Selection under partial information

Abstract

The selection problem asks to find the $k$-th smallest element in a given set $X$. Its complexity is well-known to be $Θ(|X|)$ for every fixed $k$. We consider the poset selection problem where a partial order $P$ on $X$ is given, and the underlying order of $X$ is guaranteed to be a linear extension of $P$. This generalizes several variants of the selection problem studied before. Our main result is an algorithm that performs $O( k + \operatorname{OPT}(P,k) )$ comparisons, where $\operatorname{OPT}(P,k)$ is the minimum number of comparisons required for the input $(P,k)$. If $P$ is a rooted tree, we obtain an $O( \operatorname{OPT} \log (k / \operatorname{OPT}) )$ algorithm, which can be further improved to $O( \operatorname{OPT} \log \log (k/\operatorname{OPT}) )$ or $O( \operatorname{OPT} \log \operatorname{OPT})$ under mild assumptions about $P$. Our algorithms are based on the work of Kaplan, Kozma, Zamir, and Zwick [SOSA 2019], who used Chazelle's soft heaps [J. ACM 2000] to simplify several older results. We introduce a new soft heap variant, the weighted soft heap, which allows greater control over which elements get corrupted.