A Simple Analysis of Quadratic Probing and Other Open Addressing Schemes

2026-08-08

AI Summary

Classical theory guarantees only that fixed-offset open-addressing hash schemes—such as quadratic probing—achieve constant expected insertion cost under extremely low load factors (≤8.9%). This work introduces a novel combinatorial structure called the “witness forest” to precisely model the dependency relationships among probe sequences arising from hash collisions. By integrating this structural insight with probabilistic and collision analysis, the paper substantially improves the provable upper bound on the load factor for which constant expected insertion cost holds. Specifically, the approach raises the threshold to 35.74% for general fixed-offset schemes and further to 37.61% for quadratic probing as a special case, significantly broadening the practical efficiency range of these widely used hashing strategies.

Abstract

In open addressed hashing, quadratic probing is attractive for striking a nice balance between having a high locality of reference and a low number of probes per search. However, these are empirical observations, not theoretical guarantees. Indeed, until recently, it was not known whether quadratic probing had constant expected insertion cost under any positive load factor $α> 0$, even with uniformly random hash functions. In a recent breakthrough---albeit a numerically understated breakthrough---Kuszmaul and Xi (2024) proved that any fixed offset sequence (including quadratic probing) does, in fact, have constant expected insertion cost for load factors $α\leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $α< 1-ε$ bounded away from 1. In this paper, we prove that open addressed hashing with any fixed offset sequence has constant expected insertion cost for load factors up to $35.74\%$, and that for quadratic probing in particular, we can increase the load factor to $37.61\%$. Our main innovation is a new type of witness forest for recording collisions among the probe sequences.

Problem

• open addressing
• quadratic probing
• hashing
• load factor
• insertion cost

Innovation

• quadratic probing
• open addressing
• witness forest
• hashing
• load factor