An optimal deterministic algorithm for finding a strict saddlepoint

2026-07-24

AI Summary

This work addresses the problem of efficiently determining the existence of a strict saddle point—defined as an entry that is the unique maximum in its row and the unique minimum in its column—in an $n \times n$ matrix. We present the first deterministic linear-time algorithm for this task, achieving both $O(n)$ time complexity and $O(n)$ query complexity in the matrix query model. Our approach integrates classical saddle-point search strategies with linear-time selection techniques for sorted lists, enabling exact resolution of the problem within optimal theoretical bounds. This result resolves a long-standing open question by establishing, for the first time, that a deterministic algorithm can achieve linear-time performance for strict saddle-point detection, thereby matching the information-theoretic lower bound in the deterministic setting.

Abstract

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.

Problem

• strict saddlepoint
• deterministic algorithm
• matrix queries
• optimal time complexity

Innovation

• strict saddlepoint
• deterministic algorithm
• linear-time selection
• matrix queries
• optimal complexity