Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting

We investigate one of the most basic problems in streaming algorithms: approximating the number of elements in the stream. In 1978, Morris famously gave a randomized algorithm achieving a constant-factor approximation error for streams of length at most N in space $O(\log \log N)$. We investigate the pseudo-deterministic complexity of the problem and prove a tight $\mathrm{\Omega }(\log N)$ lower bound, thus resolving a problem of Goldwasser-Grossman-Mohanty-Woodruff.