On the Advantage of Adaptivity for Sampling with Cell Probes

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 $\tilde{\mathrm{\Omega }}(N)$ 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-$\tilde{\mathrm{\Omega }}(N)$ separation for sampling with adaptive versus non-adaptive cell probes, improving upon the $2$-vs-$\tilde{\mathrm{\Omega }}(\log N)$ separation of Yu and Zhan (ITCS '24) and the $(\log N{)}^{O(1)}$-vs-$N^{\mathrm{\Omega }(1)}$ separation of Alekseev, Göös, Myasnikov, Riazanov, and Sokolov (STOC '26).