On Two-Stage Guessing

TL;DR

This paper analyzes the exponential growth rate of guessing moments in a two-stage guessing process for correlated sources, proposing bounds and a Huffman-based method to optimize guessing efficiency.

Contribution

It characterizes the minimal exponential growth rate of guessing moments in two-stage guessing and introduces a Huffman code-based approach for optimizing the function linking the stages.

Findings

Lower bounds on exponential growth rates depending on the function f

A Huffman code-based construction for f that minimizes growth rate

Analysis of guessing when the second stage may be incorrect or without source assumptions

Abstract

Stationary memoryless sources produce two correlated random sequences $X^n$ and $Y^n$. A guesser seeks to recover $X^n$ in two stages, by first guessing $Y^n$ and then $X^n$. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in $n$) of any positive $\rho$-th moment of the total number of guesses when $Y^n$ is obtained by applying a deterministic function $f$ component-wise to $X^n$. We prove that, depending on $f$, the least exponential growth rate in the two-stage setup is lower than when guessing $X^n$ directly. We further propose a simple Huffman code-based construction of a function $f$ that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the $\rho$-th moment of the total number of guesses required to recover $X^n$ when Stage 1 need not end with a correct guess of $Y^n$ and without assumptions on the stationary memoryless sources producing $X^n$ and $Y^n$.