Fast Computation on Semirings Isomorphic to $(\times, \max)$ on $\mathbb{R}_+$

Abstract: Important problems across multiple disciplines involve computations on the semiring $(\times, \max)$ (or its equivalents, the negated version $(\times, \min)$), the log-transformed version $(+, \max)$, or the negated log-transformed version $(+, \min)$): max-convolution, all-pairs shortest paths in a weighted graph, and finding the largest $k$ values in $x_i+y_j$ for two lists $x$ and $y$. However, fast algorithms such as those enabling FFT convolution, sub-cubic matrix multiplication, \emph{etc.}, require inverse operations, and thus cannot be computed on semirings. This manuscript generalizes recent advances on max-convolution: in this approach a small family of $p$-norm rings are used to efficiently approximate results on a nonnegative semiring. The general approach can be used to easily compute sub-cubic estimates of the all-pairs shortest paths in a graph with nonnegative edge weights and sub-quadratic estimates of the top $k$ values in $x_i+y_j$ when $x$ and $y$ are nonnegative. These methods are fast in practice and can benefit from coarse-grained parallelization.