On the Advantage of Adaptivity for Sampling with Cell Probes

TL;DR

This paper demonstrates a significant separation between adaptive and non-adaptive cell-probe sampling methods, showing that adaptivity can drastically reduce the number of probes needed to sample from a distribution.

Contribution

It constructs an explicit distribution that exhibits an optimal separation, improving previous bounds on the advantage of adaptivity in cell-probe sampling.

Findings

Adaptive sampling requires only two probes per bit.

Non-adaptive sampling needs nearly linear probes in N.

The separation surpasses previous known bounds.

Abstract

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from $[N]$. In contrast, any non-adaptive sampler requires $\widetilde{\Omega}(N)$ non-adaptive cell probes to generate a distribution with total variation distance less than $1-o(1)$ from $\mathbf{D}$. This provides a $2$-vs-$\widetilde{\Omega}(N)$ separation for sampling with adaptive versus non-adaptive cell probes, improving upon the $2$-vs-$\widetilde{\Omega}(\log N)$ separation of Yu and Zhan (ITCS '24) and the $(\log N)^{O(1)}$-vs-$N^{\Omega(1)}$ separation of Alekseev, G\"o\"os, Myasnikov, Riazanov, and Sokolov (STOC '26).