New Methods for the Analysis of Randomized Algorithms

在设计高效算法时，随机性是一个非常强大的资源。对随机算法的分析需要获得关于其正确性和效率的可证明保证，这带来了重大的挑战，并且关于有效计算中随机性的一般力量的许多基本问题仍然是复杂性理论中的开放问题。提出的研究旨在通过开发和利用新的算法、数学和信息论方法来设计和分析随机算法，从而克服这些重要的挑战。这些方法将为多种基本计算任务带来新的高效算法，为确定随机算法的局限性提供新的方法，并对随机性和计算之间的相互作用产生新的见解。本研究将为实现建立完整、统一、健全的随机化算法理论基础这一高层次的长期目标迈出重要的一步。

Randomness is a remarkably powerful resource in the design of efficient algorithms. The analysis of randomized algorithms required to obtain provable guarantees regarding their correctness and efficiency introduces significant challenges, and many basic questions about the general power of randomness in efficient computation remain open problems in complexity theory. The proposed research aims to overcome these important challenges by developing and exploiting new algorithmic, mathematical, and information-theoretic methods for the design and analysis of randomized algorithms. These methods will lead to new efficient algorithms for multiple fundamental computational tasks, to new methods for determining the limitations of randomized algorithms, and to new insights on the interplay between randomness and computation. This research will provide an important step in reaching the high-level long-term goal of establishing a complete, unified, and sound theoretical foundation for randomized algorithms.