We introduce the notion of metric embeddings into a similarity measure over $\mathbb{R}_{+}^{m}$, such as the weighted Jaccard coefficient. We develop average embeddings into such similarity measures for a number of metric spaces, with (appropriately defined) distortion that is smaller than the best possible or known distortion of embedding into $\ell_{1}$ or $\ell_{2}$ spaces (biLipschitz or average). We complement our embeddings with a new algorithm for Approximate Nearest Neighbor Search (ANNS) that leverages such an embedding in a black box fashion. Combining these results, we obtain new efficient algorithms for ANNS under the following two classic metrics, achieving an exponential improvement to longstanding prior work: - Edit distance over length- k strings: $\operatorname{poly}(\log k)$ approximation; - $\ell_{p}$ over $\mathbb{R}^{d}$, for $p\gt2: O(\log p)$ approximation (known to be asymptotically optimal in relevant models of computation).

Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length n can be computed exactly in time $O(n\log n)$, the complexity of estimating the (length of the) LIS in sublinear time, especially when LIS $\ll n$, is still open. We show that for any $n\in\mathbb{N}$ and $\lambda=o(1)$, there exists a (randomized) non-adaptive algorithm that, given a sequence of length n with LIS $\geq\lambda n$, approximates the LIS up to a factor of $1/\lambda^{o(1)}$ in $ n^{o(1)}/\lambda$ time. Our algorithm improves upon prior work substantially in terms of both approximation and run-time: (i) we provide the first sub-polynomial approximation for LIS in sub-linear time; and (ii) our run-time complexity essentially matches the trivial sample complexity lower bound of $\Omega(1/\lambda)$, which is required to obtain any non-trivial approximation of the LIS. As part of our solution, we develop two novel ideas which may be of independent interest. First, we define a new Genuine-LIS problem, in which each sequence element may be either genuine or corrupted. In this model, the user receives unrestricted access to the actual sequence, but does not know a priori which elements are genuine. The goal is to estimate the LIS using genuine elements only, with the minimal number of tests for genuineness. The second idea, Precision Tree, enables accurate estimations for composition of general functions from “coarse” (sub-)estimates. Precision Tree essentially generalizes classical precision sampling, which works only for summations. As a central tool, the Precision Tree is pre-processed on a set of samples, which thereafter is repeatedly used by multiple components of the algorithm, improving their amortized complexity.

Abstract. We present an algorithm for approximating the edit distance between two strings of length [Formula: see text] in time [Formula: see text] up to a constant factor for any [Formula: see text]. Our result completes a research direction set forth in the recent breakthrough paper [ CDG[Formula: see text]18 ], which showed the first constant-factor approximation algorithm with a (strongly) subquadratic running time. Recent results [ KS20b , BR20 ] have shown near-linear time algorithms that obtain an additive approximation, near-linear in [Formula: see text] (equivalently, constant-factor approximation when the edit distance value is close to [Formula: see text]). In contrast, our algorithm obtains a constant-factor approximation in near-linear time for any input string. In contrast to prior algorithms, which are mostly recursing over smaller substrings, our algorithm gradually smoothes out the local contribution to the edit distance over progressively larger substrings. To accomplish this, we iteratively construct a distance oracle data structure for the metric of edit distance on all substrings of input strings of length [Formula: see text] for [Formula: see text]. The distance oracle approximates the edit distance over these substrings in a certain average sense, just enough to estimate the overall edit distance.

We study the problem of discrete distribution testing in the two-party setting. For example, in the standard closeness testing problem, Alice and Bob each have t samples from, respectively, distributions a and b over [n], and they need to test whether a=b or a,b are epsilon-far (in the l_1 distance). This is in contrast to the well-studied one-party case, where the tester has unrestricted access to samples of both distributions. Despite being a natural constraint in applications, the two-party setting has previously evaded attention. We address two fundamental aspects of the two-party setting: 1) what is the communication complexity, and 2) can it be accomplished securely, without Alice and Bob learning extra information about each other’s input. Besides closeness testing, we also study the independence testing problem, where Alice and Bob have t samples from distributions a and b respectively, which may be correlated; the question is whether a,b are independent or epsilon-far from being independent. Our contribution is three-fold: 1) We show how to gain communication efficiency given more samples, beyond the information-theoretic bound on t. The gain is polynomially better than what one would obtain via adapting one-party algorithms. 2) We prove tightness of our trade-off for the closeness testing, as well as that the independence testing requires tight Omega(sqrt{m}) communication for unbounded number of samples. These lower bounds are of independent interest as, to the best of our knowledge, these are the first 2-party communication lower bounds for testing problems, where the inputs are a set of i.i.d. samples. 3) We define the concept of secure distribution testing, and provide secure versions of the above protocols with an overhead that is only polynomial in the security parameter.

We analyze LSH Forest [BCG05]—a popular heuristic for the nearest neighbor search—and show that a careful yet simple modification of it outperforms “vanilla” LSH algorithms. The end result is the first instance of a simple, practical algorithm that provably leverages data-dependent hashing to improve upon data-oblivious LSH. Here is the entire algorithm for the d-dimensional Hamming space. The LSH Forest, for a given dataset, applies a random permutation to all the d coordinates, and builds a trie on the resulting strings. In our modification, we further augment this trie: for each node, we store a constant number of points close to the mean of the corresponding subset of the dataset, which are compared to any query point reaching that node. The overall data structure is simply several such tries sampled independently. While the new algorithm does not quantitatively improve upon the best data-dependent hashing algorithms from [AR15] (which are known to be optimal), it is significantly simpler, being based on a practical heuristic, and is provably better than the best LSH algorithm for the Hamming space [IM98, HIM12].