Factory.Math.Implementations.Primes.SieveOfEratosthenes

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Generates the constant, conceptually infinite, list of prime-numbers, using the Sieve of Eratosthenes; https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes.
• Based on http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf.
• The implementation; has been optimised using a wheel of static, but parameterised, size; is polymorphic, but with a specialisation for type Int.

CAVEAT

The Int-specialisation is implemented by a rewrite-rule, which is very fragile.

Types

Type-synonyms

Functions

sieveOfEratosthenes :: Integral i => NPrimes -> [i]

• A refinement of the Sieve Of Eratosthenes, which pre-sieves candidates, selecting only those coprime to the specified short sequence of low prime-numbers.
• The short sequence of initial primes are represented by a PrimeWheel, of parameterised, but static, size; https://en.wikipedia.org/wiki/Wheel_factorization.
• The algorithm requires one to record multiples of previously discovered primes, allowing composite candidates to be eliminated by comparison.
• Because each list of multiples, starts with the square of the prime from which it was generated, the vast majority will be larger than the maximum prime ultimately demanded, and the effort of constructing and storing this list, is consequently wasted. Many implementations solve this, by requiring specification of the maximum prime required, thus allowing the construction of redundant lists of multiples to be avoided.
• This implementation doesn't impose that constraint, leaving a requirement for rapid storage, which is supported by appending the list of prime-multiples, to a queue. If a large enough candidate is ever generated, to match the head of the list of prime-multiples, at the head of this queue, then the whole list of prime-multiples is dropped from the queue, but the tail of this list of prime-multiples, for which there is now a high likelyhood of a subsequent match, must now be re-recorded. A queue doesn't support efficient random insertion, so a Map is used for these subsequent multiples. This solution is faster than just using a Data.PQueue.Min.
• CAVEAT: has linear O(n) space-complexity.