal pair is connected by two edge disjoint paths.

In this paper, we consider the communication of information in the presence of an online adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codewordx=(x1,...,xn) symbol-by-symbol over a communication channel. The adversarial jammer can view the transmitted symbolsxione at a time and can change up to ap-fraction of them. However, for each symbolxi, the jammer's decision on whether to corrupt it or not (and on how to change it) must depend only onxjforj≤i. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge ofx. More generally, for a delay parameter δ ∈ (0,1), we study the scenario in which the jammer's decision on the corruption ofximust depend solely onxjforj≤i-δn. In this study, the transmitted symbols are assumed to be over a sufficiently large field F. The sender and receiver do not share resources such as common randomness (though the sender is allowed to use stochastic encoding). We present a tight characterization of the amount of information one can transmit in both the 0-delay and, more generally, the δ-delay online setting. We show that for 0-delay adversaries, the achievable rate asymptotically equals that of the classical adversarial model. For positive values of δ, we consider two types of jamming: additive and overwrite. We also extend our results to a jam-or-listen online model, where the online adversary can either jam a symbol or eavesdrop on it. We present computationally efficient achievability schemes even against computationally unrestricted jammers.

In this paper, we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codewordx=(x1, ...,xn) bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bitsxione at a time and can change up to ap-fraction of them. However, the decisions of the jammer must be made in a causal manner. Namely, for each bitxi, the jammer's decision on whether to corrupt it or not must depend only onxjforj≤i. This is in contrast to the “classical” adversarial jamming situations in which the jammer has no knowledge ofx, or knowsxcompletely. In this study, we present upper bounds (that hold under both the average and maximal probability of error criteria) on the capacity which hold for both deterministic and stochastic encoding schemes.

We consider the local rank-modulation scheme, in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank-modulation is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study gray codes for the local rank-modulation scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation. Unlike the limited scope of previous works, we consider code constructions for the entire range of parameters including the code length, sliding-window size, and overlap between adjacent windows. We show that the presented codes have asymptotically optimal rate. We also provide efficient encoding, decoding, and next-state algorithms.

A generalized rewriting model is defined for flash memory that represents stored data and permitted rewrite operations by a directed graph. This model is a generalization of previously introduced rewriting models of codes, including floating codes, write-once memory codes, and buffer codes. This model is used to design a new rewriting code for flash memories. The new code, referred to as trajectory code, allows stored data to be rewritten as many times as possible without block erasures. It is proved that the trajectory codes are asymptotically optimal for a wide range of scenarios. In addition, rewriting codes that use a randomized rewriting scheme are presented that obtain good performance with high probability for all possible rewrite sequences.

In this work we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, ..., xn) bit-by-bit over a communication channel. The adversarial jammer can view the transmitted bits xione at a time, and can change up to a p-fraction of them. However, the decisions of the jammer must be made in a causal manner. Namely, for each bit xithe jammer's decision on whether to corrupt it or not must depend only on xjfor j ≤ i. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge of x. Binary channels with causal adversarial jammers have seen recent studies in which both lower bounds and upper bounds on their capacity is derived. In this work, we present improved upper bounds on the capacity which hold for both deterministic and stochastic encoding schemes.

We consider the coded cooperative data exchange problem for general graphs. In this problem, given a graph G = (V, E) representing clients in a broadcast network, each of which initially hold a (not necessarily disjoint) set of information packets; one wishes to design a communication scheme in which eventually all clients will hold all the packets of the network. Communication is performed in rounds, where in each round a single client broadcasts a single (possibly encoded) information packet to its neighbors in G. The objective is to design a broadcast scheme that satisfies all clients with the minimum number of broadcast rounds. The coded cooperative data exchange problem has seen significant research over the last few years; mostly when the graph G is the complete broadcast graph in which each client is adjacent to all other clients in the network, but also on general topologies, both in the fractional and integral setting. In this work we focus on the integral setting in general undirected topologies G. We tie the data exchange problem on G to certain well studied combinatorial properties of G and in such show that solving the problem exactly or even approximately within a multiplicative factor of log |V| is intractable (i.e., NP-Hard). We then turn to study efficient data exchange schemes yielding a number of communication rounds comparable to our intractability result. Our communication schemes do not involve encoding, and in such yield bounds on the coding advantage in the setting at hand.

In the index coding problem, the goal is to transmit an n character word over a field F to n receivers (one character per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword broadcasted to all receivers which allows each receiver to learn its character. For linear index coding, the minimum possible length is known to be equal to the minrank parameter. In this paper we initiate the study of the typical minimum length of a linear index code for the random graph G(n, p) over a field F. First, we prove that for every constant size field F and a constant p, the minimum length of a linear index code for G(n, p) over F is almost surely Ω(√n). Second, we introduce and study two special models of index coding and study their typical minimum length: Locally decodable index codes in which the receivers are required to query at most q characters from the encoded message (such codes naturally correspond to efficient decoding); and low density index codes in which every character of the broadcasted word affects at most q characters in the encoded message (such codes naturally correspond to efficient encoding procedures). We present enhanced results for these special models.

Motivated by the study of deletion channels, this work presents improved bounds on the number of subsequences obtained from a binary sting X of length n under t deletions. It is known that the number of subsequences in this setting strongly depends on the number of runs in the string X; where a run is a maximal sequence of the same character. Our improved bounds are obtained by a structural analysis of the family of r-run strings X, an analysis in which we identify the extremal strings with respect to the number of subsequences. Specifically, for every r, we present r-run strings with the minimum (respectively maximum) number of subsequences under any t deletions; and perform an exact analysis of the number of subsequences of these extremal strings.

In this work, we address the capacity region of multi-source multi-terminal network communication problems, and study the change in capacity when one moves form independent to dependent source information. Specifically, we ask whether the trade off between capacity and source independence is of continuous nature. We tie the question at hand to that of edge removal which has seen recent interest.

The Index Coding problem has recently attracted a significant attention from the research community. In this problem, a server needs to deliver data to a set of wireless clients over the broadcast channel. Each client requires one or more packets, but it might have access to the packets requested by other clients as side information. The goal is to deliver the required data to each client with minimum number of transmissions. In this paper, we focus on finding sparse solutions to the Index Coding problem. In a sparse solution each transmitted packet is a linear combination of at most two original packets. We focus both on scalar and vector versions of the problem. For the scalar case, we present a polynomial time algorithm that achieves an approximation ratio of 2-(1/√n). For the vector case, we present a polynomial time algorithm that identifies an optimal solution to the problem. Our simulation studies demonstrate that our algorithms achieve good performance in practical scenarios.

We study the Index Coding problem with side information graphs which are outerplanar. For general side information graphs, linearly solving the Index Coding problem implies a linear solution to the general (non-multicast) Network Coding problem - a central open problem in the field of network communication. For outerplanar side information graphs, we show that the Index Coding problem can be solved efficiently, and characterize its solution in terms of the clique cover size of the information graph at hand.

The Index Coding problem is one of the basic problems in wireless network coding. In this problem, a server needs to deliver a set P of packets to several clients through a noiseless broadcast channel. Each client needs to obtain a certain subset of P and has prior side information about a different subset of P. The objective is to satisfy the requirements of all clients with the minimum number of transmissions. Recently, it was shown that the Index Coding problem is NP-hard. Furthermore, this problem was shown to be hard to approximate under a widely accepted complexity assumption. In this paper, we consider a complementary problem whose goal is to maximize the number of saved transmissions, i.e., the number of transmissions that are saved by combining packets compared to the solution that does not involve coding. We refer to this problem as the the Complementary Index Coding problem. It turns out that the complementary problem can be approximated in certain cases of practical importance. We consider the multiple unicast and multiple multicast scenarios. In the multiple unicast scenario, each packet is requested by a single client; while in the multiple multicast scenario, each packet can be requested by several clients. For the multiple unicast scenario, we present approximation algorithms for finding scalar and vector linear solutions. For the multiple multicast scenario, we show that finding an approximation solution is NP-hard.

We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank-modulation is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. Unlike the limited scope of previous works, we consider code constructions for the entire range of parameters including the code length, sliding window size, and overlap between adjacent windows. We show our constructed codes have asymptotically-optimal rate. We also provide efficient encoding, decoding, and next-state algorithms.

In the online channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x =(x_1,...,x_n) in {0,1}^n bit by bit via a channel limited to at most pn corruptions. The channel is online in the sense that at the ith step the channel decides whether to flip the ith bit or not and its decision is based only on the bits transmitted so far, i.e., (x_1,...,x_i). This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge on the sent codeword x. The best known lower bound on the capacity of both the online channel and the classical adversarial channel is the well-known Gilbert-Varshamov bound. In this paper we prove a lower bound on the capacity of the online channel which beats the Gilbert-Varshamov bound for any positive p such that H(2p) < 0.5 (where H is the binary entropy function). To do so, we prove that for any such p, a code chosen at random combined with the nearest neighbor decoder achieves with high probability a rate strictly higher than the Gilbert-Varshamov bound (for the online channel).

Given a set F of n positive functions over a ground set X, we consider the problem of computing x* that minimizes the expression ∑f ∈ Ff(x), over x ∈ X. A typical application is shape fitting, where we wish to approximate a set P of n elements (say, points) by a shape x from a (possibly infinite) family X of shapes. Here, each point p ∈ P corresponds to a function f such that f(x) is the distance from p to x, and we seek a shape x that minimizes the sum of distances from each point in P. In the k-clustering variant, each x\in X is a tuple of k shapes, and f(x) is the distance from p to its closest shape in x.

We consider the local rank-modulation (LRM) scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. LRM is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the LRM scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation. We present a practical construction of codes with asymptotically-optimal rate and weight asymptotically half the length, thus having an asymptotically-optimal charge difference between adjacent cells. Next, we turn to examine the existence of optimal codes by specifically studying codes of weight 2 and 3. In the former case, we upper bound the code efficiency, proving that there are no such asymptotically-optimal cyclic codes. In contrast, for the latter case we construct codes which are asymptotically-optimal. We conclude by providing necessary conditions for the existence of cyclic and cyclic optimal Gray codes.

This work addresses the computational complexity of achieving the capacity of a general network coding instance. It has been shown [Lehman and Lehman, SODA 2005] that determining the “scalar linear” capacity of a general network coding instance is NP-hard. In this paper we address the notion of approximation in the context of both linear and nonlinear network coding. Loosely speaking, we show that given an instance of the general network coding problem of capacityC, constructing a code of rate αCfor any universal (i.e., independent of the size of the instance) constant α ≤ 1 is “hard”. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. Our results refer to scalar linear, vector linear, and nonlinear encoding functions and are the first results that address the computational complexity of achieving the network coding capacity in both the vector linear and general network coding scenarios. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., an instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.

Shannon defined channel capacity as the highest rate at which there exists a sequence of codes of block length n such that the error probability goes to zero as n goes to infinity. In this definition, it is implicit that the block length, which can be viewed as the number of available channel uses, is unlimited. This is not the case when the transmission power must be concentrated on a single transmission, most notably in military scenarios with adversarial conditions or delay-tolerant networks with random short encounters. A natural question arises: how much information can we transmit in a single use of the channel? We give a precise characterization of the one-shot capacity of discrete channels, defined as the maximum number of bits that can be transmitted in a single use of a channel with an error probability that does not exceed a prescribed value. This capacity definition is shown to be useful and significantly different from the zero-error problem statement.

In this work we consider the communication of information in the presence of a delayed adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, ..., xn) over a communication channel. The adversarial jammer can view the transmitted symbols xi one at a time, but must base its action (when changing xi) on xjfor j ≤ i - Δn, where Δ ∈ [0, 1] is a delay parameter. In this work, we study codes for a class of delayed adversaries, and for any delay Δ > 0 present a single letter characterization of the achievable communication rate in the presence of such adversaries.

NAND flash memories have become the most widely used type of non-volatile memories. In a NAND flash memory, every block of memory cells consists of numerous pages, and rewriting a single page requires the whole block to be erased. As block erasures significantly reduce the longevity, speed and power efficiency of flash memories, it is critical to minimize the number of erasures when data are reorganized. This leads to the data movement problem, where data need to be switched in blocks, and the objective is to minimize the number of block erasures. It has been shown that optimal solutions can be obtained by coding. However, coding-based algorithms with the minimum coding complexity still remain an important topic to study. In this paper, we present a very efficient data movement algorithm with coding over GF(2) and with the minimum storage requirement. We also study data movement with more auxiliary blocks and present its corresponding solution. Furthermore, we extend the study to the data aggregation problem, where data can not only be moved but also aggregated. We present both non-coding and coding-based solutions, and rigorously prove the performance gain by using coding.

We consider the problem of multicasting sums over directed acyclic networks with unit capacity edges. A set of source nodes siobserve independent unit-entropy source processes Xiand want to communicate Σ Xito a set of terminals tj. Previous work on this problem has established necessary and sufficient conditions on the si-tjconnectivity in the case when there are two sources or two terminals (Ramamoorthy '08), and in the case of three sources and three terminals (Langberg-Ramamoorthy '09). In particular the latter result establishes that each terminal can recover the sum if there are two edge disjoint paths between each si-tjpair. In this work, we provide a new and significantly simpler proof of this result, and introduce techniques that may be of independent interest in other network coding problems.

Let X be a space and F a family of 0, 1-valued functions on X. Vapnik and Chervonenkis showed that if F is “simple” (finite VC dimension), then for every probability measure μ on X and ε > 0 there is a finite set S such that for all f ∊ F, σx∊S f(x)/|S| = [∫ f (x)dμ(x)] ± ε. Think of S as a “universal ε-approximator” for integration in F. S can actually be obtained w.h.p. just by sampling a few points from μ. This is a mainstay of computational learning theory. It was later extended by other authors to families of bounded (e.g., [0, 1]-valued) real functions. In this work we establish similar “universal ε-approximators” for families of unbounded nonnegative real functions — in particular, for the families over which one optimizes when performing data classification. (In this case the ε-approximation should be multiplicative.) Specifically, let F be the family of “k-median functions” (or k-means, etc.) on ℝd with an arbitrary norm ϱ. That is, any set u1, …, uk ∊ ℝd determines an f by f(x) = (mini ϱ(x – ui))α. (Here α ≥ 0.) Then for every measure μ on ℝd there exists a set S of cardinality poly(k, d, 1/ε) and a measure ν supported on S such that for every f ∊ F, σx∊S f(x)v(x) ∊ (1 ± ε) · (∫ f(x)dμ(x)).

This paper concerns graph spanners that are resistant to vertex or edge failures. In the failure-free setting, it is known how to efficiently construct a $(2k-1)$-spanner of size $O(n^{1+1/k})$, and this size-stretch trade-off is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [C. Levcopoulos, G. Narasimhan, and M. Smid, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 186–195]. A subgraph H is an f-vertex fault tolerant k-spanner of the graph G if for any set $F\subseteq V$ of size at most f and any pair of vertices $u,v\in V\setminus F$, the distances in H satisfy $\delta_{H\setminus F}(u,v)\leq k\cdot\delta_{G\setminus F}(u,v)$. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [A. Czumaj and H. Zhao, Discrete Comput. Geom., 32 (2004), pp. 207–230]. This paper also raised as an open problem the question of whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an f-vertex fault tolerant $(2k-1)$-spanner of size $O(f^{2}k^{f+1}\cdot n^{1+1/k}\log^{1-1/k}n)$. Interestingly, the stretch of the spanner remains unchanged, while the size of the spanner increases only by a factor that depends on the stretch k, on the number of potential faults f, and on logarithmic terms in n. In addition, we consider the simpler setting of f-edge fault tolerant spanners (defined analogously). We present an f-edge fault tolerant $(2k-1)$-spanner with edge set of size $O(f\cdot n^{1+1/k})$ (only f times larger than standard spanners). For both edge and vertex faults, our results are shown to hold when the given graph G is weighted.

We study approximation algorithms, integrality gaps, and hardness of approximation of two problems related to cycles of “small” length k in a given (undirected) graph. The instance for these problems consists of a graph $G=(V,E)$ and an integer k. The k-Cycle Transversal problem is to find a minimum edge subset of E that intersects every k-cycle. The k-Cycle-Free Subgraph problem is to find a maximum edge subset of E without k-cycles. Our main result is for the k-Cycle-Free Subgraph problem with even values of k. For any $k=2r$, we give an $\Omega(n^{-\frac{1}{r}+\frac{1}{r(2r-1)}-\varepsilon})$-approximation scheme with running time $(1/\varepsilon)^{O(1/\varepsilon)}\mathsf{poly}(n)$, where $n=|V|$ is the number of vertices in the graph. This improves upon the ratio $\Omega(n^{-1/r})$ that can be deduced from extremal graph theory. In particular, for $k=4$ the improvement is from $\Omega(n^{-1/2})$ to $\Omega(n^{-1/3-\varepsilon})$. Our additional result is for odd k. The 3-Cycle Transversal problem (covering all triangles) was studied by Krivelevich [Discrete Math., 142 (1995), pp. 281–286], who presented an LP-based 2-approximation algorithm. We show that k-Cycle Transversal admits a $(k-1)$-approximation algorithm, which extends to any odd k the result that Krivelevich proved for $k=3$. Based on this, for odd k we give an algorithm for k-Cycle-Free Subgraph with ratio $\frac{k-1}{2k-3}=\frac{1}{2}+\frac{1}{4k-6}$; this improves upon the trivial ratio of $1/2$. For $k=3$, the integrality gap of the underlying LP was posed as an open problem in the work of Krivelevich. We resolve this problem by showing a sequence of graphs with integrality gap approaching 2. In addition, we show that if k-Cycle Transversal admits a $(2-\varepsilon)$-approximation algorithm, then so does the Vertex-Cover problem; thus improving the ratio 2 is unlikely. Similar results are shown for the problem of covering cycles of length $\leq k$ or finding a maximum subgraph without cycles of length $\leq k$ (i.e., with girth $>k$).

A set of k balls B 1 , …, B k in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the k - center problem for lines in high-dimensional space: Given a set of n lines l = { l 1 ,…, l n in R d , find k balls of minimum radius which cover l . We present a 2-approximation algorithm for the cases k = 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space. Our result for 3-clustering is strongly based on a new result in discrete geometry that may be of independent interest: a Helly-type theorem for collections of axis-parallel “crosses” in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets. In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.

A constrained memory is a storage device whose elements change their states under some constraints. A typical example is flash memories, in which cell levels are easy to increase but hard to decrease. In a general rewriting model, the stored data changes with some pattern determined by the application. In a constrained memory, an appropriate representation is needed for the stored data to enable efficient rewriting.

In this work we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, ..., xn) bit-by-bit over a communication channel. The adversarial jammer can view the transmitted bits xione at a time, and can change up to a p-fraction of them. However, the decisions of the jammer must be made in an online or causal manner. Namely, for each bit xithe jammer's decision on whether to corrupt it or not (and on how to change it) must depend only on xjfor j ¿ i. This is in contrast to the ¿classical¿ adversarial jammer which may base its decisions on its complete knowledge of x. We present a non-trivial upper bound on the amount of information that can be communicated. We show that the achievable rate can be asymptotically no greater than min{1 - H(p), (1 - 4p)+}. Here H(.) is the binary entropy function, and (1 - 4p)+equals 1 - 4p for p ¿ 0.25, and 0 otherwise.

We consider the network communication scenario in which a number of sources sieach holding independent information Xiwish to communicate the sum ¿Xito a set of terminals tj. In this work we consider directed acyclic graphs with unit capacity edges and independent sources of unit-entropy. The case in which there are only two sources or only two terminals was considered by the work of Ramamoorthy [ISIT 2008] where it was shown that communication is possible if and only if each source terminal pair si/tjis connected by at least a single path. In this work we study the communication problem in general, and show that even for the case of three sources and three terminals, a single path connecting source/terminal pairs does not suffice to communicate ¿Xi. We then present an efficient encoding scheme which enables the communication of ¿Xifor the three sources, three terminals case, given that each source terminal pair is connected by two edge disjoint paths. Our encoding scheme includes a structural decomposition of the network at hand which may be found useful for other network coding problems as well.

The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected n-vertex graph G=(V,E) and an integer k ≥ 1, the subgraph H=(V,E'), E'⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u,v) ≤ k· δG(u,v) for every u,v ∈ V, where δG'(u,v) denotes the distance between u and v in G'. Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k-1)-spanner of size O(n1+1/k), and this size-stretch tradeoff is conjectured to be tight.

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e. nodes that generate new packets by performing algebraic operations on packets received over incoming links.We present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph G(V,E) in time 0(\E\kh + \V\k2h2+ h4k3(k + h)), where k is the number of destinations and h is the number of packets. Our algorithm improves the best known running time for network code construction. In addition, our algorithm guarantees that the number of encoding nodes in the obtained network code is upper- bounded by 0(h3k2). Next, we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes. We prove that in the majority of settings this problem is NP-hard. However, we show that if h = O(1),k = O(1), and the underlying communication graph is acyclic, then there exists an algorithm that solves this problem in polynomial time.

This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that determining the (scalar) linear capacity of a general network coding instance is NP-hard. In this work we initiate the study of approximation in this context. Namely, we show that given an instance to the general network coding problem of linear capacity C, constructing a linear code of rate alphaC for any universal (i.e., independent of the size of the instance) constant alphales1 is ldquohardrdquo. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., a general instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.

In a recent paper, Jaggi et al., presented a distributed polynomial-time rate-optimal network-coding scheme that works in the presence of Byzantine faults.We revisit their adversarial models and augment them with three, arguably realistic, models. In each of the models, we present a distributed scheme that demonstrates the usefulness of the model. In particular, all of the schemes obtain optimal rate C-z, where C is the network capacity and z is a bound on the number of links controlled by the adversary.

We provide a novel achievability proof of the Slepian-Wolf theorem for i.i.d. sources over finite alphabets. We demonstrate that random codes that are linear over the real field achieve the classical Slepian-Wolf rate region. For finite alphabets we show that decoding is equivalent to solving an integer program. The techniques used may be of independent interest for code design for a wide class of information theory problems, and for the field of compressed sensing.

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.

Let be C={x1,...,xn} sub {0,1}nbe an [n,N] binary error correcting code (not necessarily linear). Let e{0,1}nbe an error vector. A codeword xepsiC is said to be disturbed by the error e if the closest codeword to xopluse is no longer x. Let Aebe the subset of codewords in C that are disturbed by e. In this work, we study the size of Aein random codes C (i.e., codes in which each codeword is chosen uniformly and independently at random from {0,1}n). Using recent results of Vu [Random Structures and Algorithms, vol. 20, no. 3, pp. 262-316, 2002] on the concentration of non-Lipschitz functions, we show that |Ae| is strongly concentrated for a wide range of values of N and ||e||. We apply this result in the study of communication channels we refer to as oblivious. Roughly speaking, a channel W(y|x) is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x. A family of channels Psi is said to be oblivious if every member W of the family is oblivious. In this work, we define oblivious and partially oblivious families of (not necessarily memoryless) channels and analyze their capacity. When considering the capacity of a family of channels Psi, one must address the design of error correcting codes which allow communication under the uncertainty of which channel WepsiPsi is actually used. The oblivious channels we define have connections to arbitrarily varying channels with state constraints.

We study broadcast systems that distribute a series of data updates to a large number of passive clients. The updates are sent over a broadcast channel in the form of discrete packets. We assume that clients periodically access the channel to obtain the most recent update. Such scenarios arise in many practical applications, such as distribution of traffic information and market updates to mobile wireless devices. Our goal is to design broadcast schedules that minimize the waiting time, i.e., the amount of time the client needs to wait in order to obtain the most recent update. We assume that each client has a different access pattern depending on the channel conditions, computing power, and storage capabilities. We introduce and analyze optimal universal schedules that guarantee low waiting time for any client, regardless of its behavior.

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.

Network coding can substantially improve network throughput and performance. However, these codes have a major drawback if the network contains hidden malicious nodes that can eavesdrop on transmissions and inject fake information. In this scenario, even a small amount of information injected by a single malicious hidden node could mix with and contaminate much of the information inside the network, causing a decoding error. We improve on previous work by providing a polynomial- time, rate-optimal distributed network code design that functions even in the presence of a Byzantine adversary with substantial eavesdropping capabilities. As long as the sum of the adversary's jamming rate Zoand his eavesdropping rate ZIis less than the network capacity C, (Zo+ ZIo. The network codes we design are information-theoretically secure and assume no knowledge of network topology. Prior to transmission, no honest node knows the location or strength of the adversary. In our code design, interior nodes are oblivious to the presence of adversaries and implement a classical low- complexity distributed network code design; only the source and destination need to be changed. Finally, our codes work for both wired and wireless networks.

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Let C = {X1,...,XN} ⊂ {0, 1}nbe an [n, N] binary error correcting code (not necessarily linear). Let e isin {0, 1}nbe an error vector. A codeword X isin C is said to be disturbed by the error e if the closest codeword to X oplus e is no longer X. Let Aebe the subset of codewords in C that are disturbed by e. In this work we study the size of Aein random codes C (i.e. codes in which each codeword Xiis chosen uniformly and independently at random from {0, 1}n). Using recent results of Vu [random structures and algorithms 20(3)] on the concentration of non-Lipschitz functions, we show that |Ae| is strongly concentrated for a wide range of values of N and parepar. We apply this result in the study of communication channels we refer to as oblivious. Roughly speaking, a channel W(y|x) is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x. For example, the well studied binary symmetric channel is an oblivious channel. In this work, we define oblivious and partially oblivious channels and present lower bounds on their capacity. The oblivious channels we define have connections to arbitrarily varying channels with state constraints

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying communication network G. The nodes of the multicast network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h/sup 3/k/sup 2/. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h/sup 3/k/sup 2/. We show that the number of encoding nodes may depend both on h and k by presenting acyclic coding networks that require /spl Omega/(h/sup 2/k) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2B+1)h/sup 3/k/sup 2/, where B is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an /spl Nscr/P-hard problem.

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

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying network G. The nodes of the coding network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in an acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded h3k2. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h3k2. We show that the number of encoding nodes may depend both on h and k as we present acyclic instances of the multicast network coding problem in which Omega (h2k) encoding nodes are required. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. Specifically, we prove that the number of encoding nodes is bounded by (2 B + 1)h3k2, where B is the minimum size of the feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of encoding nodes required to achieve the capacity for a given instance of the network coding problem is NP-hard

In this paper we study the distribution of dynamic data over a broadcast channel to a large number of passive clients. The data is simultaneously distributed to clients in the form of discrete packets, each packet captures the most recent state of the information source. Clients obtain the information by accessing the channel and listening for the next available packet. This scenario, referred to as discrete broadcast, has many practical applications such as the distribution of stock information to wireless mobile devices and downloading up-to-date battle information in military networks. Our goal is minimize the amount of time a client has to wait in order to obtain a new data packet, i.e., the waiting time of the client. We show that we can significantly reduce the waiting time by adding redundancy to the schedule. We identify universal schedules that guarantee low waiting time for any client, regardless of the access pattern. A key point in the design of data distribution systems is to ensure that the transmitted information is always up-to-date. Accordingly, we introduce the notion of staleness that captures the amount of time that passes from the moment the information is generated, until it is delivered to the client. We investigate the fundamental trade-off between the staleness and the waiting time. In particular, we present schedules that yield lowest possible waiting time for any given staleness constraint

Coding theory addresses the design and analysis of codes that enable communication over noisy channels. Two types of channels that have been extensively considered are the binary symmetric channel and the adversarial channel. In a binary symmetric channel each bit of the sent message is flipped independently with some probability p, implying that the noise imposed by the channel is random in nature where the amount of noise is determined by p. In an adversarial channel the message is treated as a whole, and the noise may be an arbitrary (and malicious) function of the message being sent, as long as it does not effect more that a certain fraction (say p) of the bits transmitted. Roughly speaking, any code designed for an adversarial channel can be used on a corresponding binary symmetric channel successfully, whereas the contrary is not necessarily true. In this work we present a construction that transforms the best codes for binary symmetric channels into "codes" for corresponding adversarial channels. The "codes" we present assume that the sender and the receiver of the message have a joint secret random string (which is not known to the channel). These codes are referred to as private codes. Intuitively, this private randomness allows a reduction between the random and adversarial channels. Such a reduction is simple once the size of the joint random string is /spl Theta/(n log n) (here the codes are a subset of {0,1 }/sup n/). In this work we present private codes in which the size of the joint random string is O(log n). Moreover, we show that our result is tight. Namely, to design private codes that allow communication over adversarial channels that meet the bounds achievable when communicating over binary symmetric channels, an amount of /spl Omega/(log n) shared random bits are required. To the best of our knowledge, no prior results of this nature have been presented in the past. As part of our proof we establish a connection between list decodable codes and private codes which complements a recent result of Guruswami (CCC '03) on list decoding with side information.

This paper investigates an efficient scheduling for sending dynamic data over lossless broadcast channels. A server transmits dynamic data periodically to a number of passive clients and thus the updated discrete packets are sent into a separate packet. The objective of this paper is to design universal schedules that minimize the time that passes between a client's request and the broadcast of a new item, independently of the client's behavior. From the results the optimal scheduling of high transmission rate for discrete broadcast data is obtained by considering adaptive clients.

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 < \delta < 1$. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ colors). For $k = O(\log n/\log\log n)$ we show vector k-colorable graphs that do not have independent sets of size (log n) c , for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylog n. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653--750] for this problem.

Karger Motwani and Sudan (1998) introduced the notion of a vector coloring of a graph. In particular they show that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly /spl Delta//sup 1-2/k/ colors. Here /spl Delta/ is the maximum degree in the graph. Their results play a major role in the best approximation algorithms for coloring and for maximal independent set. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n//spl Delta//sup 1-2/k/ (and hence cannot be colored with significantly less that /spl Delta//sup 1-2/k/ colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)/sup c/, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser and Ron (1998) for this problem.