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We consider the problem of efficiently estimating the size of the inner join of a collection of preprocessed relational tables from the perspective of instance optimality analysis. The run time of instance optimal algorithms is comparable to the minimum time needed to verify the correctness of a solution. Previously instance optimal algorithms were only known when the size of the join was small (as one component of their run time that was linear in the join size). We give an instance optimal algorithm for estimating the join size for all instances, including when the join size is large, by removing the dependency on the join size. As a byproduct, we show how to sample rows from the join uniformly at random in a comparable amount of time.
Motivated by crowdsourced computation, peer-grading, and recommendation systems, Braverman, Mao and Weinberg [STOC16] studied the emph{query} and emph{round} complexity of fundamental problems such as finding the maximum (textsc{max}), finding all elements above a certain value (textsc{threshold-$v$}) or computing the top$-k$ elements (textsc{Top}-$k$) in a noisy environment. For example, consider the task of selecting papers for a conference. This task is challenging due the crowdsourcing nature of peer reviews: the results of reviews are noisy and it is necessary to parallelize the review process as much as possible. We study the noisy value model and the noisy comparison model: In the emph{noisy value model}, a reviewer is asked to evaluate a single element: What is the value of paper $i$? (eg accept). In the emph{noisy comparison model} (introduced in the seminal work of Feige, Peleg, Raghavan and Upfal [SICOMP94]) a reviewer is asked to do a pairwise comparison: Is paper $i$ better than paper $j$? In this paper, we show optimal worst-case query complexity for the textsc{max},textsc{threshold-$v$} and textsc{Top}-$k$ problems. For textsc{max} and textsc{Top}-$k$, we obtain optimal worst-case upper and lower bounds on the round vs query complexity in both models. For textsc{threshold}-$v$, we obtain optimal query complexity and nearly-optimal round complexity, where $k$ is the size of the output) for both models. We then go beyond the worst-case and address the question of the importance of knowledge of the instance by providing, for a large range of parameters, instance-optimal algorithms with respect to the query complexity. Furthermore, we show that the value model is strictly easier than the comparison model.
We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 pm epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/epsilon)$ matrix-vector products. This improves on the ubiquitous Hutchinsons estimator, which requires $O(1/epsilon^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinsons estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinsons method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.
Given a point set $Psubset mathbb{R}^d$, a kernel density estimation for Gaussian kernel is defined as $overline{mathcal{G}}_P(x) = frac{1}{left|Pright|}sum_{pin P}e^{-leftlVert x-p rightrVert^2}$ for any $xinmathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimation of $P$ can be approximated by the kernel density estimation of $Q$. This subset $Q$ is called coreset. The primary technique in this work is to construct $pm 1$ coloring on the point set $P$ by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyks Theorem. When $d>1$ is constant, our construction gives a coreset of size $Oleft(frac{1}{varepsilon}right)$ as opposed to the best-known result of $Oleft(frac{1}{varepsilon}sqrt{logfrac{1}{varepsilon}}right)$. It is the first to give a breakthrough on the barrier of $sqrt{log}$ factor even when $d=2$.
From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size $k$ that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of on-line reservoir sampling, thinking of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, $varoptk$, that dominates all previous schemes in terms of estimation quality. $varoptk$ provides {em variance optimal unbiased estimation of subset sums}. More precisely, if we have seen $n$ items of the stream, then for {em any} subset size $m$, our scheme based on $k$ samples minimizes the average variance over all subsets of size $m$. In fact, the optimality is against any off-line scheme with $k$ samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worst-case bounds on the variance of {em particular} subsets than previously possible. It is efficient, handling each new item of the stream in $O(log k)$ time. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting.
We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP. Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances.