# Tight Thresholds for Cuckoo Hashing via XORSAT

## TL;DR

This paper determines the precise thresholds for successful offline cuckoo hashing with multiple choices by linking it to the known thresholds of the random k-XORSAT problem, providing a comprehensive understanding of the problem's phase transition.

## Contribution

It establishes the exact thresholds for offline cuckoo hashing for all k > 2 by connecting them to the thresholds of the random k-XORSAT problem and extends these results to variable choice scenarios.

## Key findings

- Thresholds match those of the random k-XORSAT problem.
- Successfully extended results to keys with differing number of choices.
- Provides evidence and an algorithm for a conjecture on multi-key bucket storage.

## Abstract

We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k >= 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds alpha_k such that, as n goes to infinity, if n/m <= alpha for some alpha < alpha_k then a hash table can be constructed successfully with high probability, and if n/m >= alpha for some alpha > alpha_k a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of k > 2 by showing that they are in fact the same as the previously known thresholds for the random k-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.