AI 中文总结
研究如何在\(n\times n\)矩阵中找严格鞍点，此前虽有相关算法，但能否确定性地在\(O(n)\)时间内找到未解决。本文提出简单确定性算法，结合前人方法要素与线性时间选择，能在最优\(O(n)\)时间内找到严格鞍点或报告不存在。
AI 中文摘要
给定一个\(n\times n\)矩阵\(A\)，鞍点是在其行中最大且在其列中最小的元素。若其行或列中没有其他元素具有相同值，则为严格鞍点。在最坏情况下，找到一个非严格鞍点需要\(\Theta(n^2)\)次矩阵查询，而找到一个严格鞍点仅需\(O(n)\)次查询。1991年，Bienstock等人以及Byrne和Vaserstein表明可以用\(O(n)\)次矩阵查询在\(O(n\log n)\)时间内找到严格鞍点。2024年，Dallant等人给出了\(O(n\log^* n)\)时间算法，随后还有一个以高概率在\(O(n)\)时间运行的最优随机算法。这些工作未解决是否能确定性地在\(O(n)\)时间内找到严格鞍点的问题。本文通过提出一种简单的确定性算法解决了该问题，该算法能在最优的\(O(n)\)时间内找到严格鞍点或报告不存在。我们的算法将先前方法的基本要素与从排序列表集合中进行线性时间选择相结合。
英文摘要
Given an $n\times n$ matrix $A$, a saddlepoint of $A$ is an entry that is the maximum in its row and the minimum in its column. It is a strict saddlepoint if no other entry in its row or column has the same value. Finding a non-strict saddlepoint requires $Θ(n^2)$ matrix queries in the worst case. In contrast, a strict saddlepoint can be found with only $O(n)$ queries. In 1991, Bienstock, Chung, Fredman, Schäffer, Shor, and Suri---and, independently, Byrne and Vaserstein---showed that one can find a strict saddlepoint (or certify that none exists) in $O(n\log n)$ time using $O(n)$ matrix queries. In 2024, Dallant, Haagensen, Jacob, Kozma, and Wild gave an $O(n\log^* n)$-time algorithm, followed shortly after by an optimal randomized algorithm running in $O(n)$ time with high probability. Whether $O(n)$ time could also be achieved deterministically was left open by these works. Here we resolve this question by presenting a simple deterministic algorithm that finds a strict saddlepoint, or reports that none exists, in optimal $O(n)$ time. Our algorithm combines elementary ingredients from previous approaches with linear-time selection from a collection of sorted lists.
