The parity search problem

TL;DR

This paper proves that a logarithmic number of parity-based queries suffices to identify up to d marked items among n, matching the lower bound and solving an open problem in combinatorial search theory.

Contribution

It establishes an asymptotically optimal non-adaptive parity search scheme for identifying up to d marked items among n, answering an open question in the field.

Findings

Logarithmic parity queries suffice for identification

Matching lower bound confirms optimality

Solution to an open problem in search theory

Abstract

We prove that for any positive integers $n$ and $d$ there exists a collection consisting of $f=d\log n+O(1)$ subsets $A_1, A_2, \ldots, A_f$ of $[n]$ such that for any two distinct subsets $X$ and $Y$ of $[n]$ whose size is at most $d$ there is an index $i\in [f]$ for which $| A_i\cap X|$ and $|A_i\cap Y|$ have different parity. Here we think of $d$ as fixed whereas $n$ is thought of as tending to infinity, and the base of the logarithm is $2$. Translated into the language of combinatorial search theory, this tells us that \[ d \log n+O(1) \] queries suffice to identify up to $d$ marked items from a totality of $n$ items if the answers one gets are just whether an even or an odd number of marked elements has been queried, even if the search is performed non-adaptively. Since the entropy method easily yields a matching lower bound for the adaptive version of this problem, our result is asymptotically best possible. This answers a question posed by D\'aniel Gerbner and Bal\'azs Patk\'os in Gyula O.H. Katona's Search Theory Seminar at the R\'enyi institute.