Portability | Non-portable (GHC extensions) |
---|---|
Stability | Provisional |
Maintainer | Daniel Fischer <daniel.is.fischer@googlemail.com> |
Prime generation using a sieve. Currently, an enhanced sieve of Eratosthenes is used, switching to an Atkin sieve is planned (if I get around to implementing it and it's not slower).
The sieve used is segmented, with a chunk size chosen to give good (enough)
cache locality while still getting something substantial done per chunk.
However, that means we must store data for primes up to the square root of
where sieving is done, thus sieving primes up to n
requires
O(sqrt n/log n)
space.
- primes :: [Integer]
- sieveFrom :: Integer -> [Integer]
- data PrimeSieve
- primeSieve :: Integer -> PrimeSieve
- psieveList :: [PrimeSieve]
- psieveFrom :: Integer -> [PrimeSieve]
- primeList :: PrimeSieve -> [Integer]
Limitations
There are three factors limiting the range of these sieves.
- Memory
- Overflow
- The internal representation of the state
An Eratosthenes type sieve needs to store the primes up to the square root of
the currently sieved region, thus requires O(sqrt n/log n)
space.We store 16
bytes
of information per prime, thus a Gigabyte of memory takes you to about 1.6*10^18
.
The log
doesn't change much in that range, so as a first approximation, doubling
the storage increases the sieve range by a factor of four.
On a 64-bit system, this is (currently) the only limitation to be concerned with, but
with more than four Terabyte of memory, the fact that the internal representation
currently limits the sieve range to about 6.8*10^25
could become relevant.
Overflow in array indexing doesn't become a concern before memory and internal
representation would allow to sieve past 10^37
.
On a 32-bit system, the internal representation imposes no additional limits,
but overflow has to be reckoned with. On the one hand, the fact that arrays are
Int
-indexed restricts the size of the prime store, on the other hand, overflow
in calculating the indices to cross off multiples is possible before running out
of memory. The former limits the upper bound of the monolithic primeSieve
to
shortly above 8*10^9
, the latter limits the range of the segmented sieves to
about 1.7*10^18
.
Sieves and lists
List of primes.
Since the sieve uses unboxed arrays, overflow occurs at some point.
On 64-bit systems, that point is beyond the memory limits, on
32-bit systems, it is at about 1.7*10^18
.
data PrimeSieve Source
Compact store of primality flags.
primeSieve :: Integer -> PrimeSieveSource
Sieve primes up to (and including) a bound. For small enough bounds, this is more efficient than using the segmented sieve.
Since arrays are Int
-indexed, overflow occurs when the sieve
size comes near
, that corresponds to an
upper bound near maxBound
:: Int
15/8*
. On maxBound
32
-bit systems, that
is often within memory limits, so don't give bounds larger than
8*10^9
there.
psieveList :: [PrimeSieve]Source
List of primes in the form of a list of PrimeSieve
s, more compact than
primes
, thus it may be better to use
than keeping the list of primes alive during the entire run.
psieveList
>>= primeList
psieveFrom :: Integer -> [PrimeSieve]Source
creates the list of psieveFrom
nPrimeSieve
s starting roughly
at n
. Due to the organisation of the sieve, the list may contain
a few primes less than n
.
This form uses less memory than [
, hence it may be preferable
to use this if it is to be reused.
Integer
]
primeList :: PrimeSieve -> [Integer]Source
Generate a list of primes for consumption from a
PrimeSieve
.