DeepAI

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

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 ๐ณ = (z_1,โฆ,z_n) โ{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 โ_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 ฮฉ(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 ฮฉ(โ(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.

• 8 publications
• 5 publications
โ 07/12/2020

Graph Connectivity and Single Element Recovery via Linear and OR Queries

We study the problem of finding a spanning forest in an undirected, n-ve...
โ 06/08/2020

An Optimal Tester for k-Linear

A Boolean function f:{0,1}^nโ{0,1} is k-linear if it returns the sum (ov...
โ 12/31/2020

The Query Complexity of Local Search and Brouwer in Rounds

We study the query complexity of local search and Brouwer fixed-point co...
โ 12/11/2021

A tight bound for the clique query problem in two rounds

We consider a problem introduced by Feige, Gamarnik, Neeman, Rรกcz and Te...
โ 06/10/2019

The Demand Query Model for Bipartite Matching

We introduce a `concrete complexity' model for studying algorithms for m...
โ 03/21/2023