Difference between revisions of "Prime numbers miscellaneous"
(→One-liners: simplify) |
(→Prime Wheels: remove apostrophes in names; c/e) |
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<haskell> |
<haskell> |
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primes :: [Integer] |
primes :: [Integer] |
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− | primes = 2:3: |
+ | primes = 2:3:prs |
where |
where |
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1:p:candidates = [6*k+r | k <- [0..], r <- [1,5]] |
1:p:candidates = [6*k+r | k <- [0..], r <- [1,5]] |
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− | + | prs = p : filter isPrime candidates |
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isPrime n = all (not . divides n) |
isPrime n = all (not . divides n) |
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− | $ takeWhile (\p -> p*p <= n) |
+ | $ takeWhile (\p -> p*p <= n) prs |
divides n p = n `mod` p == 0 |
divides n p = n `mod` p == 0 |
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</haskell> |
</haskell> |
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− | Here, <hask> |
+ | Here, <hask>prs</hask> is the list of primes greater than 3 and <hask>isPrime</hask> does not test for divisibility by 2 or 3 because the <hask>candidates</hask> by construction don't have these numbers as factors. We also need to exclude 1 from the candidates and mark the next one as prime to start the recursion. |
Such a scheme to generate candidate numbers first that avoid a given set of primes as divisors is called a '''prime wheel'''. Imagine that you had a wheel of circumference 6 to be rolled along the number line. With spikes positioned 1 and 5 units around the circumference, rolling the wheel will prick holes exactly in those positions on the line whose numbers are not divisible by 2 and 3. |
Such a scheme to generate candidate numbers first that avoid a given set of primes as divisors is called a '''prime wheel'''. Imagine that you had a wheel of circumference 6 to be rolled along the number line. With spikes positioned 1 and 5 units around the circumference, rolling the wheel will prick holes exactly in those positions on the line whose numbers are not divisible by 2 and 3. |
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</haskell> |
</haskell> |
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We can create a larger wheel by rolling a smaller wheel of circumference <hask>n</hask> along a rim of circumference <hask>p*n</hask> while excluding spike positions at multiples of <hask>p</hask>. |
We can create a larger wheel by rolling a smaller wheel of circumference <hask>n</hask> along a rim of circumference <hask>p*n</hask> while excluding spike positions at multiples of <hask>p</hask>. |
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+ | |||
<haskell> |
<haskell> |
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nextSize (Wheel n rs) p = |
nextSize (Wheel n rs) p = |
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− | Wheel (p*n) [ |
+ | Wheel (p*n) [r2 | k <- [0..(p-1)], r <- rs, |
− | + | let r2 = n*k+r, r2 `mod` p /= 0] |
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</haskell> |
</haskell> |
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+ | |||
Combining both, we can make wheels that prick out numbers that avoid a given list <hask>ds</hask> of divisors. |
Combining both, we can make wheels that prick out numbers that avoid a given list <hask>ds</hask> of divisors. |
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<haskell> |
<haskell> |
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It's a pretty big wheel with a circumference of 210 and allows us to calculate the first 10000 primes in convenient time. |
It's a pretty big wheel with a circumference of 210 and allows us to calculate the first 10000 primes in convenient time. |
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− | A fixed size wheel is fine, but |
+ | A fixed size wheel is fine, but adapting the wheel size while generating prime numbers quickly becomes impractical, because the circumference grows very fast, as primorial, but the returns quickly diminish, the improvement being just <code>(p-1)/p</code>. See [[Prime_numbers#Euler.27s_Sieve | Euler's Sieve]], or the [[Research papers/Functional pearls|functional pearl]] titled [http://citeseer.ist.psu.edu/runciman97lazy.html Lazy wheel sieves and spirals of primes] for more. |
== Using IntSet for a traditional sieve == |
== Using IntSet for a traditional sieve == |
Revision as of 12:00, 26 January 2015
For a context to this, please see Prime numbers.
Implicit Heap
The following is an original implicit heap implementation for the sieve of
Eratosthenes, kept here for historical record. Also, it implements more sophisticated, lazier scheduling. The Prime_numbers#Tree merging with Wheel section simplifies it, removing the People a
structure altogether, and improves upon it by using a folding tree structure better adjusted for primes processing, and a wheel optimization.
See also the message threads Re: "no-coding" functional data structures via lazyness for more about how merging ordered lists amounts to creating an implicit heap and Re: Code and Perf. Data for Prime Finders for an explanation of the People a
structure that makes it work.
data People a = VIP a (People a) | Crowd [a]
mergeP :: Ord a => People a -> People a -> People a
mergeP (VIP x xt) ys = VIP x $ mergeP xt ys
mergeP (Crowd xs) (Crowd ys) = Crowd $ merge xs ys
mergeP xs@(Crowd (x:xt)) ys@(VIP y yt) = case compare x y of
LT -> VIP x $ mergeP (Crowd xt) ys
EQ -> VIP x $ mergeP (Crowd xt) yt
GT -> VIP y $ mergeP xs yt
merge :: Ord a => [a] -> [a] -> [a]
merge xs@(x:xt) ys@(y:yt) = case compare x y of
LT -> x : merge xt ys
EQ -> x : merge xt yt
GT -> y : merge xs yt
diff xs@(x:xt) ys@(y:yt) = case compare x y of
LT -> x : diff xt ys
EQ -> diff xt yt
GT -> diff xs yt
foldTree :: (a -> a -> a) -> [a] -> a
foldTree f ~(x:xs) = x `f` foldTree f (pairs xs)
where pairs ~(x: ~(y:ys)) = f x y : pairs ys
primes, nonprimes :: [Integer]
primes = 2:3:diff [5,7..] nonprimes
nonprimes = serve . foldTree mergeP . map multiples $ tail primes
where
multiples p = vip [p*p,p*p+2*p..]
vip (x:xs) = VIP x $ Crowd xs
serve (VIP x xs) = x:serve xs
serve (Crowd xs) = xs
nonprimes
effectively implements a heap, exploiting lazy evaluation.
Prime Wheels
The idea of only testing odd numbers can be extended further. For instance, it is a useful fact that every prime number other than 2 and 3 must be of the form or . Thus, we only need to test these numbers:
primes :: [Integer]
primes = 2:3:prs
where
1:p:candidates = [6*k+r | k <- [0..], r <- [1,5]]
prs = p : filter isPrime candidates
isPrime n = all (not . divides n)
$ takeWhile (\p -> p*p <= n) prs
divides n p = n `mod` p == 0
Here, prs
is the list of primes greater than 3 and isPrime
does not test for divisibility by 2 or 3 because the candidates
by construction don't have these numbers as factors. We also need to exclude 1 from the candidates and mark the next one as prime to start the recursion.
Such a scheme to generate candidate numbers first that avoid a given set of primes as divisors is called a prime wheel. Imagine that you had a wheel of circumference 6 to be rolled along the number line. With spikes positioned 1 and 5 units around the circumference, rolling the wheel will prick holes exactly in those positions on the line whose numbers are not divisible by 2 and 3.
A wheel can be represented by its circumference and the spiked positions.
data Wheel = Wheel Integer [Integer]
We prick out numbers by rolling the wheel.
roll (Wheel n rs) = [n*k+r | k <- [0..], r <- rs]
The smallest wheel is the unit wheel with one spike, it will prick out every number.
w0 = Wheel 1 [1]
We can create a larger wheel by rolling a smaller wheel of circumference n
along a rim of circumference p*n
while excluding spike positions at multiples of p
.
nextSize (Wheel n rs) p =
Wheel (p*n) [r2 | k <- [0..(p-1)], r <- rs,
let r2 = n*k+r, r2 `mod` p /= 0]
Combining both, we can make wheels that prick out numbers that avoid a given list ds
of divisors.
mkWheel ds = foldl nextSize w0 ds
Now, we can generate prime numbers with a wheel that for instance avoids all multiples of 2, 3, 5 and 7.
primes :: [Integer]
primes = small ++ large
where
1:p:candidates = roll $ mkWheel small
small = [2,3,5,7]
large = p : filter isPrime candidates
isPrime n = all (not . divides n)
$ takeWhile (\p -> p*p <= n) large
divides n p = n `mod` p == 0
It's a pretty big wheel with a circumference of 210 and allows us to calculate the first 10000 primes in convenient time.
A fixed size wheel is fine, but adapting the wheel size while generating prime numbers quickly becomes impractical, because the circumference grows very fast, as primorial, but the returns quickly diminish, the improvement being just (p-1)/p
. See Euler's Sieve, or the functional pearl titled Lazy wheel sieves and spirals of primes for more.
Using IntSet for a traditional sieve
module Sieve where
import qualified Data.IntSet as I
-- findNext - finds the next member of an IntSet.
findNext c is | I.member c is = c
| c > I.findMax is = error "Ooops. No next number in set."
| otherwise = findNext (c+1) is
-- mark - delete all multiples of n from n*n to the end of the set
mark n is = is I.\\ (I.fromAscList (takeWhile (<=end) (map (n*) [n..])))
where
end = I.findMax is
-- primes - gives all primes up to n
primes n = worker 2 (I.fromAscList [2..n])
where
worker x is
| (x*x) > n = is
| otherwise = worker (findNext (x+1) is) (mark x is)
(doesn't look like it runs very efficiently).
One-liners
primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..n-1]]]
primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..min j (n`div`j)]]]
primes = nubBy (((>1).).gcd) [2..]
primes = [n | n<-[2..], all ((> 0).rem n) [2..n-1]]
primes = 2 : [n | n<-[3,5..], all ((> 0).rem n) [3,5..floor.sqrt$fromIntegral n]]
primes = zipWith (flip (!!)) [0..]
. scanl1 minus . scanl1 (zipWith(+)) $ repeat [2..] -- APL-style
primes = tail . concat . unfoldr (\(a:b:r)-> let (h,t)=span (< head b) a in
Just (h, minus t b : r)) . scanl1 (zipWith(+) . tail) $ tails [1..]
primes = 2 : [n | n<-[3..], all ((> 0).rem n) $ takeWhile ((<= n).(^2)) primes]
primes = 2 : 3 : [n | n<-[5,7..],
foldr (\p r-> p*p>n || (rem n p>0 && r)) True $ tail primes]
primes = 2 : fix (\xs-> 3 : [n | n<-[5,7..],
foldr (\p r-> p*p>n || (rem n p>0 && r)) True xs])
primes = map head $ iterate (\(x:xs)-> filter ((> 0).(`rem`x)) xs) [2..]
primes = 2 : unfoldr (\(x:xs)-> Just(x, filter ((> 0).(`rem`x)) xs)) [3,5..]
primesTo n = foldl (\r x-> r `minus` [x*x, x*x+2*x..]) (2:[3,5..n])
[3,5..floor.sqrt$fromIntegral n]
primesTo n = 2 : foldr (\r z-> if (head r^2) <= n then head r : z else r) []
(iterate (\(p:t)-> minus t [p*p, p*p+2*p..]) [3,5..n])
primes = 2 : concat (unfoldr (\(xs,p:ps)-> let (h,t)=span (< p*p) xs in
Just (h, (filter ((> 0).(`rem`p)) t, ps))) ([3,5..],[3,5..]))
primes = 2 : 3 : concat (unfoldr (\(xs,p:ps)-> let (h,t)=span (< p*p) xs in
Just (h, (t `minus` [p*p, p*p+2*p..], ps))) ([5,7..],tail primes))
primes = 2 : _Y (\ps-> concatMap snd $ iterate (\((fs:ft, x, p:t),_) ->
((ft,p*p+2,t), [x | x <- [x, x+2 .. p*p-2],
all ((/= 0).rem x) fs])) ((inits ps, 5, ps), [3]) )
primes = 2 : _Y (\ps-> concatMap snd $ iterate (\((fs:ft, x, p:t),_) ->
((ft,p*p+2,t), minus [x, x+2 .. p*p-2]
$ foldi union [] [[o, o+2*i .. p*p-2] | i <- fs,
let o=x+mod(i-x)(2*i)])) ((inits ps, 5, ps), [3]) )
primes = let { sieve (x:xs) = x : sieve [n | n <- xs, rem n x > 0] } in sieve [2..]
primes = let { sieve xs (p:ps) = let (h,t)=span (< p*p) xs in
h ++ sieve (filter ((> 0).(`rem`p)) t) ps }
in 2 : 3 : sieve [5,7..] (tail primes)
primes = let { sieve xs (p:ps) = let (h,t)=span (< p*p) xs in
h ++ sieve (t `minus` [p*p, p*p+2*p..]) ps }
in 2 : 3 : sieve [5,7..] (tail primes)
primes = 2 : minus [3..] (foldr (\(x:xs)->(x:).union xs) []
$ map (\x->[x*x, x*x+x..]) primes)
primes = 2 : minus [3,5..] (foldi (\(x:xs)->(x:).union xs) []
$ map (\x->[x*x, x*x+2*x..]) [3,5..])
primes = 2 : _Y ( (3:) . minus [5,7..] -- unbounded Sieve of Eratosthenes
. foldi (\(x:xs) ys-> x:union xs ys) []
. map (\p->[p*p, p*p+2*p..]) )
_Y g = g (_Y g)
foldi
is an infinitely right-deepening tree folding function found here. minus
of course is on the main page here.