Quadratic Probing Revisited: Smoothed Analysis and the Fall of Robin Hood

2026-07-14

AI Summary

This work addresses a long-standing gap in theoretical guarantees for the query performance of quadratic probing hash tables under different key ordering strategies. Introducing a smoothed analysis framework, it establishes—for the first time—that anti-Robin Hood ordering achieves an expected query time of Θ(log ε⁻¹) under smoothed quadratic probing at load factor 1 − ε, substantially outperforming Robin Hood’s Θ(ε⁻¹/²). The result generalizes to arbitrary d-probe schemes and further demonstrates that nearly all fixed-offset, quadratic-like probing strategies attain logarithmic expected query time under anti-Robin Hood ordering. These findings resolve a fundamental open problem in the theory of open-addressing hashing by providing strong theoretical backing for the efficacy of anti-Robin Hood ordering in practical probing schemes.

Abstract

Quadratic probing is one of the most widely used open-addressing hash-table schemes in practice, but after more than half a century, even its most basic performance guarantees remain poorly understood. In this paper, we revisit quadratic probing through the lens of a smoothed variant in which each key follows a random probe sequence where its $k$th probe is expected at offset $Θ(k^2)$. This is simultaneously a toy model for better understanding regular quadratic probing and a natural hashing scheme in its own right. We analyse smoothed quadratic probing for both Robin Hood ordering and anti-Robin Hood ordering and reveal a surprising separation: At load factor $1-\varepsilon$, anti-Robin Hood achieves an expected query time of $Θ(\log \varepsilon^{-1})$, which matches the conjectured expected average successful query time for regular quadratic probing, while Robin Hood falls short at $Θ(\varepsilon^{-1/2})$. Our analysis generalises to degree-$d$ probing for any $d \ge 1$ with expected query time $O(\max(\log \varepsilon^{-1}, \varepsilon^{1-2/d}))$ for anti-Robin Hood and $Θ(\varepsilon^{-1/d})$ for Robin Hood. Finally, we go beyond smoothed analysis: using the probabilistic method, we show that for every $d \ge 2$, almost every random fixed-offset degree-$d$ probing sequence achieves expected query time $O(\log \varepsilon^{-1})$ under anti-Robin Hood ordering, simultaneously over all admissible table sizes and load factors. Thus, while quadratic probing itself remains elusive, we prove that essentially all quadratic-probing-like fixed-offset schemes achieve the ideal performance under the anti-Robin Hood ordering.

Problem

• quadratic probing
• hash tables
• Robin Hood hashing
• smoothed analysis
• query time

Innovation

• smoothed analysis
• quadratic probing
• anti-Robin Hood ordering
• hash tables
• probabilistic method