New Methods for the Analysis of Randomized Algorithms

在高效算法的设计中，随机性是一个非常强大的资源。对随机化算法的分析需要获得关于其正确性和效率的可证明保证，这带来了重大挑战，而且许多关于随机性在有效计算中的一般能力的基本问题在复杂性理论中仍然是悬而未决的问题。拟议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。这些方法将为多个基本计算任务带来新的高效算法，为确定随机化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。这项研究将为实现为随机化算法建立完整、统一和健全的理论基础这一高水平的长期目标提供重要的一步。

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.