Sub-linear Upper Bounds on Fourier dimension of Boolean Functions in terms of Fourier sparsity

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of Tsang~\etal~\cite{tsang}, that for every Boolean function of sparsity $s$ there is an affine subspace of $\mathbb{F}_2^n$ of co-dimension O(\poly\log s) restricted to which the function is constant. This conjectured bound is tight upto poly-logarithmic factors as the Fourier dimension and sparsity of the address function are quadratically separated. We obtain these bounds by observing that the Fourier dimension of a Boolean function is equivalent to its non-adaptive parity decision tree complexity, and then bounding the latter

An adaptivity hierarchy theorem for property testing

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i'th round may depend on the answers obtained in the previous i-1 rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n in N and 0 <= k <= n^{0.99} there exists a property Pi_{n,k} of functions for which (1) there exists a k-adaptive tester for Pi_{n,k} with query complexity tilde O(k), yet (2) any (k-1)-adaptive tester for Pi_{n,k} must make Omega(n) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs

Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication

This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party communication model that enables "accelerated", error-preserving simulations of dynamic data structures. We use this technique to prove an $\Omega(n(\log n/\log\log n)^2)$ cell-probe lower bound for the dynamic 2D weighted orthogonal range counting problem (2D-ORC) with $n/\mathrm{poly}\log n$ updates and $n$ queries, that holds even for data structures with $\exp(-\tilde{\Omega}(n))$ success probability. This result not only proves the highest amortized lower bound to date, but is also tight in the strongest possible sense, as a matching upper bound can be obtained by a deterministic data structure with worst-case operational time. This is the first demonstration of a "sharp threshold" phenomenon for dynamic data structures. Our broader motivation is that cell-probe lower bounds for exponentially small success facilitate reductions from dynamic to static data structures. As a proof-of-concept, we show that a slightly strengthened version of our lower bound would imply an $\Omega((\log n /\log\log n)^2)$ lower bound for the static 3D-ORC problem with $O(n\log^{O(1)}n)$ space. Such result would give a near quadratic improvement over the highest known static cell-probe lower bound, and break the long standing $\Omega(\log n)$ barrier for static data structures

Adaptivity Helps for Testing Juntas

We give a new lower bound on the query complexity of any non-adaptive algorithm for testing whether an unknown Boolean function is a k-junta versus epsilon-far from every k-junta. Our lower bound is that any non-adaptive algorithm must make Omega(( k * log*(k)) / ( epsilon^c * log(log(k)/epsilon^c))) queries for this testing problem, where c is any absolute constant <1. For suitable values of epsilon this is asymptotically larger than the O(k * log(k) + k/epsilon) query complexity of the best known adaptive algorithm [Blais,STOC\u2709] for testing juntas, and thus the new lower bound shows that adaptive algorithms are more powerful than non-adaptive algorithms for the junta testing problem

Settling the Query Complexity of Non-Adaptive Junta Testing

We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f is a k-junta or epsilon-far from every k-junta must make ~Omega(k^{3/2}/ epsilon) many queries for a wide range of parameters k and epsilon. Our result dramatically improves previous lower bounds from [BGSMdW13,STW15], and is essentially optimal given Blais\u27s non-adaptive junta tester from [Blais08], which makes ~O(k^{3/2})/epsilon queries. Combined with the adaptive tester of [Blais09] which makes O(k log k + k / epsilon) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing