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1. We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far frommore »
2. Consider an algorithm performing a computation on a huge random object. Is it necessary to generate the entire object up front, or is it possible to provide query access to the object and sample it incrementally "on-the-fly"? Such an implementation should emulate the object by answering queries in a manner consistent with a random instance sampled from the true distribution. Our first set of results focus on undirected graphs with independent edge probabilities, under certain assumptions. Then, we use this to obtain the first efficient implementations for the Erdos-Renyi model and the Stochastic Block model. As in previous local-access implementationsmore »
3. A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions overmore »
4. There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform frommore »
5. In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution p over a poset is monotone if, for any pair of domain elements x and y such that x ⪯ y, p(x) ≤ p(y). To understand the sample complexity of this problem, we introduce a new property called bigness over a finite domain, where the distribution is T-big if the minimum probability for any domain element is at least T. We establish a lower bound of Ω(n/ log n) for testing bigness of distributions on domains of size n. Wemore »
6. There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size n, distinguishing the uniform distribution from distributions that are far from uniform in ℓ1-distance uses only O(n−−√) samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is ϵ-close to uniform frommore »
7. Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity. Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson,more »
8. Statistical tests are at the heart of many scientific tasks. To validate their hypothesis, researchers in medical and social sciences use individuals' data. The sensitivity of participants' data requires the design of statistical tests that ensure the privacy of the individuals in the most efficient way. In this paper, we use the framework of property testing to design algorithms to test the properties of the distribution that the data is drawn from with respect to differential privacy. In particular, we investigate testing two fundamental properties of distributions: (1) testing the equivalence of two distributions when we have unequal numbers ofmore »
9. A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a singlemore »
10. We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with ˜O(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the ˜O (plog Δ)-round algorithm of Ghaffari [PODC’17]. • OurO(log logn)-round (1+ε)-approximate maximum matching algorithm simplifies ormore »