Optimal $k$-Secretary with Logarithmic Memory
Mingda Qiao, Wei Zhang
TL;DR
This work studies memory-bounded algorithms for the $k$-secretary problem and proves that $O(\log k)$ memory suffices to achieve Kleinberg's optimal competitive ratio $1 - O(1/\sqrt{k})$. The central approach is a reduction from $k$-secretary to random-order quantile estimation, showing that a comparison-based quantile estimator with $O(k^{\alpha})$ rank error yields a $k$-secretary algorithm with competitive ratio $1 - O(1/k^{1-\alpha})$ while using only $m+O(1)$ memory. To realize this, the paper introduces two quantile estimation results: (i) a memory- efficient algorithm using $O(\log k)$ memory that attains $O(\sqrt{k})$ expected rank error, and (ii) an exact $k$-th largest selection method that uses $O(\sqrt{k})$ memory with high probability, generalizing Munro and Paterson's classical results. Together, these contributions yield memory-optimal procedures for the $k$-secretary problem that are practically relevant in streaming and routing contexts where storage is limited, while preserving near-optimal competitiveness.
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 $Ω(k)$ 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 $O(\sqrt{k})$ 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).