arXiv subjects

Swagato Sanyal

Publications and source records attributed to Swagato Sanyal.

23 records · Page 2Linked to original sources

arXiv2017

A Composition Theorem for Randomized Query Complexity

Let the randomized query complexity of a relation for error probability $ε$ be denoted by $R_ε(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $R_{1/3}(f\circ g^n) = Ω(R_{4/9}(f)\cdot R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=Ω(\log n \cdot R_{4/9}(f) \cdot R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.

cs.CC↗

arXiv2016

The zero-error randomized query complexity of the pointer function

The pointer function of G{ö}{ö}s, Pitassi and Watson \cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexities, exact and approximate polynomial degrees, etc. In particular, the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity, have been obtained by considering {\em variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function. However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the precise zero-error complexity of the original pointer function was not known. We show a lower bound of $\widetildeΩ(n^{3/4})$ on the zero-error randomized query complexity of the pointer function on $Θ(n \log n)$ bits; since an $\widetilde{O}(n^{3/4})$ upper bound is already known \cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a factor of $\polylog\, n$.

cs.CC↗

arXiv2015

Towards Better Separation between Deterministic and Randomized Query Complexity

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by Saks and Wigderson that for any Boolean function $f$, $R_0(f)=Ω({D(f)})^{0.753..}$. This also shows widest separation between $R_1(f)$ and $D(f)$ for any Boolean function. The function $F$ was defined by G{ö}{ö}s, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function $F$ certify optimal (quadratic) separation between $D(f)$ and $R_0(f)$, and polynomial separation between $R_0(f)$ and $R_1(f)$. Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considerd in the work of Ambainis et al are different variants of $F$, we work with the original function $F$ itself.

cs.CC↗

arXiv2014

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.

cs.CC↗

arXiv2009

On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids

We theoretically analyze the Cops and Robber Game for the first time in a multidimensional grid. It is shown that for an $n$-dimensional grid, at least $n$ cops are necessary to ensure capture of the robber. We also present a set of cop strategies for which $n$ cops are provably sufficient to catch the robber. Further, for two-dimensional grid, we provide an efficient cop strategy for which the robber is caught even by a single cop under certain conditions.

cs.DM↗