We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protocol while tolerating an optimal corruption of a @math fraction of all messages while incurring a moderate blowup of @math in the communication complexity. Our result is the first fully distributed interactive coding scheme in which the topology of the communication network is not known in advance. Prior work required either a coordinating node to be connected to all other nodes in the network or assumed a synchronous network in which all nodes already know the complete topology of the network.
Performing computations over noisy channels is the heart of , initiated by Schulman @cite_7 @cite_14 . A long line of work considers the 2-party case and obtains various coding schemes, as well as bounds on their capabilities in various settings and noise models @cite_17 @cite_32 @cite_38 @cite_23 @cite_6 @cite_34 @cite_0 @cite_30 @cite_4 @cite_15 . See @cite_13 for a survey on interactive coding.
@cite_30 @cite_38 @cite_14 @cite_4 @cite_7 @cite_32 @cite_6 @cite_0 @cite_23 @cite_15 @cite_34 @cite_13 @cite_17
2754061954 2077242989 2026670837 2072979474
We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol. In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and tolerates noise rates of up to 1 18 - epsilon. To this end we develop a novel primitive we name edit distance tree code. The edit distance tree code is designed to replace the Hamming distance constraints in Schulman's tree codes (STOC 93), with a stronger edit distance requirement. However, the straightforward generalization of tree codes to edit distance does not seem to yield a primitive that suffices for communication in the presence of synchronization problems. Giving the "right" definition of edit distance tree codes is a main conceptual contribution of this work. In this dissertation, we prove a coding theorem for distributed network protocols. We show that any distributed protocol which runs on a noiseless network in time T, can be simulated on an identical noisy network with a slow-down factor proportional to log(d + 1), where d is the maximum degree in the network, and with exponentially small probability of error. Our theorem is a network analog of the classical coding theorem of Shannon (1948), which addresses the issue of the transmission of data over a noisy link. Although Shannon's original theorem is non-constructive--it proves the existence of codes that transfer data at asymptotically optimal rates--the subsequent development of coding theory makes this result constructive. Shannon's theory, however, does not address issues that arise when noise occurs in the communications of a wide area computational network with a number of links available for simultaneous use. This question of what happens when computation is interleaved with noisy communications was proposed and first addressed by Schulman in the special case of a pair of computers connected by a bidirectional communication link. In this case, Schulman obtains an analog of Shannon's theorem. Like those of Shannon and Schulman, the results we exhibit are non constructive. However, in our case, it is possible to obtain constructive simulations by compromising the reliability with which they run. For a particular problem, the 'telephone' game, we exhibit a constructive solution for which this compromise is unnecessary. Work reported in this dissertation stems from that done jointly with Leonard Schulman. We revisit the problem of reliable interactive communication over a noisy channel, and obtain the first fully explicit (randomized) efficient constant-rate emulation procedure for reliable interactive communication. Our protocol works for any discrete memory less noisy channel with constant capacity, and fails with exponentially small probability in the total length of the protocol. Following a work by Schulman [Schulman 1993] our simulation uses a tree-code, yet as opposed to the non-constructive absolute tree-code used by Schulman, we introduce a relaxation in the notion of goodness for a tree code and define a potent tree code. This relaxation allows us to construct an explicit emulation procedure for any two-party protocol. Our results also extend to the case of interactive multiparty communication. We show that a randomly generated tree code (with suitable constant alphabet size) is an efficiently decodable potent tree code with overwhelming probability. Furthermore we are able to partially derandomize this result by means of epsilon-biased distributions using only O(N) random bits, where N is the depth of the tree. In this work, we study the problem of constructing interactive protocols that are robust to noise, a problem that was originally considered in the seminal works of Schulman (FOCS '92, STOC '93), and has recently regained popularity. Robust interactive communication is the interactive analogue of error correcting codes: Given an interactive protocol which is designed to run on an error-free channel, construct a protocol that evaluates the same function (or, more generally, simulates the execution of the original protocol) over a noisy channel. As in (non-interactive) error correcting codes, the noise can be either stochastic, i.e. drawn from some distribution, or adversarial, i.e. arbitrary subject only to a global bound on the number of errors. We show how to simulate any interactive protocol in the presence of constant-rate noise, while incurring only a constant blow-up in the communication complexity (CC). Our simulator is randomized, and succeeds in simulating the original protocol with probability at least @math .