Fourier meets Möbius: fast subset convolution
Författare
Summary, in English
We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/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(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in Õ(2n log M) time; the notation Õ suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in Õ(2n M) time.
To demonstrate the applicability of fast subset convolution, wepresent the first Õ(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the Õ(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent Õ(2n)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).
To demonstrate the applicability of fast subset convolution, wepresent the first Õ(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the Õ(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent Õ(2n)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).
Avdelning/ar
- Institutionen för datavetenskap
- Computer Science
- Parallella System
Publiceringsår
2007
Språk
Engelska
Sidor
67-74
Publikation/Tidskrift/Serie
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Dokumenttyp
Konferensbidrag
Ämne
- Computer Science
Conference name
Annual ACM Symposium on Theory of Computing
Conference date
0001-01-02
Conference place
San Diego, California, United States
Status
Published
Projekt
- Exact algorithms
Forskningsgrupp
- Algorithms
ISBN/ISSN/Övrigt
- ISBN: 978-1-59593-631-8