Universal algorithms, mathematics of semirings and parallel computations

TL;DR

This survey explores how semiring mathematics underpins universal algorithms and parallel computations, highlighting applications in optimization, idempotent linear algebra, and interval analysis, emphasizing the benefits of linearity over semirings.

Contribution

It provides a comprehensive overview of the role of semirings in developing universal algorithms and their applications in numerical analysis and parallel computing.

Findings

Semiring mathematics enables linearization of nonlinear problems.

Idempotent semirings facilitate parallel computations.

Applications include optimization and interval analysis.

Abstract

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.