Manaswi Paraashar — Page 2

arXiv2020

Disjointness through the Lens of Vapnik-Chervonenkis Dimension: Sparsity and Beyond

The disjointness problem - where Alice and Bob are given two subsets of $\{1, \dots, n\}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be $Θ(n)$, it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik-Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by $d$, we analyze how large can the deterministic and randomized communication complexities be, as a function of $d$ and $n$. In this paper, we construct two natural set systems of VC dimension $d$, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be $\widetildeΘ\left(d\log \left( n/d \right)\right)$ for set systems of VC dimension $d$ and this matches the deterministic upper bound for all set systems of VC dimension $d$. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension $d$ such that both deterministic and randomized (one-way and multi-round) complexity for the set intersection problem can be as high as $Θ\left( d\log \left( n/d \right) \right)$, and this is tight among all set systems of VC dimension $d$.

cs.CC↗

arXiv2019

Quantum Query-to-Communication Simulation Needs a Logarithmic Overhead

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function $f : \{-1, 1\}^n \to \{-1, 1\}$ and $\bullet : \{-1, 1\}^2 \to \{-1, 1\}$ the two-party bounded-error quantum communication complexity of $(f \circ \bullet)$ is $O(Q(f) \log n)$, where $Q(f)$ is the bounded-error quantum query complexity of $f$. Note that the bounded-error randomized communication complexity of $(f \circ \bullet)$ is bounded by $O(R(f))$, where $R(f)$ denotes the bounded-error randomized query complexity of $f$. Thus, the BCW simulation has an extra $O(\log n)$ factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Høyer and de Wolf (STACS'02) showed that for the Set-Disjointness function, this can be reduced to $c^{\log^* n}$ for some constant $c$, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is $\mathsf{NOR}_n \circ \wedge$) is $O(Q(\mathsf{NOR}_n))$. Perhaps somewhat surprisingly, we show that when $ \bullet = \oplus$, then the extra $\log n$ factor in the BCW simulation is unavoidable. In other words, we exhibit a total function $F : \{-1, 1\}^n \to \{-1, 1\}$ such that $Q^{cc}(F \circ \oplus) = Θ(Q(F) \log n)$. To the best of our knowledge, it was not even known prior to this work whether there existed a total function $F$ and 2-bit function $\bullet$, such that $Q^{cc}(F \circ \bullet) = ω(Q(F))$.

quant-ph↗