Optimal $k$-Secretary with Logarithmic Memory
For researchers in online algorithms and streaming, this resolves the memory complexity of the k-secretary problem, showing that logarithmic memory suffices for optimal performance.
The paper presents a memory-efficient algorithm for the k-secretary problem that achieves the optimal competitive ratio of 1 - O(1/√k) using only O(log k) words of memory, compared to the previous Ω(k) memory requirement. This is achieved via a reduction to quantile estimation, for which they introduce a new algorithm with O(√k) expected error using O(log k) memory.
We study memory-bounded algorithms for the $k$-secretary problem. The algorithm of Kleinberg (SODA 2005) achieves an optimal competitive ratio of $1 - O(1/\sqrt{k})$, 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).