Tight Streaming Lower Bounds for Deterministic Approximate Counting

TL;DR

The paper proves tight deterministic streaming lower bounds for the k-counter approximate counting problem, showing that any worst-case algorithm using read-once branching programs requires at least $Ω(k\log(n/k))$ space when outputting additive-approximate counts with $Δ=\frac{n}{3(k-1)}$. The core technique blends rectangle-labeling with a novel potential function to force large ROBP width, and is extended from the {0,1}-case to the general $k$-counter and to the $k$-parallel setting, yielding strong lower bounds ($Ω((n/k)^{k-1})$ and $Ω(n^{Ω(k)})$ respectively) and tightness in several regimes. The authors also present non-trivial deterministic algorithms that nearly match the lower bounds in certain parameter regions, and derive broad implications for other streaming problems, including heavy hitters and quantile sketches, thereby establishing the optimality of several classical approaches in the deterministic setting. Overall, the work clarifies the exact limitations of deterministic streaming for approximate counting and related tasks, highlighting a sharp separation from randomized models.

Abstract

We study the streaming complexity of $k$-counter approximate counting. In the $k$-counter approximate counting problem, we are given an input string in $[k]^n$, and we are required to approximate the number of each $j$'s ($j\in[k]$) in the string. Typically we require an additive error $\leq\frac{n}{3(k-1)}$ for each $j\in[k]$ respectively, and we are mostly interested in the regime $n\gg k$. We prove a lower bound result that the deterministic and worst-case $k$-counter approximate counting problem requires $Ω(k\log(n/k))$ bits of space in the streaming model, while no non-trivial lower bounds were known before. In contrast, trivially counting the number of each $j\in[k]$ uses $O(k\log n)$ bits of space. Our main proof technique is analyzing a novel potential function. Our lower bound for $k$-counter approximate counting also implies the optimality of some other streaming algorithms. For example, we show that the celebrated Misra-Gries algorithm for heavy hitters [MG82] has achieved optimal space usage.

Figures (2)

• Figure 1: An example for the dynamic programming process.
• Figure 2: An example for the definition of the potential function. In this example we take $k=3$ and $t=8$. Let $\mathcal{R}_t=\{R_1,R_2,R_3,R_4,R_5\}$ and the values of $R_1,\cdots,R_5$ are shown in the figure. We calculate $\phi_t(2,0)$ and $\phi_t(2,3)$ as an example: for $\phi_t(2,0)$, the max operator in \ref{['eq.k-counter-phi']} is taken over $R\in\{R_1,R_3\}$, and the maximum is $4$, which is achieved when $R=R_3$; for $\phi_t(2,3)$, the max operator in \ref{['eq.k-counter-phi']} is taken over $R\in\{R_1,R_4\}$, and the maximum is $3$, which is achieved when $R=R_4$.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 3.1
• Definition 3.2
• Definition 3.3