有界宽度偏序集中极小元查找的随机查询复杂度
The Randomized Query Complexity of Finding Minimal Elements in Bounded-Width Posets
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中文总结 AI 辅助
本研究针对宽度至多为w的n元偏序集极小元查找问题，证明了其零误差随机查询复杂度的下界，填补了前期上界与下界首项常数的渐近间隙，确定了该问题的渐近查询复杂度。
中文摘要 AI 辅助
我们研究宽度至多为w的未知n元偏序集的所有极小元查找的零误差随机查询复杂度。Daskalakis、Karp、Mossel、Riesenfeld和Verbin的前期工作给出了首项为((w+1)/2)n的随机上界，而对应的下界在首项常数上存在乘法间隙，当w增大时该间隙趋近于因子2。我们证明有限下界R^{LV}_{n,w}≥((w+1)/2)n - (w(w+3))/4 + w(1 - 1/w)^n + (w(w-1))/4(1 - 2/w)^n。因此，对每个固定的w，R^{LV}_{n,w}=((w+1)/2 + o(1))n。由此可知，已知的随机上界对每个固定宽度都具有正确的渐近首项常数。该论证基于随机链困难分布下不可比查询的成对计数，利用分量翻转对合和不可比比较的唯一所有权性质。本手稿的准备过程中使用了生成式AI。
英文摘要
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.
