We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are -far from having the property, that is, differ from every function with the property on at least an fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f .We consider functions over domains {0, 1} d , {1, . . . , n} and {1, . . . , n} d , equipped with 1 distance.We design efficient testers of the Lipschitz property for functions of the form f : {0, 1} d → δZ, where δ ∈ (0, 1] and δZ is the set of integer multiples of δ, and of the form f : {1, . . . , n} → R, where R is (discretely) metrically convex. In the first case, the tester runs in time O(d · min {d, r} /δ ), where r is the diameter of the image of f ; in the second, in time O((log n)/ ). We give corresponding lower bounds of Ω(d) and Ω(log n) on the query complexity (in the second case, only for nonadaptive 1-sided error testers). Our lower bound for functions over {0, 1} d is tight for the case of the {0, 1, 2} range and constant . The first tester implies an algorithm for functions of the form f : {0, 1} d → R that distinguishes Lipschitz functions from functions that are -far from (1 + δ)-Lipschitz. We also present a local filter of the Lipschitz property for functions of the form f : {1, . . . , n} d → R with lookup complexity O((log n + 1) d ). For functions of the form {0, 1} d , we show that every nonadaptive local filter has lookup complexity exponential in d.The testers that we developed have applications to programs analysis. The reconstructors have applications to data privacy. For the first application, the Lipschitz property of the function computed by a program corresponds to a notion of robustness to noise in the data. The application to privacy is based on the fact that a function f of entries in a database of sensitive information can be released with noise of magnitude proportional to a Lipschitz constant of f , while preserving the privacy of individuals whose data is stored in the database (Dwork, McSherry, Nissim and Smith, TCC 2006). We give a differentially private mechanism, based on local filters, for releasing a function f when a Lipschitz constant of f is provided by a distrusted client. We show that when no reliable Lipschitz constant of f is given, previously known differentially private mechanisms either have a substantially higher running time or have a higher expected error for a large class of symmetric functions f .

No abstract

We design a space-efficient algorithm that approximates the transitivity (global clustering coefficient) and total triangle count with only a single pass through a graph given as a stream of edges. Our procedure is based on the classic probabilistic result, the birthday paradox. When the transitivity is constant and there are more edges than wedges (common properties for social networks), we can prove that our algorithm requires O( √ n) space (n is the number of vertices) to provide accurate estimates. We run a detailed set of experiments on a variety of real graphs and demonstrate that the memory requirement of the algorithm is a tiny fraction of the graph. For example, even for a graph with 200 million edges, our algorithm stores just 40,000 edges to give accurate results. Being a single pass streaming algorithm, our procedure also maintains a real-time estimate of the transitivity/number of triangles of a graph by storing a minuscule fraction of edges. ACM Reference Format:Madhav Jha, C. Seshadhri, and Ali Pinar. 2015. A space-efficient streaming algorithm for estimating transitivity and triangle counts using the birthday paradox. ACM Trans. Knowl. Discov. Data 9, 3, Article 15 (February 2015), 21 pages.

Given a directed acyclic graph (DAG) Gn = (Vn, E), a function on Gn is given by f : Vn → R. Such a function is monotone if f (x) ≤ f (y) for all (x, y) ∈ E. A local monotonicity reconstructor for Gn, introduced by Saks and Seshadhri (SICOMP 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f : Vn → R on Gn, can quickly evaluate a related function g : Vn → R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner.Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitiveclosure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH ) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing and data structures.In this paper, we show a connection between 2-TC-spanners of Gn and local monotonicity reconstructors for Gn. We show that an efficient local monotonicity reconstructor for Gn implies a sparse 2-TC-spanner of Gn, providing a new technique for proving lower bounds for local monotonicity reconstructors. We present tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid, DAGs which are very-well studied in this area. These bounds imply lower bounds for local monotonicity reconstructors for the hypergrid (hypercube) that nearly match the known upper bounds.

No abstract

The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is more a matter of convenience than of necessity, and it is important to investigate distances induced by more general distributions. In this paper, we make significant strides in this direction. We give simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing.We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show this sum-of-depths is also a lower bound. A fundamental technical contribution of this work is an optimal dimension reduction theorem for all bounded derivative properties, which relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity for the past 15 years, and our theorem is an exponential improvement to the previous best known result.This means the rth-partial derivative of f is bounded by quantities that only depend on the rth coordinate. Note that this dependence is completely arbitrary, and different dimensions can have completely different bounds. This forms a rich class of properties which includes monotonicity and c-Lipschitz continuity. To get monotonicity, simply set l r (y) = 0 and u r (y) = ∞ for all r. To get c-Lipschitz continuity, set l r (y) = −c and u r (y) = +c for all r. The class also includes the property demanding monotonicity for some (fixed) coordinates and the c-Lipschitz continuity for others; and the non-uniform Lipschitz property that demands different Lipschitz constants for different coordinates.Definition 1.2. Fix a bounding family B and product distribution D = r≤d D r . Define dist D (f, g) = Pr x∼D [f (x) = g(x)]. A property tester for P(B) with respect to D takes as input proximity parameter ε > 0 and has query access to function f . If f ∈ P(B), the tester accepts with probability > 2/3. If dist D (f, P(B)) > ε, the tester rejects with probability > 2/3.

Abstract. A function f (x1, ..., x d ), where each input is an integer from 1 to n and output is a real number, is Lipschitz if changing one of the inputs by 1 changes the output by at most 1. In other words, Lipschitz functions are not very sensitive to small changes in the input. Our main result is an efficient tester for the Lipschitz property of functions f : [n] d → δZ, where δ ∈ (0, 1] and δZ is the set of integer multiples of δ. The main tool in the analysis of our tester is a smoothing procedure that makes a function Lipschitz by modifying it at a few points. Its analysis is already nontrivial for the 1-dimensional version, which we call Bubble Smooth, in analogy to Bubble Sort. In one step, Bubble Smooth modifies two values that violate the Lipschitz property, i.e., differ by more than 1, by transferring δ units from the larger to the smaller. We define a transfer graph to keep track of the transfers, and use it to show that the 1 distance between f and BubbleSmooth(f ) is at most twice the 1 distance from f to the nearest Lipschitz function. Bubble Smooth has other important properties, which allow us to obtain a dimension reduction, i.e., a reduction from testing functions on multidimensional domains to testing functions on the 1-dimensional domain, that incurs only a small multiplicative overhead in the running time and thus avoids the exponential dependence on the dimension.

Abstract. In the past few years, the focus of research in the area of statistical data privacy has been in designing algorithms for various problems which satisfy some rigorous notions of privacy. However, not much effort has gone into designing techniques to computationally verify if a given algorithm satisfies some predefined notion of privacy. In this work, we address the following question: Can we design algorithms which tests if a given algorithm satisfies some specific rigorous notion of privacy (e.g., differential privacy)?We design algorithms to test privacy guarantees of a given algorithm A when run on a dataset x containing potentially sensitive information about the individuals. More formally, we design a computationally efficient algorithm Tpriv that verifies whether A satisfies differential privacy on typical datasets (DPTD) guarantee in time sublinear in the size of the domain of the datasets. DPTD, a similar notion to generalized differential privacy first proposed by [3], is a distributional relaxation of the popular notion of differential privacy [14].To design algorithm Tpriv, we show a formal connection between the testing of privacy guarantee for an algorithm and the testing of the Lipschitz property of a related function. More specifically, we show that an efficient algorithm for testing of Lipschitz property can be used as a subroutine in Tpriv that tests if an algorithm satisfies differential privacy on typical datasets.Apart from formalizing the connection between the testing of privacy guarantee and testing of the Lipschitz property, we generalize the work of [21] to the setting of property testing under product distribution. More precisely, we design an efficient Lipschitz tester for the case where the domain points are drawn from hypercube according to some fixed but unknown product distribution instead of the uniform distribution.