Search using queries on indistinguishable items

TL;DR

This paper analyzes the query complexity for identifying a set of indistinguishable integers through noisy comparisons, establishing tight bounds using information theory and noisy binary search techniques.

Contribution

It provides the first tight bounds for the number of queries needed in this indistinguishable items search problem, extending understanding of search under uncertainty.

Findings

Query complexity is Θ(k^3 log n) for k ≤ n.

Query complexity is Θ(n^2 k log n) for k > n.

Information theory and noisy binary search are key tools used.

Abstract

We investigate the problem of determining a set S of k indistinguishable integers in the range [1,n]. The algorithm is allowed to query an integer $q\in [1,n]$, and receive a response comparing this integer to an integer randomly chosen from S. The algorithm has no control over which element of S the query q is compared to. We show tight bounds for this problem. In particular, we show that in the natural regime where $k\le n$, the optimal number of queries to attain $n^{-\Omega(1)}$ error probability is $\Theta(k^3 \log n)$. In the regime where $k>n$, the optimal number of queries is $\Theta(n^2 k \log n)$. Our main technical tools include the use of information theory to derive the lower bounds, and the application of noisy binary search in the spirit of Feige, Raghavan, Peleg, and Upfal (1994). In particular, our lower bound technique is likely to be applicable in other situations that involve search under uncertainty.