A set of n players, each holding a private input bit, communicate over a noisy broadcast channel. Their mutual goal is for all players to learn all inputs. At each round one of the players broadcasts a bit to all the ...
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ISBN:
(纸本)9781450355599
A set of n players, each holding a private input bit, communicate over a noisy broadcast channel. Their mutual goal is for all players to learn all inputs. At each round one of the players broadcasts a bit to all the other players, and the bit received by each player is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed? This problem was first suggested by El Gamal in 1984. In 1988, Gallager gave an elegant noise-resistant protocol requiring only O(n log logn) rounds. The problem got resolved in 2005 by a seminal paper of Goyal, Kindler, and Saks, proving that Gallager's protocol is essentially optimal. We revisit the above noisy broadcast problem and showthat O(n) rounds suffice. This is possible due to a relaxation of the model assumed by the previous works. We no longer demand that exactly one player broadcasts in every round, but rather allow any number of players to broadcast. However, if it is not the case that exactly one player chooses to broadcast, each of the other players gets an adversely chosen bit. We generalized the above result and initiate the study of interactive coding over the noisy broadcast channel. We show that any interactive protocol that works over the noiseless broadcast channel can be simulated over our restrictive noisy broadcast model with constant blowup of the communication.
The problem of constructing error-resilient interactive protocols was introduced in the seminal works of Schulman (FOGS 1992, STOC 1993). These works show how to convert any two-party interactive protocol into one tha...
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ISBN:
(纸本)9781450333337
The problem of constructing error-resilient interactive protocols was introduced in the seminal works of Schulman (FOGS 1992, STOC 1993). These works show how to convert any two-party interactive protocol into one that is resilient to constant-fraction of adversarial error, while blowing up the communication by only a constant factor. In this work we extend the work of Schulman to the multiparty setting. We show how to convert any (non-adaptive) n-party protocol into one that is resilient to Theta(1/n)-fraction of adversarial error, while blowing up the communication by only a constant factor. One might hope to get resilience to constant-fraction of errors, by restricting the adversary's error distribution, and allowing him to make at most a constant-fraction of errors per party. We present a black-box lower bound, showing that we cannot be resilient to such adversarial errors, even if we increase the communication by an arbitrary polynomial factor, assuming the error-resilient protocol has a fixed (non-adaptive) speaking order.
interactive coding, pioneered by Schulman (FOCS '92, STOC '93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small consta...
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interactive coding, pioneered by Schulman (FOCS '92, STOC '93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversary's discretion, as they pass through the communication channel. Braverman et al. proposed a far-reaching generalization of this model, whereby the adversary can additionally manipulate the channel by removing and inserting symbols. For any epsilon > 0, they showed how to faithfully simulate any protocol in this model with corruption rate up to 1/18 - epsilon, using a constant-size alphabet and a constant-factor overhead in communication. We give an optimal simulation of any protocol in this generalized model of substitutions, insertions, and deletions, tolerating a corruption rate up to 1/4 - epsilon, while keeping the alphabet to a constant size and the communication overhead to a constant factor. This resolves a question due to Gelles (2015). Our corruption tolerance matches an impossibility result for corruption rate 1/4 which holds even for substitutions alone (Braverman and Rao, STOC '11).
Consider two parties who wish to communicate in order to execute some interactive protocol pi. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channe...
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Consider two parties who wish to communicate in order to execute some interactive protocol pi. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channel and can change a constant fraction of the bits arbitrarily, thus interrupting the execution of (which was designed for an error-free channel). If pi only contains a single long message, then a good error correcting code would overcome the noise with only a constant overhead in communication. However, this solution is not applicable to interactive protocols consisting of many short messages. Schulman [1992, 1993] introduced the notion of interactive coding: A simulator that, given any protocol pi, is able to simulate it (i.e., produce its intended transcript) even in the presence of constant rate adversarial channel errors, and with only constant (multiplicative) communication overhead. However, the running time of Schulman's simulator, and of all simulators that followed, has been exponential (or subexponential) in the communication complexity of pi (which we denote by N). In this work, we present three efficient simulators, all of which are randomized and have a certain failure probability (over the choice of coins). The first runs in time poly(N), has failure probability roughly 2(-N), and is resilient to 1/32-fraction of adversarial error. The second runs in time 0(N log N), has failure probability roughly 2-N, and is resilient to some constant fraction of adversarial error. The third runs in time 0(N), has failure probability 1/poly(N), and is resilient to some constant fraction of adversarial error. (Computational complexity is measured in the RAM model.) The first two simulators can be made deterministic if they are a priori given a random string (which may be known to the adversary ahead of time). In particular, the simulators can be made to be nonuniform and deterministic (with equivalent performance). Categories and Subject Descriptors: E.4 [
We consider computations over networks with multiple broadcast channels that intersect at a single party. Each broadcast link suffers from random bit-flip noise that affects the receivers independently. We design inte...
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We consider computations over networks with multiple broadcast channels that intersect at a single party. Each broadcast link suffers from random bit-flip noise that affects the receivers independently. We design interactive coding schemes that successfully perform any computation over these noisy networks and strive to reduce their communication overhead with respect to the original (noiseless) computation. A simple variant of a coding scheme by Rajagopalan and Schulman (STOC 1994) shows that any (noiseless) protocol of R rounds can be reliably simulated in O(Rlog n) rounds over a network with n = n(1)n(2) + 1 parties in which a single party is connected to n(2) noisy broadcast channels, each of which connects n(1) distinct parties. We design new coding schemes with improved overheads. Our approach divides the network into four regimes according to the relationship between n(1) and n(2). We employ a two-layer coding where the inner code protects each broadcast channel and is tailored to the specific conditions of the regime in consideration. The outer layer protects the computation in the network and is generally based on the scheme of Rajagopalan and Schulman, adapted to the case of broadcast channels. The overhead we obtain ranges from O(log log n(2)) to O(log n(2) log log n(1)/log n(1)) and beats the trivial O(log n) overhead in all four regimes.
In the field of interactive coding, two or more parties wish to carry out a distributed computation over a communication network that may be noisy. The ultimate goal is to develop efficient coding schemes that can tol...
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ISBN:
(纸本)9781450362177
In the field of interactive coding, two or more parties wish to carry out a distributed computation over a communication network that may be noisy. The ultimate goal is to develop efficient coding schemes that can tolerate a high level of noise while increasing the communication by only a constant factor (i.e., constant rate). In this work we consider synchronous communication networks over an arbitrary topology, in the powerful adversarial insertion-deletion noise model. Namely, the noisy channel may adversarially alter the content of any transmitted symbol, as well as completely remove a transmitted symbol or inject a new symbol into the channel. We provide efficient, constant rate schemes that successfully conduct any computation with high probability as long as the adversary corrupts at most epsilon/m fraction of the total communication, where m is the number of links in the network and epsilon is a small constant. This scheme assumes an oblivious adversary which is independent of the parties' inputs and randomness. We can remove this assumption and resist a worst-case adversary at the price of being resilient to epsilon/m log m errors. While previous work considered the insertion-deletion noise model in the two-party setting, to the best of our knowledge, our scheme is the first multiparty scheme that is resilient to insertions and deletions. Furthermore, our scheme is the first computationally efficient scheme in the multiparty setting that is resilient to adversarial noise.
The field of interactive coding studies how an interactive protocol can be made resilient to channel errors. Even though this field has received abundant attention since Schulman's seminal paper (FOCS 92), constru...
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ISBN:
(纸本)9781728196213
The field of interactive coding studies how an interactive protocol can be made resilient to channel errors. Even though this field has received abundant attention since Schulman's seminal paper (FOCS 92), constructing interactive coding schemes that are both deterministic and efficient, and at the same time resilient to adversarial errors (with constant information and error rates), remains an elusive open problem. An appealing approach towards resolving this problem is to construct an efficiently encodable and decodable combinatorial object called a tree code (Schulman, STOC 93). After a lot of effort in this direction, the current state of the art has deterministic constructions of tree codes that are efficiently encodable but require an alphabet of size logarithmic (instead of constant) in the depth of the tree code (Cohen, Haeupler, and Schulman, STOC 18). We emphasize that we still lack (even heuristic) candidate constructions that are efficiently decodable. In this work, we show that tree codes that are efficiently encodable, but not efficiently decodable, also imply deterministic and efficient interactive coding schemes that are resilient to adversarial errors. Our result immediately implies a deterministic and efficient interactive coding scheme with a logarithmic alphabet (i.e., 1/log log rate). We show this result using a novel implementation of hashing through deterministic tree codes that is powerful enough to yield interactive coding schemes.
interactive coding, pioneered by Schulman (FOCS '92, STOC '93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small consta...
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ISBN:
(纸本)9781538634646
interactive coding, pioneered by Schulman (FOCS '92, STOC '93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversary's discretion, as they pass through the communication channel. Braverman, Gelles, Mao, and Ostrovsky (2015) proposed a far-reaching generalization of this model, whereby the adversary can additionally manipulate the channel by removing and inserting symbols. They showed how to faithfully simulate any protocol in this model with corruption rate up to 1/18, using a constant-size alphabet and a constant-factor overhead in communication. We give an optimal simulation of any protocol in this generalized model of substitutions, insertions, and deletions, tolerating a corruption rate up to 1/4 while keeping the alphabet to a constant size and the communication overhead to a constant factor. Our corruption tolerance matches an impossibility result for corruption rate 1/4 which holds even for substitutions alone (Braverman and Rao, STOC '11).
A two-terminal interactive function computation problem with alternating messages is studied within the framework of distributed block source coding theory. For any finite number of messages, a single-letter character...
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A two-terminal interactive function computation problem with alternating messages is studied within the framework of distributed block source coding theory. For any finite number of messages, a single-letter characterization of the sum-rate-distortion function was established in previous works using standard information-theoretic techniques. This, however, does not provide a satisfactory characterization of the infinite-message limit, which is a new, unexplored dimension for asymptotic analysis in distributed block source coding involving potentially an infinite number of infinitesimal-rate messages. In this paper, the infinite-message sum-rate-distortion function, viewed as a functional of the joint source distribution and the distortion levels, is characterized as the least element of a partially ordered family of functionals having certain convex-geometric properties. The new characterization does not involve evaluating the infinite-message limit of a finite-message sum-rate-distortion expression. This characterization leads to a family of lower bounds for the infinite-message sum-rate-distortion expression and a simple criterion to test the optimality of any achievable infinite-message sum-rate-distortion expression. The new convex-geometric characterization is used to develop an iterative algorithm for evaluating any finite-message sum-rate-distortion function. It is also used to construct the first examples which demonstrate that for lossy source reproduction, two messages can strictly improve the one-message Wyner-Ziv rate-distortion function settling an unresolved question from a 1985 paper. It is shown that a single backward message of arbitrarily small rate can lead to an arbitrarily large gain in the sum-rate.
A problem of interactive function computation in a collocated network is studied in a distributed block source coding framework. With the goal of computing samples of a desired function of sources at the sink, the sou...
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A problem of interactive function computation in a collocated network is studied in a distributed block source coding framework. With the goal of computing samples of a desired function of sources at the sink, the source nodes exchange messages through a sequence of error-free broadcasts. For any function of independent sources, a computable characterization of the set of all feasible message coding rates-the rate region-isderived in terms of single-letter information measures. In the limit as the number of messages tends to infinity, the infinite-message minimum sum rate, viewed as a functional of the joint source probability mass function, is characterized as the least element of a partially ordered family of functionals having certain convex-geometric properties. This characterization leads to a family of lower bounds for the infinite-message minimum sum rate and a simple criterion to test the optimality of any achievable infinite-message sum rate. An iterative algorithm for evaluating the infinite-message minimum sum-rate functional is proposed and is demonstrated through an example of computing the minimum function of three Bernoulli sources. Based on the characterizations of the rate regions, it is shown that when computing symmetric functions of binary sources, the sink will inevitably learn certain additional information that is not demanded in computing the function. This conceptual understanding leads to new improved bounds for the minimum sum rate. The new bounds are shown to be orderwise better than those based on cut-sets as the network scales. The scaling law of the minimum sum rate is explored for different classes of symmetric functions and source parameters.
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