A distributed prime sieving algorithm based on Scheduling by Multiple Edge Reversal

Abstract

This paper presents a new distributed approach for generating all prime numbers in a given interval of integers. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the current generation of parallel computers, which have permitted to reach larger bounds on the interval or to obtain previous results in a shorter time, prime numbers generation still represents an attractive domain of research and plays a central role in cryptography. We propose a fully distributed algorithm for finding all primes in the interval $[2\ldots, n]$, based on the \emph{wheel sieve} and the SMER (\emph{Scheduling by Multiple Edge Reversal}) multigraph dynamics. Given a multigraph $\mathcal{M}$ of arbitrary topology, having $N$ nodes, a SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of $\mathcal{M}$, and by the global period length of all "arc reversals" in $\mathcal{M}$. The new prime number generation method inherits the distributed and parallel nature of SMER and requires at most $n + \lfloor \sqrt{n}\rfloor$ time steps. The message complexity achieves at most $nΔ_N + \lfloor \sqrt{n}\rfloor Δ_N$, where $1\le Δ_N\le N - 1$ is the maximal multidegree of $\mathcal{M}$, and the maximal amount of memory space required per process is $\mathcal{O}(n)$ bits.