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### Fourier meets Möbius: fast subset convolution

(2007) Annual ACM Symposium on Theory of Computing p.67-74
Abstract
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... (More)
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). (Less)
author
; ; and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
pages
67 - 74
conference name
Annual ACM Symposium on Theory of Computing
conference location
San Diego, California, United States
conference dates
0001-01-02
external identifiers
• scopus:35449001611
ISBN
978-1-59593-631-8
DOI
10.1145/1250790.1250801
project
Exact algorithms
language
English
LU publication?
yes
id
d23b0760-54e7-470f-a423-3e7c167049ba (old id 623051)
2016-04-04 13:46:58
date last changed
2021-05-06 19:28:10
```@inproceedings{d23b0760-54e7-470f-a423-3e7c167049ba,
abstract     = {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.<br/><br>
<br/><br>
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).},
author       = {Björklund, Andreas and Husfeldt, Thore and Kaski, Petteri and Koivisto, Mikko},
booktitle    = {Proceedings of the thirty-ninth annual ACM symposium on Theory of computing},
isbn         = {978-1-59593-631-8},
language     = {eng},
pages        = {67--74},
title        = {Fourier meets Möbius: fast subset convolution},
url          = {http://dx.doi.org/10.1145/1250790.1250801},
doi          = {10.1145/1250790.1250801},
year         = {2007},
}

```