The noisy broadcast model was first studied in [Gallager, TranInf'88] where an @math -character input is distributed among @math processors, so that each processor receives one input bit. Computation proceeds in rounds, where in each round each processor broadcasts a single character, and each reception is corrupted independently at random with some probability @math . [Gallager, TranInf'88] gave an algorithm for all processors to learn the input in @math rounds with high probability. Later, a matching lower bound of @math was given in [Goyal, Kindler, Saks; SICOMP'08]. We study a relaxed version of this model where each reception is erased and replaced with a ?' independently with probability @math . In this relaxed model, we break past the lower bound of [Goyal, Kindler, Saks; SICOMP'08] and obtain an @math -round algorithm for all processors to learn the input with high probability. We also show an @math -round algorithm for the same problem when the alphabet size is @math .
We note that it follows from the lower bound in @cite_3 that in a variant of our model where one considers substitution errors instead of erasure errors, any protocol from which all the processors learn the entire input must take @math rounds. In light of this lower bound, our result of an @math protocol is interesting, as it shows a fundamental difference between substitution and erasure errors in this broadcast model.
@cite_3
2047805348 2030003887 1986293063 2144590858
We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts. In the renaming task n + 1 processes start with unique input names taken from a large space and must choose unique output names taken from a smaller name space, 0, 1, . . . , K. To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process id. showed in 1990 that renaming has a wait-free solution when K ≥ 2n. Several proofs of a lower bound stating that no such protocol exists when K < 2n have been published. We presented in the ACM PODC 2008 conference the following two results. First, we presented the first completely combinatorial lower bound proof stating that no such a protocol exists when K < 2n. This bound holds for infinitely many values of n. Second, for the other values of n, we proved that the lower bound for K < 2n is incorrect, exhibiting a wait-free renaming protocol for K = 2n−1. More precisely, we presented a theorem stating that there exists a wait-free renaming protocol for K < 2n if and only if the set of integers ( n+1 i+1 | 0 i n-1 2 ) are relatively prime. This paper is the first part of the full version of the results presented in the ACM PODC 2008 conference. It includes only the lower bound. Namely, we show here that no protocol for renaming exists when K < 2n, if n is such that ( n+1 i+1 | 0 i n-1 2 ) are not relatively prime. We prove this result using the known equivalence of K-renaming for K = 2n−1 and the weak symmetry breaking task. In this task processes have no input values and the output values are 0 or 1, and it is required that in every execution in which all processes participate, at least one process decides 0 and at least one process decides 1. The full version of the upper bound appears in a companion paper [10]. We give efficient protocols and matching accuracy lower bounds for frequency estimation in the local model for differential privacy. In this model, individual users randomize their data themselves, sending differentially private reports to an untrusted server that aggregates them. We study protocols that produce a succinct histogram representation of the data. A succinct histogram is a list of the most frequent items in the data (often called "heavy hitters") along with estimates of their frequencies; the frequency of all other items is implicitly estimated as 0. If there are n users whose items come from a universe of size d, our protocols run in time polynomial in n and log(d). With high probability, they estimate the accuracy of every item up to error O(√ log(d) (e2n) ). Moreover, we show that this much error is necessary, regardless of computational efficiency, and even for the simple setting where only one item appears with significant frequency in the data set. Previous protocols (Mishra and Sandler, 2006; Hsu, Khanna and Roth, 2012) for this task either ran in time Ω(d) or had much worse error (about √[6] log(d) (e2n) ), and the only known lower bound on error was Ω(1 √ n ). We also adapt a result of (2010) to the local setting. In a model with public coins, we show that each user need only send 1 bit to the server. For all known local protocols (including ours), the transformation preserves computational efficiency. We consider a fault tolerance broadcast network of n processors each holding one bit of information. The goal is to compute a given Boolean function on the n bits. In each step, a processor may broadcast one bit of information. Each listening processor receives the bit that was broadcasted with error probability bounded by a fixed constant spl epsi . The errors in different steps, as well as for different receiving processors in the same step, are mutually independent. The protocols that are considered in this model are oblivious protocols: At each step, the processors that broadcast are fixed in advanced and independent of the input and the outcome of previous steps. The primal complexity measure in this model is the total number of broadcasts that is performed by the protocol. We present here the first linear complexity protocols for several classes of Boolean functions, including the OR function, functions that have O(l)-minterm (maxterm) size, functions that have linear size AC sub 0 formulae and some other functions. This answer an open question of Yao (1997), considering this fault tolerance model of El-Gamal (1984) and Gallager (1988).