A Simple Analysis of Quadratic Probing and Other Open Addressing Schemes

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 $\alpha > 0$, 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 $\alpha \leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $\alpha < 1-\epsilon$ 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.