The Randomized Query Complexity of Finding Minimal Elements in Bounded-Width Posets

2026-08-27

AI Summary

本文研究了在宽度为w的未知n元素偏序集中查找所有最小元素的随机查询复杂度问题，通过一种基于随机链分布的新方法解决了之前上下界之间的差距。

Abstract

We study the zero-error randomized query complexity of finding all minimal elements in an unknown $n$-element poset of width at most $w$. Previous work of Daskalakis, Karp, Mossel, Riesenfeld, and Verbin established a randomized upper bound with leading term $\frac{w+1}{2}n$, while the corresponding lower bound left a multiplicative gap in the leading constant that approaches a factor of 2 as $w$ grows. We prove the finite lower bound \( R^{\mathrm{LV}}_{n,w}\ge \frac{w+1}{2}n-\frac{w(w+3)}4 +w\left(1-\frac1w\right)^n +\frac{w(w-1)}4\left(1-\frac2w\right)^n. \) Consequently, for every fixed $w$, \( R^{\mathrm{LV}}_{n,w} = \left(\frac{w+1}{2}+o(1)\right)n. \) Thus the known randomized upper bound has the correct asymptotic leading constant for every fixed width. The argument is based on a pairwise accounting of incomparable queries under a random-chain hard distribution, using a component-flip involution and a unique ownership property for incomparable comparisons. Generative AI was used in the preparation of this manuscript.

Problem

• Randomized Query Complexity
• Minimal Elements
• Bounded-Width Posets

Innovation

• randomized query complexity
• minimal elements
• bounded-width posets
• component-flip involution
• unique ownership property