Optimal Pure Differentially Private Sparse Histograms in Deterministic Linear Time

We present an algorithm that releases a pure differentially private (under the replacement neighboring relation) sparse histogram for $n$ participants over a domain of size $d \gg n$. Our method achieves the optimal $\ell_\infty$-estimation error and runs in strictly $O(n)$ time in the Word-RAM model, improving upon the previous best deterministic-time bound of $\tilde{O}(n^2)$ and resolving the open problem of breaking this quadratic barrier (Balcer and Vadhan, 2019). Moreover, the algorithm admits an efficient circuit implementation, enabling the first near-linear communication and computation cost pure DP histogram MPC protocol with optimal $\ell_\infty$-estimation error. Central to our algorithm is a novel **private item blanket** technique with target-length padding, which hides differences in the supports of neighboring histograms while remaining efficiently implementable.