Alice and Bob want to run a protocol over a noisy channel, where a certain number of bits are flipped adversarially. Several results take a protocol requiring @math bits of noise-free communication and make it robust over such a channel. In a recent breakthrough result, Haeupler described an algorithm that sends a number of bits that is conjectured to be near optimal in such a model. However, his algorithm critically requires @math knowledge of the number of bits that will be flipped by the adversary. We describe an algorithm requiring no such knowledge. If an adversary flips @math bits, our algorithm sends @math bits in expectation and succeeds with high probability in @math . It does so without any @math knowledge of @math . Assuming a conjectured lower bound by Haeupler, our result is optimal up to logarithmic factors. Our algorithm critically relies on the assumption of a private channel. We show that privacy is necessary when the amount of noise is unknown.
The closest prior work to ours is that of Haeupler @cite_21 . His work assumes a fixed and known adversarial noise rate @math , the fraction of bits flipped by the adversary. Communication efficiency is measured by which is @math divided by the total number of bits sent. Haeupler @cite_21 describes an algorithm that achieves a communication rate of @math , which he conjectures to be optimal. We compare our work to his in .
@cite_21
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We study efficient, deterministic interactive coding schemes that simulate any interactive protocol both under random and adversarial errors, and can achieve a constant communication rate independent of the protocol length. For channels that flip bits independently with probability e For channels in which an adversary controls the noise pattern our coding scheme can tolerate Ω(1 log n) fraction of errors with rate approaching 1. Once more, all previously known nontrivial deterministic schemes (either efficient or not) in the adversarial setting had a rate bounded away from 1, and no nontrivial efficient deterministic coding schemes were known with any constant rate. Essential to both results is an explicit, efficiently encodable and decodable systematic tree code of length n that has relative distance Ω(1 log n) and rate approaching 1, defined over an O(log n)-bit alphabet. No nontrivial tree code (either efficient or not) was known to approach rate 1, and no nontrivial distance bound was known for any efficient constant rate tree code. The fact that our tree code is systematic, turns out to play an important role in obtaining rate 1 - O([EQUATION]) in the random error model, and approaching rate 1 in the adversarial error model. We provide tight upper and lower bounds on the noise resilience of interactive communication over noisy channels with feedback. In this setting, we show that the maximal fraction of noise that any nonadaptive protocol can withstand is 1 3. In addition, we provide a simple and efficient nonadaptive coding scheme that succeeds as long as the fraction of noise is at most @math . Surprisingly, both bounds hold regardless of whether the parties send bits or symbols from an arbitrarily large alphabet. We also consider interactive communication over erasure channels. We provide a coding scheme that withstands the optimal tolerable erasure rate of @math [ , IEEE Trans. Info. Theory , 2015], but operates in a much simpler and more efficient way than the previous schemes. Our coding scheme works with an alphabet of size 4, in contrast to prior schemes in which the alphabet size grows as @math . Building on the above algorithm with a fixed alphabet size, we are able to devise a protocol for binary erasure channels that tolerates erasure rates of up to @math . We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size @math that halts in time @math with probability @math , each message containing @math bit: thus its bit complexity per channel is @math (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is @math per channel (it halts in time @math and the size of each message is @math ). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg. A tunable measure for information leakage called maximal a-leakage is introduced. This measure quantifies the maximal gain of an adversary in refining a tilted version of its prior belief of any (potentially random) function of a dataset conditioning on a disclosed dataset. The choice of @math determines the specific adversarial action ranging from refining a belief for @math to guessing the best posterior for @math , and for these extremal values this measure simplifies to mutual information (MI) and maximal leakage (MaxL), respectively. For all other @math this measure is shown to be the Arimoto channel capacity. Several properties of this measure are proven including: (i) quasi-convexity in the mapping between the original and disclosed datasets; (ii) data processing inequalities; and (iii) a composition property. A full version of this paper is in [1].