Approximating Approximate Pattern Matching

Given a text $T$ of length $n$ and a pattern $P$ of length $m$, the approximate pattern matching problem asks for computation of a particular \emph{distance} function between $P$ and every $m$-substring of $T$. We consider a $(1\pm\varepsilon)$ multiplicative approximation variant of this problem, for $\ell_p$ distance function. In this paper, we describe two $(1+\varepsilon)$-approximate algorithms with a runtime of $\widetilde{O}(\frac{n}{\varepsilon})$ for all (constant) non-negative values of $p$. For constant $p \ge 1$ we show a deterministic $(1+\varepsilon)$-approximation algorithm. Previously, such run time was known only for the case of $\ell_1$ distance, by Gawrychowski and Uzna\'nski [ICALP 2018] and only with a randomized algorithm. For constant $0 \le p \le 1$ we show a randomized algorithm for the $\ell_p$, thereby providing a smooth tradeoff between algorithms of Kopelowitz and Porat [FOCS~2015, SOSA~2018] for Hamming distance (case of $p=0$) and of Gawrychowski and Uzna\'nski for $\ell_1$ distance

Quantum Chebyshev's Inequality and Applications

In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments $F_k$ of order $k \geq 3$ in the multi-pass streaming model with updates (turnstile model). We design a $P$-pass quantum streaming algorithm with memory $M$ satisfying a tradeoff of $P^2 M = \tilde{O}(n^{1-2/k})$, whereas the best classical algorithm requires $P M = \Theta(n^{1-2/k})$. Then, we study the problem of estimating the number $m$ of edges and the number $t$ of triangles given query access to an $n$-vertex graph. We describe optimal quantum algorithms that perform $\tilde{O}(\sqrt{n}/m^{1/4})$ and $\tilde{O}(\sqrt{n}/t^{1/6} + m^{3/4}/\sqrt{t})$ queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems. For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev's inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependency is quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [BHMT02] and of previous quantum algorithms for the mean estimation problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. For our applications, we also adapt the variable-time amplitude amplification technique of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.Comment: 27 pages; v3: better presentation, lower bound in Theorem 4.3 is ne

A Framework for Adversarially Robust Streaming Algorithms

We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems. In this work, we show that the answer is positive for various important streaming problems in the insertion-only model, including distinct elements and more generally $F_p$-estimation, $F_p$-heavy hitters, entropy estimation, and others. For all of these problems, we develop adversarially robust $(1+\varepsilon)$-approximation algorithms whose required space matches that of the best known non-robust algorithms up to a $\text{poly}(\log n, 1/\varepsilon)$ multiplicative factor (and in some cases even up to a constant factor). Towards this end, we develop several generic tools allowing one to efficiently transform a non-robust streaming algorithm into a robust one in various scenarios.Comment: Conference version in PODS 2020. Version 3 addressing journal referees' comments; improved exposition of sketch switchin

Finding heavy hitters from lossy or noisy data

Abstract. Motivated by Dvir et al. and Wigderson and Yehudayoff [3

Taylor Polynomial Estimator for Estimating Frequency Moments

We present a randomized algorithm for estimating the $p$th moment $F_p$ of the frequency vector of a data stream in the general update (turnstile) model to within a multiplicative factor of $1 \pm \epsilon$, for $p > 2$, with high constant confidence. For $0 < \epsilon \le 1$, the algorithm uses space $O( n^{1-2/p} \epsilon^{-2} + n^{1-2/p} \epsilon^{-4/p} \log (n))$ words. This improves over the current bound of $O(n^{1-2/p} \epsilon^{-2-4/p} \log (n))$ words by Andoni et. al. in \cite{ako:arxiv10}. Our space upper bound matches the lower bound of Li and Woodruff \cite{liwood:random13} for $\epsilon = (\log (n))^{-\Omega(1)}$ and the lower bound of Andoni et. al. \cite{anpw:icalp13} for $\epsilon = \Omega(1)$.Comment: Supercedes arXiv:1104.4552. Extended Abstract of this paper to appear in Proceedings of ICALP 201