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Michael Langberg

Publications and source records attributed to Michael Langberg.

At least 37 records · Page 2Linked to original sources
arXiv2022

Group Testing on General Set-Systems

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 \emph{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 {\em 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 $e\in E$ 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 significantly reduce the number of required 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 $e \in E$. 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 $O(\log{|E|}+d\log^2{d})$ that comes close to the lower bound of $Ω(\log{|E|} + d)$.

cs.IT↗
arXiv2022

On the Benefit of Cooperation in Relay Networks

This work addresses the cooperation facilitator (CF) model, in which network nodes coordinate through a rate limited communication device. For independent 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 $C_{CF}$, sometimes has an infinite slope at $C_{CF}=0$. This work studies the question of whether cooperation through a CF can yield similar infinite-slope benefits when applied to internal network encoders in which dependence among MAC transmitters can be established without the help of the CF. Towards this end, this work studies the CF model when applied to relay nodes of a single-source, single-terminal, diamond network consisting of 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 component when studied in isolation.

cs.IT↗
arXiv2021

Minimizing the alphabet size in codes with restricted error sets

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 hypergraph coloring.

cs.IT↗
arXiv2021

Every Bit Counts: Second-Order Analysis of Cooperation in the Multiple-Access Channel

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, there is a sum-rate benefit of order $\sqrt{\log{K}/n}$. 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 $1/\sqrt{n}$.

cs.IT↗
arXiv2021

Beyond Capacity: The Joint Time-Rate Region

The traditional notion of capacity studied in the context of memoryless network communication builds on the concept of block-codes and requires that, for sufficiently large blocklength n, all receiver nodes simultaneously decode their required information after n channel uses. In this work, we generalize the traditional capacity region by exploring communication rates achievable when some receivers are required to decode their information before others, at different predetermined times; referred here as the "time-rate" region. Through a reduction to the standard notion of capacity, we present an inner-bound on the time-rate region. The time-rate region has been previously studied and characterized for the memoryless broadcast channel (with a sole common message) under the name "static broadcasting".

cs.IT↗
arXiv2020

Edge removal in undirected networks

The edge-removal problem asks whether the removal of a $λ$-capacity edge from a given network can decrease the communication rate between source-terminal pairs by more than $λ$. In this short manuscript, we prove that for undirected networks, removing a $λ$-capacity edge decreases the rate by $O(λ)$. 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.

cs.IT↗
arXiv2020

Minimizing the alphabet size of erasure codes with restricted decoding sets

A Maximum Distance Separable code over an alphabet $F$ is defined via an encoding function $C:F^k \rightarrow F^n$ that allows to retrieve a message $m \in F^k$ from 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.

cs.IT↗
arXiv2020

Topology Dependent Bounds For FAQs

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries (FAQs) 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 is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special 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 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 the finite field of two elements) using a novel min-entropy based argument.

cs.DC↗
arXiv2020

Towards an Operational Definition of Group Network Codes

Group network codes are a generalization of linear codes that have seen several studies over the last decade. When studying network codes, operations performed at internal network nodes called local encoding functions, are of significant interest. While local encoding functions of linear codes are well understood (and of operational significance), no similar operational definition exists for group network codes. To bridge this gap, we study the connections between group network codes and a family of codes called Coordinate-Wise-Linear (CWL) codes. CWL codes generalize linear codes and, in addition, can be defined locally (i.e., operationally). In this work, we study the connection between CWL codes and group codes from both a local and global encoding perspective. We show that Abelian group codes can be expressed as CWL codes and, as a result, they inherit an operational definition.

cs.IT↗
arXiv2019

Negligible Cooperation: Contrasting the Maximal- and Average-Error Cases

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve network 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 numbers that grow sublinearly in the blocklength, can never yield a non-negligible gain in the average-error sum-capacity.

cs.IT↗
arXiv2019

Secure Network Coding in the Setting in Which a Non-Source Node May Generate Random Keys

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.

cs.IT↗
arXiv2019

A Local Perspective on the Edge Removal Problem

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.

cs.IT↗
arXiv2018

The Birthday Problem and Zero-Error List Codes

As an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this paper studies the performance of randomly generated codebooks over discrete memoryless channels under a zero-error list-decoding constraint. This study allows the application of tools from one area to the other. Furthermore, it leads to an information-theoretic formulation of the birthday problem, which is concerned with the probability that in a given population, a fixed number of people have the same birthday. Due to the lack of a closed-form expression for this probability when the distribution of birthdays is not uniform, the resulting expression is not simple to analyze; in the information-theoretic formulation, however, the asymptotic behavior of this probability can be characterized exactly for all distributions.

cs.IT↗
arXiv2018

Quadratically Constrained Channels with Causal Adversaries

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 $\mathbf{x}=x_1,...,x_n$ through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, an adversarial jammer James chooses a real-valued length-n noise sequence $\mathbf{s}=s_1,..,s_n$ in a causal manner, i.e., each $s_t (1<=t<=n)$ can only depend on $x_1,...,x_t$. Bob receives $\mathbf{y}$, the sum of Alice's transmission $\mathbf{x}$ and James' jamming vector $\mathbf{s}$, 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 $N>0$. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values of a series of optimizations. Upper and lower bounds on the optimal values 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.

cs.IT↗
arXiv2018

Can Negligible Cooperation Increase Network Capacity? The Average-Error Case

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 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.

cs.IT↗
arXiv2017

The Benefit of Encoder Cooperation in the Presence of State Information

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.

cs.IT↗
arXiv2016

Tight Network Topology Dependent Bounds on Rounds of Communication

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.

cs.CC↗