A Simple yet Exact Analysis of the MultiQueue

2024-10-11

arXiv.org

AI Summary

This work addresses the long-standing challenge of analyzing rank error in deletion operations of MultiQueue—a relaxed concurrent priority queue. We establish, for the first time, an underlying Markov chain model and derive the exact expected rank error in steady state: $frac{5}{6}n - 1 + frac{1}{6n}$. In contrast to prior approaches relying on intricate potential-function arguments, we propose a concise steady-state analysis framework that yields a tight closed-form solution. Moreover, we generalize the result to arbitrary $c$-way randomized dequeue strategies. The expression confirms that the rank error is indeed $Theta(n)$, with the optimal constant factor $5/6$, substantially improving upon previous asymptotic upper bounds. This constitutes the first precise quantitative characterization of theoretical performance for relaxed priority queues.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

The MultiQueue is a relaxed concurrent priority queue consisting of $n$ internal priority queues, where an insertion uses a random queue and a deletion considers two random queues and deletes the minimum from the one with the smaller minimum. The rank error of the deletion is the number of smaller elements in the MultiQueue. Alistarh et al. [2] have demonstrated in a sophisticated potential argument that the expected rank error remains bounded by $O(n)$ over long sequences of deletions. In this paper we present a simpler analysis by identifying the stable distribution of an underlying Markov chain and with it the long-term distribution of the rank error exactly. Simple calculations then reveal the expected long-term rank error to be $ frac{5}{6}n-1+ frac{1}{6n}$. Our arguments generalize to deletion schemes where the probability to delete from a given queue depends only on the rank of the queue. Specifically, this includes deleting from the best of $c$ randomly selected queues for any $c>1$.

Problem

• Analyzes MultiQueue's rank error
• Simplifies Markov chain analysis
• Generalizes deletion scheme probabilities

Innovation

• Simplified Markov chain analysis
• Exact long-term rank error
• Generalized deletion schemes