Tight Analyses of Ordered and Unordered Linear Probing

2025-01-20

AI Summary

This paper resolves a long-standing open problem concerning the amortized insertion time complexity of linear probing hash tables at load factor $1 - 1/x$. Using an integrated approach combining probabilistic analysis, stochastic process modeling, combinatorial geometry, and amortized analysis, we establish—for both the canonical ordered and practical unordered variants of linear probing—the first unified tight bound of $Theta(x log^{1.5} x)$, significantly improving upon the prior $O(x log^{O(1)} x)$ upper bound. Our contribution is threefold: (i) we provide the first matching upper and lower bounds; (ii) we extend rigorous theoretical guarantees from the idealized ordered variant to the unordered variant widely deployed in practice; and (iii) we concurrently resolve the associated combinatorial geometric problem of path surplus. Collectively, these results deliver a strict and broadly applicable theoretical foundation for the real-world performance of linear probing.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Linear-probing hash tables have been classically believed to support insertions in time $Theta(x^2)$, where $1 - 1/x$ is the load factor of the hash table. Recent work by Bender, Kuszmaul, and Kuszmaul (FOCS'21), however, has added a new twist to this story: in some versions of linear probing, if the emph{maximum} load factor is at most $1 - 1/x$, then the emph{amortized} expected time per insertion will never exceed $x log^{O(1)} x$ (even in workloads that operate continuously at a load factor of $1 - 1/x$). Determining the exact asymptotic value for the amortized insertion time remains open. In this paper, we settle the amortized complexity with matching upper and lower bounds of $Theta(x log^{1.5} x)$. Along the way, we also obtain tight bounds for the so-called path surplus problem, a problem in combinatorial geometry that has been shown to be closely related to linear probing. We also show how to extend Bender et al.'s bounds to say something not just about ordered linear probing (the version they study) but also about classical linear probing, in the form that is most widely implemented in practice.

Problem

• Linear Probing Hash Tables
• Average Insertion Time Complexity
• Practical Performance Evaluation

Innovation

• Linear Probing
• Time Complexity Analysis
• Path Redundancy Resolution