The @math 'th frequency moment of a sequence of integers is defined as @math , where @math is the number of times that @math occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate @math up to relative error @math . We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate @math . We describe quantum algorithms for @math , @math and @math in this model which outperform the best possible classical algorithms almost quadratically.
( @math ) Alon, Matias and Szegedy gave an @math single-pass streaming algorithm @cite_35 , and also showed an @math lower bound. An @math lower bound for single-pass streaming algorithms was proven by Woodruff @cite_37 , which was similarly extended to an @math lower bound for @math -pass streaming algorithms by Chakrabarti and Regev @cite_39 .
@cite_35 @cite_37 @cite_39
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We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest. We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is 4.91+e. In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms. At [email protected]?10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most (1+3) [email protected] 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a (1+2) 2 1.207 lower bound given by Agarwal and Sharathkumar. We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(dlogn) expected amortized time per insertion deletion. We prove a lower bound of Ω(n1 3) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f: 0,1 n→ 0,1 is monotone versus far from monotone. This improves the recent lower bound of Ω(n1 4) for the same problem by Belovs and Blais (STOC'16). Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity we prove a lower bound of Ω(√n) for two-sided, adaptive algorithms and a lower bound of Ω(n) for one-sided, non-adaptive algorithms for testing unateness, a natural generalization of monotonicity. The latter matches the linear upper bounds by Khot and Shinkar (RANDOM'16) and by Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri (2017).