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