We show how to compute any symmetric Boolean function on n variables over any field (as well as the integers) with a probabilistic polynomial of degree O(a#x221A;n log(1 a#x03B5;))) and error at most a#x03B5;. The degree dependence on n and a#x03B5; is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution. This polynomial construction is combined with other algebraic ideas to give the first sub quadratic time algorithm for computing a (worst-case) batch of Hamming distances in super logarithmic dimensions, exactly. To illustrate, let c(n): N -> N. Suppose we are given a database D of n vectors in 0, 1 (c(n) log n) and a collection of n query vectors Q in the same dimension. For all u in Q, we wish to compute a v in D with minimum Hamming distance from u. We solve this problem in n(2-1 O(c(n) log2 c(n))) randomized time. Hence, the problem is in "truly sub quadratic" time for O(log n) dimensions, and in sub quadratic time for d = o((log2 n) (log log n)2). We apply the algorithm to computing pairs with maximum inner product, closest pair in l1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.
The planted'' case of Hamming distance has been studied extensively in learning theory and cryptography. In this setting, all vectors are chosen uniformly at random, except for a planted pair of vectors with Hamming distance much smaller than the expected distance between two random vectors. Two recent references are notable: G. Valiant @cite_2 gave a breakthrough @math time algorithm, which is of the vector dimension and the Hamming distance of the planted pair. Valiant also gives a @math -approximation to the closest pair problem in Hamming distance running in @math time. See @cite_11 for very recent work on batch Hamming distance computations in cryptoanalysis.
@cite_11 @cite_2
2263882035 1985526537 2131709403 2023683873
Given a set of n d-dimensional Boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have Pearson correlation coefficient ρ (Hamming distance dċ 1−ρf2), how quickly can one find the two correlated vectorsq We present an algorithm which, for any constant e>0, and constant ρ>0, runs in expected time O(n5mωf4mωpe pnd) Applications and extensions of this basic algorithm yield significantly improved algorithms for several other problems. Approximate Closest Pair. For any sufficiently small constant e>0, given n d-dimensional vectors, there exists an algorithm that returns a pair of vectors whose Euclidean (or Hamming) distance differs from that of the closest pair by a factor of at most 1pe, and runs in time O(n2mΘ(se)). The best previous algorithms (including Locality Sensitive Hashing) have runtime O(n2mO(e)). Learning Sparse Parities with Noise. Given samples from an instance of the learning parities with noise problem where each example has length n, the true parity set has size at most k « n, and the noise rate is η, there exists an algorithm that identifies the set of k indices in time nωpef3 k poly(1f1m2η) 0.4), improves upon the results of [2011] that give a runtime of n(1p(2 η)2 p o(1))kf2 poly(1f1m2η). Learning k-Juntas with Noise. Given uniformly random length n Boolean vectors, together with a label, which is some function of just k « n of the bits, perturbed by noise rate η, return the set of relevant indices. Leveraging the reduction of [2009], our result for learning k-parities implies an algorithm for this problem with runtime nωpef3 k poly(1f1m2η) Learning k-Juntas without Noise. Given uniformly random length n Boolean vectors, together with a label, which is some function of k « n of the bits, return the set of relevant indices. Using a modification of the algorithm of [2004], and employing our algorithm for learning sparse parities with noise via the reduction of [2009], we obtain an algorithm for this problem with runtime nωp ef4 k poly(n) We prove an optimal @math lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive @math -bit strings @math and @math , respectively. They are promised that the Hamming distance between @math and @math is either at least @math or at most @math , and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283--289], it had been conjectured that the naive protocol, which uses @math bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is de... The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity. Given a set of @math @math -dimensional Boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have Pearson -- correlation @math (Hamming distance @math ), how quickly can one find the two correlated vectors? We present an algorithm which, for any constants @math and @math finds the correlated pair with high probability, and runs in time @math given @math vectors in @math , our algorithm returns a pair of vectors whose Euclidean distance differs from that of the closest pair by a factor of at most @math and runs in time @math . The best previous algorithms (including LSH) have runtime @math Learning Sparse Parity with Noise: Given samples from an instance of the learning parity with noise problem where each example has length @math , the true parity set has size at most @math . Learning @math -Juntas without Noise: Our results for learning sparse parities with noise imply an algorithm for learning juntas without noise with runtime $n^ + 4 k poly(n) which improves on the runtime n !+1 ! poly(n) n0:7kpoly(n) of [13].