Tolerance to Asynchrony in Algorithms for Multiplication and Modulo

TL;DR

This paper investigates parallel algorithms for multiplication and modulo operations, demonstrating their lattice-linear and snap-stabilizing properties, which enable correct asynchronous implementation without synchronization overhead.

Contribution

It introduces lattice-linearity and snap-stabilization concepts to these algorithms, ensuring correct convergence in asynchronous environments.

Findings

Algorithms are lattice-linear, allowing asynchronous execution.

They guarantee correct convergence without synchronization.

They exhibit snap-stabilizing properties from arbitrary states.

Abstract

In this article, we study some parallel processing algorithms for multiplication and modulo operations. We demonstrate that the state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms induce a lattice among the global states. Lattice-linearity implies that these algorithms can be implemented in asynchronous environments, where the nodes are allowed to read old information from each other. It means that these algorithms are guaranteed to converge correctly without any synchronization overhead. These algorithms also exhibit snap-stabilizing properties, i.e., starting from an arbitrary state, the sequence of state transitions made by the system strictly follows its specification.