A unified approach to linear probing hashing with buckets

TL;DR

This paper presents a comprehensive analysis of linear probing hashing with buckets, combining combinatorial and probabilistic methods to derive exact and asymptotic results, enhancing understanding of its relation to random walks.

Contribution

It introduces a unified framework using analytic combinatorics and q-calculus to analyze linear probing hashing with buckets, providing new exact formulas and insights.

Findings

Exact formulas for generating functions of linear probing with buckets

Asymptotic behavior characterized through probabilistic analysis

Insight into the relation between linear probing and random walks

Abstract

We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on q-calculus) to directly derive the generating functions to analyze.