Optimal $k$-Secretary with Logarithmic Memory

2025-02-14

AI Summary

This paper studies the $k$-secretary problem under strict memory constraints, aiming to achieve optimal competitive ratio with minimal space. To address the excessive $O(k)$ memory requirement of classical approaches, we establish, for the first time, a generic reduction between the $k$-secretary problem and random-order quantile estimation. Leveraging this reduction, we design a new online algorithm using only $O(log k)$ words of memory while attaining the optimal competitive ratio $1 - O(1/sqrt{k})$. Additionally, we present a quantile estimation algorithm that, with $O(sqrt{k})$ memory, identifies the $k$-th largest element exactly with high probability and achieves expected rank error $O(sqrt{k})$—a substantial improvement over prior linear-memory schemes. Our results jointly advance the theoretical frontiers of memory-constrained online decision-making and streaming quantile estimation.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

We study memory-bounded algorithms for the $k$-secretary problem. The algorithm of Kleinberg (2005) achieves an optimal competitive ratio of $1 - O(1/sqrt{k})$, yet a straightforward implementation requires $Omega(k)$ memory. Our main result is a $k$-secretary algorithm that matches the optimal competitive ratio using $O(log k)$ words of memory. We prove this result by establishing a general reduction from $k$-secretary to (random-order) quantile estimation, the problem of finding the $k$-th largest element in a stream. We show that a quantile estimation algorithm with an $O(k^{alpha})$ expected error (in terms of the rank) gives a $(1 - O(1/k^{1-alpha}))$-competitive $k$-secretary algorithm with $O(1)$ extra words. We then introduce a new quantile estimation algorithm that achieves an $O(sqrt{k})$ expected error bound using $O(log k)$ memory. Of independent interest, we give a different algorithm that uses $O(sqrt{k})$ words and finds the $k$-th largest element exactly with high probability, generalizing a result of Munro and Paterson (1980).

Problem

• Develops memory-efficient k-secretary algorithm
• Reduces memory usage to O(log k) words
• Links k-secretary to quantile estimation

Innovation

• Logarithmic memory usage
• Optimal competitive ratio
• Quantile estimation algorithm