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

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 ELEMENT-EXTRACTION in a query model that combines both: linear measurements with bounded adaptivity.In the ELEMENT-EXTRACTION problem, one is given a nonzero vector z = (z 1 , . . . , z n ) ∈ {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 ℓ 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 ELEMENT-EXTRACTION algorithm must spend Ω(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 Ω( √ 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 zero-sum problem as well as recent improvements to the sunflower lemma.

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

“…For these reasons, Assadi et al [ACK21] and Chakrabarti and Stoeckl [CS21] have recently sought to reduce query adaptivity under a variety of global query models, including Linear, OR, Cut and BIS queries. These works study the single element recovery problem, which is a weaker variant of uniform edge sampling, requiring that the algorithm return a single edge in G. Assadi et al also study the problem of checking connectivity, presenting a BIS query algorithm making O(n) queries and using three rounds of adaptivity.…”