One to rule them all: a general method for fast computation on semirings isomorphic to $(\times, \max)$ on $\mathbb{R}_+$

Important problems across multiple disciplines involve computations on the semiring (×,max) (or its equivalents, the negated version (×,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 xi + yj for two lists x and y. However, fast algorithms such as those enabling FFT convolution, sub-cubic matrix multiplication, 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 xi + yj when x and y are nonnegative. These methods are fast in practice and can benefit from coarse-grained parallelization.