Fourier 真的遇见 Möbius 了吗？基于 FFT 的快速子集卷积
Did Fourier Really Meet Möbius? Fast Subset Convolution via FFT
AI总结：
针对子集卷积现有算法因集合函数变换导致实际不可用的问题，提出一种基于 FFT 的快速算法，消除变换需求并保持原运行时间，使高效 FFT 实现的应用成为可能。
摘要：
在其关于子集卷积的开创性工作中，Björklund、Husfeldt、Kaski 和 Koivisto 提出了如今广为人知的在求和-乘积环中 $O(2^n n^2)$ 时间的子集卷积计算方法。这引发了一系列针对基础问题的显著成果，例如最小 Steiner 树和色数。然而，尽管具有理论改进，由于其集合函数变换中交替的加减法导致的庞大中间输出和浮点精度误差，使得该算法在实践中无法使用。我们提供了一种简单的基于 FFT 的算法，完全消除了对集合函数变换的需求，并保持了原始算法的运行时间。这使得利用近六十年来关于高效 FFT 实现的研究成果成为可能。
英文摘要：
In their seminal work on subset convolution, Björklund, Husfeldt, Kaski and Koivisto introduced the now well-known $O(2^n n^2)$-time evaluation of the subset convolution in the sum-product ring. This sparked a wave of remarkable results for fundamental problems, such as the minimum Steiner tree and the chromatic number. However, in spite of its theoretical improvement, large intermediate outputs and floating-point precision errors due to alternating addition and subtraction in its set function transforms make the algorithm unusable in practice. We provide a simple FFT-based algorithm that completely eliminates the need for set function transforms and maintains the running time of the original algorithm. This makes it possible to take advantage of nearly sixty years of research on efficient FFT implementations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
