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. Famously, [Mor78] 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 $\Omega(\log N)$ lower bound, thus resolving a problem of [GGMW20].

I. Introduction

The study of streaming algorithms originated with the seminal paper of Morris [Mor78], which gave a low-memory randomized algorithm to approximately count a number of elements which arrive online in a stream. Roughly speaking, the idea is to have a counter which approximates , where is the number of elements seen in the stream so far. Each time the algorithm encounters an element from the stream, it increases the counter with probability about . As later proved by [Fla85], Morris’s algorithm achieves a constant-factor approximation error for streams of length at most N in space .