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 ) Flajolet and Martin gave a single-pass streaming algorithm which uses @math bits of space and computes @math up to a constant factor @cite_15 . Alon, Matias and Szegedy improved this by replacing the randomness used in the Flajolet-Martin algorithm with a family of simple hash functions @cite_35 . Bar- gave several different algorithms for approximating @math up to a @math factor, using as little as @math space @cite_42 . Kane, Nelson and Woodruff have now completed this line of research by giving a single-pass streaming algorithm which approximates @math using @math space @cite_17 . This is optimal for single-pass streaming algorithms; a space lower bound of @math was shown by Alon, Matias and Szegedy @cite_35 , and a lower bound of @math was shown by Woodruff @cite_37 . This was generalised to an @math lower bound for @math -pass streaming algorithms by Chakrabarti and Regev @cite_39 .
@cite_35 @cite_37 @cite_42 @cite_39 @cite_15 @cite_17
2950256099 1995437102 2069278600 1515635142
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. There is growing interest in algorithms for processing and querying continuous data streams (i.e., data seen only once in a fixed order) with limited memory resources. In its most general form, a data stream is actually an update stream, i.e., comprising data-item deletions as well as insertions. Such massive update streams arise naturally in several application domains (e.g., monitoring of large IP network installations or processing of retail-chain transactions). Estimating the cardinality of set expressions defined over several (possibly distributed) update streams is perhaps one of the most fundamental query classes of interest; as an example, such a query may ask “what is the number of distinct IP source addresses seen in passing packets from both router R1 and R 2 but not router R3?”. Earlier work only addressed very restricted forms of this problem, focusing solely on the special case of insert-only streams and specific operators (e.g., union). In this paper, we propose the first space-efficient algorithmic solution for estimating the cardinality of full-fledged set expressions over general update streams. Our estimation algorithms are probabilistic in nature and rely on a novel, hash-based synopsis data structure, termed ”2-level hash sketch”. We demonstrate how our 2-level hash sketch synopses can be used to provide low-error, high-confidence estimates for the cardinality of set expressions (including operators such as set union, intersection, and difference) over continuous update streams, using only space that is significantly sublinear in the sizes of the streaming input (multi-)sets. Furthermore, our estimators never require rescanning or resampling of past stream items, regardless of the number of deletions in the stream. We also present lower bounds for the problem, demonstrating that the space usage of our estimation algorithms is within small factors of the optimal. Finally, we propose an optimized, time-efficient stream synopsis (based on 2-level hash sketches) that provides similar, strong accuracy-space guarantees while requiring only guaranteed logarithmic maintenance time per update, thus making our methods applicable for truly rapid-rate data streams. Our results from an empirical study of our synopsis and estimation techniques verify the effectiveness of our approach. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13.