The Element Extraction Problem and the Cost of Determinism and Limited Adaptivity in Linear Queries

TL;DR

This paper investigates the fundamental limits of element extraction using linear measurements with bounded adaptivity, establishing tight lower bounds on the number of queries needed in various settings.

Contribution

It introduces tight lower bounds for element extraction with limited adaptivity, connecting the problem to classic combinatorial problems and advancing understanding of query complexity.

Findings

Deterministic k-round algorithms require (n^{1/k}) queries, matching upper bounds.

Two-round integer arithmetic algorithms have a (\u221a{n}) lower bound, tight up to polylogarithmic factors.

The proofs leverage zero-sum problems and sunflower lemma results.

Abstract

Two widely-used computational paradigms for sublinear algorithms are using linear measurements to perform computations on a high dimensional input and using structured queries to access a massive input. Typically, algorithms in the former paradigm are non-adaptive whereas those in the latter are highly adaptive. This work studies the fundamental search problem of \textsc{element-extraction} in a query model that combines both: linear measurements with bounded adaptivity. In the \textsc{element-extraction} problem, one is given a nonzero vector $\mathbf{z} = (z_1,\ldots,z_n) \in \{0,1\}^n$ and must report an index $i$ where $z_i = 1$. The input can be accessed using arbitrary linear functions of it with coefficients in some ring. This problem admits an efficient nonadaptive randomized solution (through the well known technique of $\ell_0$-sampling) and an efficient fully adaptive deterministic solution (through binary search). We prove that when confined to only $k$ rounds of adaptivity, a deterministic \textsc{element-extraction} algorithm must spend $\Omega(k (n^{1/k} -1))$ queries, when working in the ring of integers modulo some fixed $q$. This matches the corresponding upper bound. For queries using integer arithmetic, we prove a $2$-round $\widetilde{\Omega}(\sqrt{n})$ lower bound, also tight up to polylogarithmic factors. Our proofs reduce to classic problems in combinatorics, and take advantage of established results on the {\em zero-sum problem} as well as recent improvements to the {\em sunflower lemma}.