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arXiv · 0809.2489

The fast intersection transform with applications to counting paths

Andreas Björklund·Thore Husfeldt·Petteri Kaski·Mikko Koivisto
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Abstract

We present an algorithm for evaluating a linear ``intersection transform'' of a function defined on the lattice of subsets of an $n$-element set. In particular, the algorithm constructs an arithmetic circuit for evaluating the transform in ``down-closure time'' relative to the support of the function and the evaluation domain. As an application, we develop an algorithm that, given as input a digraph with $n$ vertices and bounded integer weights at the edges, counts paths by weight and given length $0\leq\ell\leq n-1$ in time $O^*(\exp(n\cdot H(\ell/(2n))))$, where $H(p)=-p\log p-(1-p)\log(1-p)$, and the notation $O^*(\cdot)$ suppresses a factor polynomial in $n$.

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Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto. 2008-09-15. The fast intersection transform with applications to counting paths. https://arxiv.org/abs/0809.2489

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