POLYNOMIAL DATA STRUCTURE LOWER BOUNDS IN THE GROUP MODEL

Alexander Golovnev; Gleb Posobin; Oded Regev; Omri Weinstein

doi:10.1137/20M1381988

POLYNOMIAL DATA STRUCTURE LOWER BOUNDS IN THE GROUP MODEL

Alexander Golovnev, Gleb Posobin, Oded Regev, Omri Weinstein

Research output: Contribution to journal › Article › peer-review

Abstract

Proving superlogarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early '80s. We prove a polynomial (n^Ω(1)) lower bound for an explicit range counting problem of n³ convex polygons in R² (each with n^Õ⁽¹⁾ facets/semialgebraic complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing-in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime (k = n^1-δ). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.

Original language	English
Pages (from-to)	FOCS20-74-FOCS20-101
Journal	SIAM Journal on Computing
Volume	53
Issue number	6
DOIs	https://doi.org/10.1137/20M1381988
State	Published - 2024
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

Keywords

compressed sensing
computational geometry
data structures
pseudorandomness

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10.1137/20M1381988

Cite this

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abstract = "Proving superlogarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early '80s. We prove a polynomial (nΩ(1)) lower bound for an explicit range counting problem of n3 convex polygons in R2 (each with n{\~O}(1) facets/semialgebraic complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing-in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime (k = n1-δ). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.",

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POLYNOMIAL DATA STRUCTURE LOWER BOUNDS IN THE GROUP MODEL

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