arXiv subjects
Publications and source records attributed to Michael Langberg.
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 problem of secure unicast transmission between two nodes in a directed graph, where an adversary eavesdrops/jams a subset of nodes. This adversarial setting is in contrast to traditional ones where the adversary controls a subset of links. In particular, we study, in the main, the class of routing-only schemes (as opposed to those allowing coding inside the network). Routing-only schemes usually have low implementation complexity, yet a characterization of the rates achievable by such schemes was open prior to this work. We first propose an LP based solution for secure communication against eavesdropping, and show that it is information-theoretically rate-optimal among all routing-only schemes. The idea behind our design is to balance information flow in the network so that no subset of nodes observe "too much" information. Interestingly, we show that the rates achieved by our routing-only scheme are always at least as good as, and sometimes better, than those achieved by "naïve" network coding schemes (i.e. the rate-optimal scheme designed for the traditional scenario where the adversary controls links in a network rather than nodes.) We also demonstrate non-trivial network coding schemes that achieve rates at least as high as (and again sometimes better than) those achieved by our routing schemes, but leave open the question of characterizing the optimal rate-region of the problem under all possible coding schemes. We then extend these routing-only schemes to the adversarial node-jamming scenarios and show similar results. During the journey of our investigation, we also develop a new technique that has the potential to derive non-trivial bounds for general secure-communication schemes.
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.
We consider the network communication scenario, over directed acyclic networks with unit capacity edges in which a number of sources $s_i$ each holding independent unit-entropy information $X_i$ wish to communicate the sum $\sum{X_i}$ to a set of terminals $t_j$. 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 $s_i/t_j$ is 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 $\sum{X_i}$. We then present an efficient encoding scheme which enables the communication of $\sum{X_i}$ for the three sources, three terminals case, given that each source terminal pair is connected by {\em two} edge disjoint paths.
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 $(x_1,...,x_n)$ bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bits $x_i$ one 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 $x_i$ the jammer's decision on whether to corrupt it or not must depend only on $x_j$ for $j \leq i$. This is in contrast to the "classical" adversarial jamming situations in which the jammer has no knowledge of $(x_1,...,x_n)$, or knows $(x_1,...,x_n)$ completely. In this work, 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 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 one while preserving feasibility. Previous connections were restricted to the linear case.
This paper shows determining the secrecy capacity of a unicast network with uniform wiretap sets is at least as difficult as the k-unicast problem. In particular, we show that a general k-unicast problem can be reduced to the problem of finding the secrecy capacity of a corresponding single unicast network with uniform link capacities and one arbitrary wiretap link.
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 this work we study zero vs. epsilon-error capacity in network coding instances. For multicast network coding it is well known that all rates that can be delivered with arbitrarily small error probability can also be delivered with zero error probability; that is, the epsilon-error multicast capacity region and zero-error multicast capacity region are identical. For general network coding instances in which all sources originate at the same source node, Chan and Grant recently showed [ISIT 2010] that, again, epsilon-error communication has no rate advantage over zero-error communication. We start by revisiting the setting of co-located sources, where we present an alternative proof to that given by Chan and Grant. While the new proof is based on similar core ideas, our constructive strategy complements the previous argument.We then extend our results to the setting of index coding, which is a special and representative form of network coding that encapsulates the "source coding with side information" problem. Finally, we consider the "edge removal" problem (recently studied by Jalali, Effros, and Ho in [Allerton 2010] and [ITA 2011]) that aims to quantify the loss in capacity associated with removing a single edge from a given network. Using our proof for co-located sources, we tie the "zero vs. epsilon-error" problem in general network coding instances with the "edge removal" problem. Loosely speaking, we show that the two problem are equivalent.
A sender wishes to broadcast an n character word x in F^n (for a field F) to n receivers R_1,...,R_n. Every receiver has some side information on x consisting of a subset of the characters of x. The side information of the receivers is represented by a graph G on n vertices in which {i,j} is an edge if R_i knows x_j. In the index coding problem the goal is to encode x using a minimum number of characters in F in a way that enables every R_i to retrieve the ith character x_i using the encoded message and the side information. An index code is linear if the encoding is linear, and in this case the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS'06). Several bounds on the minimum length of an index code for side information graphs G were shown in the study of index coding. However, the minimum length of an index code for the random graph G(n,p) is far from being understood. In this paper we initiate the study of the typical minimum length of a linear index code for 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 Omega(\sqrt{n}). Second, we introduce and study the following two restricted models of index coding: 1. A locally decodable index code is an index code in which the receivers are allowed to query at most q characters from the encoded message. 2. A low density index code is a linear index code in which every character of the word x affects at most q characters in the encoded message. Equivalently, it is a linear code whose generator matrix has at most q nonzero entries in each row.
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).
Flash memory is well-known for its inherent asymmetry: the flash-cell charge levels are easy to increase but are hard to decrease. In a general rewriting model, the stored data changes its value with certain patterns. The patterns of data updates are determined by the data structure and the application, and are independent of the constraints imposed by the storage medium. Thus, an appropriate coding scheme is needed so that the data changes can be updated and stored efficiently under the storage-medium's constraints. In this paper, we define the general rewriting problem using a graph model. It extends many known rewriting models such as floating codes, WOM codes, buffer codes, etc. We present a new rewriting scheme for flash memories, called the trajectory code, for rewriting the stored data as many times as possible without block erasures. We prove that the trajectory code is asymptotically optimal in a wide range of scenarios. We also present randomized rewriting codes optimized for expected performance (given arbitrary rewriting sequences). Our rewriting codes are shown to be asymptotically optimal.
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.
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, rate-optimal distributed network codes for this scenario, improving on the rates achievable in previous work. Our main contribution shows that as long as the sum of the adversary's jamming rate Zo and his eavesdropping rate Zi is less than the network capacity C, (i.e., Zo+Zi<C), 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 to design codes that allow communication at the optimal source rate of C-Zo-Zi, while keeping the communicated message 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. In proving our results we correct an error in prior work by a subset of the authors in this work.
Given a graph G = (V,E) and a parameter k, we consider the problem of finding a subset U in V of size k that maximizes the number of induced edges (DkS). We improve upon the previously best known approximation ratio for DkS, a ratio that has not seen any progress during the last decade. Specifically, we improve the approximation ratio from n^{0.32258} to n^{0.3159}. The improved ratio is obtained by studying a variant to the DkS problem in which one considers the problem of finding a subset U in V of size at most k that maximizes the number of induced edges. Finally, we study the DkS variant in which one considers the problem of finding a subset U in V of size at least k that maximizes the number of induced edges.
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=(x_1,...,x_n) bit-by-bit over a communication channel. The adversarial jammer can view the transmitted bits x_i one 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 x_i the jammer's decision on whether to corrupt it or not (and on how to change it) must depend only on x_j for 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.
In this work 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 codeword x=x_1,...,x_n symbol-by-symbol over a communication channel. The adversarial jammer can view the transmitted symbols x_i one 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. More generally, for a delay parameter 0<d<1, we study the scenario in which the jammer's decision on the corruption of x_i must depend solely on x_j for j < i - dn. In this work, we initiate the study of codes for online adversaries, and present a tight characterization of the amount of information one can transmit in both the 0-delay and, more generally, the d-delay online setting. We prove tight results for both additive and overwrite jammers when the transmitted symbols are assumed to be over a sufficiently large field F. Finally, we extend our results to a jam-or-listen online model, where the online adversary can either jam a symbol or eavesdrop on it. We again provide a tight characterization of the achievable rate for several variants of this model. The rate-regions we prove for each model are informational-theoretic in nature and hold for computationally unbounded adversaries. The rate regions are characterized by "simple" piecewise linear functions of p and d. The codes we construct to attain the optimal rate for each scenario are computationally efficient.