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Optimal k-Secretary with Logarithmic Memory

Authors Mingda Qiao , Wei Zhang

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Streaming algorithms
  • online algorithms
  • secretary problem

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Abstract

We study memory-bounded algorithms for the k-secretary problem. The algorithm of Kleinberg (SODA 2005) achieves an optimal competitive ratio of 1 - O(1/√k), yet a straightforward implementation requires Ω(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^{α}) expected error (in terms of the rank) gives a (1 - O(1/k^{1-α}))-competitive k-secretary algorithm with O(1) extra words. We then introduce a new quantile estimation algorithm that achieves an O(√k) expected error bound using O(log k) memory. Of independent interest, we give a different algorithm that uses O(√k) words and finds the k-th largest element exactly with high probability, generalizing a result of Munro and Paterson (1980).

Cite As Get BibTex

Mingda Qiao and Wei Zhang. Optimal k-Secretary with Logarithmic Memory. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 150:1-150:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.150

Author Details

Mingda Qiao
  • University of Massachusetts Amherst, MA, USA
Wei Zhang
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

We thank the anonymous reviewers, whose suggestions have helped improved this paper.

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