On extracting common random bits from correlated sources on large alphabets

TL;DR

This paper investigates the limits of extracting common random bits from correlated sources over large alphabets, revealing that as alphabet size increases, the agreement probability rate diminishes and Hamming ball methods become less effective.

Contribution

It extends the analysis of common randomness extraction from binary to large alphabets, establishing upper bounds and showing the inefficacy of Hamming ball strategies for large s.

Findings

Agreement probability rate decreases as alphabet size increases.

Hamming ball based constructions are less effective than trivial algorithms for large s.

Upper bounds show no strategy can surpass a certain agreement probability as s grows.

Abstract

Suppose Alice and Bob receive strings $X=(X_1,...,X_n)$ and $Y=(Y_1,...,Y_n)$ each uniformly random in $[s]^n$ but so that $X$ and $Y$ are correlated . For each symbol $i$, we have that $Y_i = X_i$ with probability $1-\eps$ and otherwise $Y_i$ is chosen independently and uniformly from $[s]$. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from $[s]^k$ but without communicating. How well can they do? The trivial strategy of outputting the first $k$ symbols yields an agreement probability of $(1 - \eps + \eps/s)^k$. In a recent work by Bogdanov and Mossel it was shown that in the binary case where $s=2$ and $k = k(\eps)$ is large enough then it is possible to extract $k$ bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability $(k\eps)^{-1/2} \cdot 2^{-k\eps/(2(1 - \eps/2))}$ using a random construction based on Hamming balls, and this is optimal up to lower order terms. In the current paper we consider the same problem over larger alphabet sizes $s$ and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular we show no strategy can achieve agreement probability better than $(1-\eps)^k (1+\delta(s))^k$ where $\delta(s) \to 0$ as $s \to \infty$. We also show that Hamming ball based constructions have {\em much lower} agreement probability rate than the trivial algorithm as $s \to \infty$. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.