On perfect hashing of numbers with sparse digit representation via multiplication by a constant

TL;DR

This paper demonstrates the existence of optimal multipliers for perfect hashing of sparse digit vectors and convolution-based integer sets, connecting hashing efficiency with algebraic properties of Schur functions.

Contribution

It introduces a method to find 'magic' multipliers for perfect hashing of sparse digit representations and explores algebraic properties related to Schur functions in positive characteristic.

Findings

Existence of optimal multipliers for perfect hashing.

Optimal multipliers transfer sparse coefficients into consecutive digits.

Non-degeneracy results for Schur functions in positive characteristic.

Abstract

Consider the set of vectors over a field having non-zero coefficients only in a fixed sparse set and multiplication defined by convolution, or the set of integers having non-zero digits (in some base $b$) in a fixed sparse set. We show the existence of an optimal (resp. almost-optimal in the latter case) `magic' multiplier constant that provides a perfect hash function which transfers the information from the given sparse coefficients into consecutive digits. Studying the convolution case we also obtain a result of non-degeneracy for Schur functions as polynomials in the elementary symmetric functions in positive characteristic.