Fast Computation on Semirings Isomorphic to (×, ) on R_+ [article]

Oliver Serang
2016 arXiv   pre-print
Important problems across multiple disciplines involve computations on the semiring (×, ) (or its equivalents, the negated version (×, )), the log-transformed version (+, ), or the negated log-transformed version (+, )): 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, etc., require inverse operations, and thus cannot
more » ... e 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.
arXiv:1511.05690v2 fatcat:lop6h2ejjrehxaal372cjxigae