On sample-based testers

The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, sample-based testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were defined by Goldreich, Goldwasser, and Ron (JACM 1998), most research in property testing is focused on query-based testers. In this work, we advance the study of sample-based property testers by providing several general positive results as well as by revealing relations between variants of this testing model. In particular: • We show that certain types of query-based testers yield sample-based testers of sublinear sample complexity. For example, this holds for a natural class of proximity oblivious testers. • We study the relation between distribution-free sample-based testers and one-sided error sample-based testers w.r.t the uniform distribution. While most of this work ignores the time complexity of testing, one part of it does focus on this aspect. The main result in this part is a sublinear-time sample-based tester for k-Colorability, for any k ≥ 2

Sublinear-Time Computation in the Presence of Online Erasures

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase $t$ input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant $t$ with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of $t$, showing that the query complexity is $\Theta(\log t)$. In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for $t=1$. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

On the Download Rate of Homomorphic Secret Sharing

A homomorphic secret sharing (HSS) scheme is a secret sharing scheme that supports evaluating functions on shared secrets by means of a local mapping from input shares to output shares. We initiate the study of the download rate of HSS, namely, the achievable ratio between the length of the output shares and the output length when amortized over $\ell$ function evaluations. We obtain the following results. * In the case of linear information-theoretic HSS schemes for degree-$d$ multivariate polynomials, we characterize the optimal download rate in terms of the optimal minimal distance of a linear code with related parameters. We further show that for sufficiently large $\ell$ (polynomial in all problem parameters), the optimal rate can be realized using Shamir's scheme, even with secrets over $\mathbb{F}_2$. * We present a general rate-amplification technique for HSS that improves the download rate at the cost of requiring more shares. As a corollary, we get high-rate variants of computationally secure HSS schemes and efficient private information retrieval protocols from the literature. * We show that, in some cases, one can beat the best download rate of linear HSS by allowing nonlinear output reconstruction and $2^{-\Omega(\ell)}$ error probability

Space-Efficient Algorithms and Verification Schemes for Graph Streams

Structured data-sets are often easy to represent using graphs. The prevalence of massive data-sets in the modern world gives rise to big graphs such as web graphs, social networks, biological networks, and citation graphs. Most of these graphs keep growing continuously and pose two major challenges in their processing: (a) it is infeasible to store them entirely in the memory of a regular server, and (b) even if stored entirely, it is incredibly inefficient to reread the whole graph every time a new query appears. Thus, a natural approach for efficiently processing and analyzing such graphs is reading them as a stream of edge insertions and deletions and maintaining a summary that can be (a) stored in affordable memory (significantly smaller than the input size) and (b) used to detect properties of the original graph. In this thesis, we explore the strengths and limitations of such graph streaming algorithms under three main paradigms: classical or standard streaming, adversarially robust streaming, and streaming verification. In the classical streaming model, an algorithm needs to process an adversarially chosen input stream using space sublinear in the input size and return a desired output at the end of the stream. Here, we study a collection of fundamental directed graph problems like reachability, acyclicity testing, and topological sorting. Our investigation reveals that while most problems are provably hard for general digraphs, they admit efficient algorithms for the special and widely-studied subclass of tournament graphs. Further, we exhibit certain problems that become drastically easier when the stream elements arrive in random order rather than adversarial order, as well as problems that do not get much easier even under this relaxation. Furthermore, we study the graph coloring problem in this model and design color-efficient algorithms using novel parameterizations and establish complexity separations between different versions of the problem. The classical streaming setting assumes that the entire input stream is fixed by an adversary before the algorithm reads it. Many randomized algorithms in this setting, however, fail when the stream is extended by an adaptive adversary based on past outputs received. This is the so-called adversarially robust streaming model. We show that graph coloring is significantly harder in the robust setting than in the classical setting, thus establishing the first such separation for a ``natural\u27\u27 problem. We also design a class of efficient robust coloring algorithms using novel techniques. In classical streaming, many important problems turn out to be ``intractable\u27\u27, i.e., provably impossible to solve in sublinear space. It is then natural to consider an enhanced streaming setting where a space-bounded client outsources the computation to a space-unbounded but untrusted cloud service, who replies with the solution and a supporting ``proof\u27\u27 that the client needs to verify. This is called streaming verification or the annotated streaming model. It allows algorithms or verification schemes for the otherwise intractable problems using both space and proof length sublinear in the input size. We devise efficient schemes that improve upon the state of the art for a variety of fundamental graph problems including triangle counting, maximum matching, topological sorting, maximal independent set, graph connectivity, and shortest paths, as well as for computing frequency-based functions such as distinct items and maximum frequency, which have broad applications in graph streaming. Some of our schemes were conjectured to be impossible, while some others attain smooth and optimal tradeoffs between space and communication costs

Learning sparse polynomial functions

Abstract We study the question of learning a sparse multivariate polynomial over the real domain. In particular, for some unknown polynomial f ( x) of degree-d and k monomials, we show how to reconstruct f , within erro