A Simple Search Problem

TL;DR

This paper analyzes a basic search problem involving N boxes with a known probability distribution for the prize's location, establishing bounds on the minimal expected work using entropy and Hölder norms, and exploring problem composition.

Contribution

It introduces bounds on minimal search work based on probability density properties and defines the concept of Cartesian product of problems for asymptotic analysis.

Findings

Bounds on minimal work in terms of entropy and Hölder norm

Introduction of Cartesian product of problems

Asymptotic behavior of minimal work for problem powers

Abstract

A simple problem is studied in which there are N boxes and a prize known to be in one of the boxes. Furthermore, the probability that the prize is in any box is given. It is desired to find the prize with minimal expected work, where it takes one unit of work to open a box and look inside. The paper establishes bounds on the minimal work in terms of the $p=1/2$ H\"older norm of the probability density and in terms of the entropy of the probability density. We also introduce the notion of "Cartesian product" of problems, and determine the asymptotic behavior of the minimal work for the $n$th power of a problem. (This article is a newly typeset version of an internal publication written in 1984. The second author passed away on November 12, 2020, and his estate has approved the submission of this paper.)