# Linear Hashing is Awesome

## TL;DR

This paper proves that a specific class of linear hash functions, when used with chaining in hashing tables, results in a longest chain length of approximately n^{1/3}, demonstrating the effectiveness of linear hashing.

## Contribution

It provides a rigorous analysis of linear hash functions' performance, establishing a bound on the expected maximum chain length in hashing with chaining.

## Key findings

- Expected longest chain length is n^{1/3} for linear hash functions.
- The analysis extends to multiply-shift hash functions.
- Supports the effectiveness of linear hashing in practical applications.

## Abstract

We consider the hash function $h(x) = ((ax+b) \bmod p) \bmod n$ where $a,b$ are chosen uniformly at random from $\{0,1,\ldots,p-1\}$. We prove that when we use $h(x)$ in hashing with chaining to insert $n$ elements into a table of size $n$ the expected length of the longest chain is $\tilde{O}\!\left(n^{1/3}\right)$. The proof also generalises to give the same bound when we use the multiply-shift hash function by Dietzfelbinger et al. [Journal of Algorithms 1997].