傅立叶满足莫比乌斯：快速子集卷积
Fourier meets Möbius: fast subset convolution
摘要：
给出了子集卷积问题的一个快速算法：给定定义在n元集合N的子集格上的函数f和g，计算它们的子集卷积f*g，定义为（f * g）（S）= \subseteq f（T）g（S \setminus T），其中加法和乘法在任意环中进行.通过莫比乌斯变换和求逆，我们的算法在O（n^2 2^n）的加法和乘法中计算子集卷积，大大改进了直接的O（3^n）算法。具体地，如果输入函数具有整数范围{-M，-M+1，.，M}，它们在普通和积环上的子集卷积可以在O^*（2^n log M）时间内计算;符号O^* 抑制了多对数因子。此外，使用标准的嵌入技术，我们可以在O^*（2^nM）时间内计算最大和或最小和半环上的子集卷积。为了证明快速子集卷积的适用性，我们针对具有n个顶点、k个端点和m条边且具有有界整数权重的图中的最小Steiner树问题提出了第一个O^*（2^k n^2 + n m）算法，改进了经典Dreyfus-Wagner算法的O^*（3^k n + 2^k n^2 + n m）时间界。我们还讨论了最近的O^*（2^n）时间算法的扩展覆盖和划分问题（Björklund和Husfeldt，FOCS 2006; Koivisto，FOCS 2006）。
英文摘要：
We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f*g, defined for all S\subseteq N by (f * g)(S) = \sum_{T \subseteq S}f(T) g(S\setminus T), where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n^2 2^n) additions and multiplications, substantially improving upon the straightforward O(3^n) algorithm. Specifically, if the input functions have an integer range {-M,-M+1,...,M}, their subset convolution over the ordinary sum-product ring can be computed in O^*(2^n log M) time; the notation O^* suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max-sum or min-sum semiring in O^*(2^n M) time. To demonstrate the applicability of fast subset convolution, we present the first O^*(2^k n^2 + n m) algorithm for the minimum Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O^*(3^k n + 2^k n^2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O^*(2^n)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).
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