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Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting

Ofer Grossman, Meghal Gupta, Mark Sellke

TL;DR

The pseudo-deterministic complexity of the problem is investigated and a tight $\Omega(\log N)$ lower bound is proved, thus resolving a problem of [GGMW20].

Abstract

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 $Ω(\log N)$ lower bound, thus resolving a problem of Goldwasser-Grossman-Mohanty-Woodruff.

Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting

TL;DR

The pseudo-deterministic complexity of the problem is investigated and a tight lower bound is proved, thus resolving a problem of [GGMW20].

Abstract

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 . We investigate the pseudo-deterministic complexity of the problem and prove a tight lower bound, thus resolving a problem of Goldwasser-Grossman-Mohanty-Woodruff.
Paper Structure (20 sections, 16 theorems, 55 equations)

This paper contains 20 sections, 16 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Motivation for Pseudo-Determinism.
  3. Related Work
  4. Prior Work on Approximate Counting in a Stream.
  5. Prior Work on Pseudo-Determinism.
  6. Main Result
  7. Markov Chain Formulation
  8. Technical Overview
  9. Illustrative Examples
  10. Example 1: Threshold Morris Counter.
  11. Example 2: Prime Cycles.
  12. Proof Outline
  13. Recurrent Behavior on Moderate Time-Scales.
  14. Decomposition into Periodic Parts.
  15. Analysis of Periodic Decomposition.
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $N<M$, a pseudo-deterministic streaming algorithm to distinguish between streams of length at most $N$ and at least $M$ must use $\Omega(\log N)$ space.

Theorems & Definitions (35)

  • Theorem 1.1: Informal
  • Definition 1
  • Definition 2: Pseudo-Deterministic Approximate Threshold Problem
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 3: Markov Chain Solution to Pseudo-Deterministic Approximate Thresholding
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • ...and 25 more