arXiv subjects

Ronitt Rubinfeld

Publications and source records attributed to Ronitt Rubinfeld.

59 records · Page 4Linked to original sources

arXiv2011

Fast Local Computation Algorithms

For input $x$, let $F(x)$ denote the set of outputs that are the "legal" answers for a computational problem $F$. Suppose $x$ and members of $F(x)$ are so large that there is not time to read them in their entirety. We propose a model of {\em local computation algorithms} which for a given input $x$, support queries by a user to values of specified locations $y_i$ in a legal output $y \in F(x)$. When more than one legal output $y$ exists for a given $x$, the local computation algorithm should output in a way that is consistent with at least one such $y$. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of $k$-wise independent random variables and Beck's analysis in his algorithmic approach to the Lov{á}sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in {\em polylogarithmic} time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying $k$-SAT formulas.

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arXiv2011

Approximating the Influence of a monotone Boolean function in O(\sqrt{n}) query complexity

The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$, \else $f: \bitset^n \to \bitset$, \fi which we denote by $I[f]$. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $(1\pm \eps)$ by performing $O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $ queries. % \mnote{D: say something about technique?} We also prove a lower bound of % $Ω(\frac{\sqrt{n/\log n}}{I[f]})$ $Ω(\frac{\sqrt{n}}{\log n \cdot I[f]})$ on the query complexity of any constant-factor approximation algorithm for this problem (which holds for $I[f] = Ω(1)$), % and $I[f] = O(\sqrt{n}/\log n)$), hence showing that our algorithm is almost optimal in terms of its dependence on $n$. For general functions we give a lower bound of $Ω(\frac{n}{I[f]})$, which matches the complexity of a simple sampling algorithm.

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arXiv2010

Testing Closeness of Discrete Distributions

Given samples from two distributions over an $n$-element set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in $n$, specifically, $O(n^{2/3}ε^{-8/3}\log n)$, independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than $\max\{ε^{4/3}n^{-1/3}/32, εn^{-1/2}/4\}$) or large (more than $ε$) in $\ell_1$ distance. This result can be compared to the lower bound of $Ω(n^{2/3}ε^{-2/3})$ for this problem given by Valiant. Our algorithm has applications to the problem of testing whether a given Markov process is rapidly mixing. We present sublinear for several variants of this problem as well.

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arXiv2009

Sublinear Time Algorithms for Earth Mover's Distance

We study the problem of estimating the Earth Mover's Distance (EMD) between probability distributions when given access only to samples. We give closeness testers and additive-error estimators over domains in $[0, Δ]^d$, with sample complexities independent of domain size - permitting the testability even of continuous distributions over infinite domains. Instead, our algorithms depend on other parameters, such as the diameter of the domain space, which may be significantly smaller. We also prove lower bounds showing the dependencies on these parameters to be essentially optimal. Additionally, we consider whether natural classes of distributions exist for which there are algorithms with better dependence on the dimension, and show that for highly clusterable data, this is indeed the case. Lastly, we consider a variant of the EMD, defined over tree metrics instead of the usual L1 metric, and give optimal algorithms.

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arXiv2007

Sublinear Algorithms for Approximating String Compressibility

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to Lempel-Ziv to the number of distinct short substrings contained in it. In addition, we show that approximating the compressibility with respect to LZ is related to approximating the support size of a distribution.

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