From Independence to Expansion and Back Again

TL;DR

This paper presents near-optimal, efficient recursive constructions of k-independent hash functions and small, fast-expanding graph representations by exploiting the deep connection between these problems.

Contribution

It introduces a recursive method that achieves near-optimal space-time tradeoffs for k-independent hash functions and expander graph representations, improving upon previous quasipolynomial time solutions.

Findings

Achieves logarithmic factor from Siegel's lower bound in construction time.

Provides small, efficiently computable expander graph representations.

Establishes a strong connection between hash functions and expander graphs.

Abstract

We consider the following fundamental problems: (1) Constructing $k$-independent hash functions with a space-time tradeoff close to Siegel's lower bound. (2) Constructing representations of unbalanced expander graphs having small size and allowing fast computation of the neighbor function. It is not hard to show that these problems are intimately connected in the sense that a good solution to one of them leads to a good solution to the other one. In this paper we exploit this connection to present efficient, recursive constructions of $k$-independent hash functions (and hence expanders with a small representation). While the previously most efficient construction (Thorup, FOCS 2013) needed time quasipolynomial in Siegel's lower bound, our time bound is just a logarithmic factor from the lower bound.