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99 questions/Solutions/39

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</haskell>
 
</haskell>
   
If we are challenged to give all primes in the range between a and b we simply take all numbers from a up to b and filter the primes out.
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If we are challenged to give all primes in the range between a and b we simply take all numbers from a up to b and filter all the primes through.
  +
  +
This is good for ''very narrow ranges'' as Q.31's <code>isPrime</code> tests by ''trial division'' using (up to<math>\sqrt b</math>) a memoized primes list produced by sieve of Eratosthenes to which it refers internally. So it'll be slower, but immediate.
   
 
'''Solution 2:'''
 
'''Solution 2:'''
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</haskell>
 
</haskell>
   
Another way to compute the claimed list is done by using the ''Sieve of Eratosthenes''. The <code>primes</code> function generates a list of all (!) prime numbers using this algorithm and <code>primesR</code> filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the ''Sieve of Eratosthenes'' can be implemented in Haskell :)]
+
Another way to compute the claimed list is done by using the ''Sieve of Eratosthenes''. The <code>primes</code> function generates a list of all (!) prime numbers using this algorithm and <code>primesR</code> filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the ''Sieve of Eratosthenes'' can be implemented in Haskell :)]
  +
  +
''this is of course a famous case of executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code''.
   
 
'''Solution 3:'''
 
'''Solution 3:'''
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union a b = a ++ b
 
union a b = a ++ b
 
</haskell>
 
</haskell>
''(This turned out to be quite a project, with a few quite subtle points.)'' It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the square root of ''b'' and ''a''. To wit,
+
''(This turned out to be quite a project, with some quite subtle points.)'' It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the ''square root of b'' and ''a''. To wit,
 
<haskell>
 
<haskell>
 
> primesR 10100 10200 -- Sol.3
 
> primesR 10100 10200 -- Sol.3
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(testing with Hugs of Nov 2002).
 
(testing with Hugs of Nov 2002).
   
This is good for narrow ranges. For wide ranges, we're better off just using the primes sequencs itself, because of its immediate productivity:
+
This solution is faster but not immediate. It has a certain preprocessing stage but then goes on fast to produce the whole range. To illustrate, to produce the 49 primes in 1000-wide range above 120200300100 it takes about 18 seconds on my oldish notebook for the 1st version, with the first number produced almost immediately (~ 0.4 sec); but this version spews up all 49 primes in one go after just under 1 sec.
  +
  +
'''Solution 4.'''
  +
  +
For ''very wide'' ranges, specifically when <math>a < \sqrt{b}</math>, we're better off just using the primes sequence itself, without any post-processing:
   
 
<haskell>
 
<haskell>
primes = primesTME
+
primes :: Integral a => [a]
primesWideR a b = takeWhile (<= b) $ dropWhile (< a) primes
+
primes = primesTME -- of Q.31
  +
  +
primesR :: Integral a => a -> a -> [a]
  +
primesR a b = takeWhile (<= b) . dropWhile (< a) $ primes
 
</haskell>
 
</haskell>

Revision as of 10:59, 2 June 2011

(*) A list of prime numbers.

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range.

Solution 1:

primesR :: Integral a => a -> a -> [a]
primesR a b = filter isPrime [a..b]

If we are challenged to give all primes in the range between a and b we simply take all numbers from a up to b and filter all the primes through.

This is good for very narrow ranges as Q.31's isPrime tests by trial division using (up to\sqrt b) a memoized primes list produced by sieve of Eratosthenes to which it refers internally. So it'll be slower, but immediate.

Solution 2:

primes :: Integral a => [a]
primes = let sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] 
         in sieve [2..]
 
primesR :: Integral a => a -> a -> [a]
primesR a b = takeWhile (<= b) $ dropWhile (< a) primes

Another way to compute the claimed list is done by using the Sieve of Eratosthenes. The primes function generates a list of all (!) prime numbers using this algorithm and primesR filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the Sieve of Eratosthenes can be implemented in Haskell :)]

this is of course a famous case of executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code.

Solution 3:

Use the proper Sieve of Eratosthenes from e.g. 31st question's solution (instead of the above sieve of Turner), adjusted to start its multiples production from the given start point:

{-# OPTIONS_GHC -O2 -fno-cse #-}
-- tree-merging Eratosthenes sieve, primesTME of haskellwiki/prime_numbers, 
--  adjusted to produce primes in a given range (inclusive)
primesR a b 
  | b<a || b<2 = []
  | otherwise  = 
     (if a <= 2 then [2] else []) ++
     gaps a' (join [[x,x+step..b] | p <- takeWhile (<= z) primes'
                    , let q = p*p ; step = 2*p
                          x = if a' <= q then q else snapUp a' q step ])
  where
    primes' = tail primesTME        -- external unbounded list of primes
    a'      = if a<=3 then 3 else (if even a then a+1 else a)
    z       = floor $ sqrt $ fromIntegral b + 1
    join  (xs:t)    = union xs (join (pairs t))
    join  []        = []
    pairs (xs:ys:t) = (union xs ys) : pairs t
    pairs  t        = t
    gaps k xs@(x:t) | k==x  = gaps (k+2) t 
                    | True  = k : gaps (k+2) xs
    gaps k []       = [k,k+2..b]
    snapUp v origin step = let r = rem (v-origin) step
                           in if r==0 then v else v-r+step
    -- duplicates-removing union of two ordered increasing lists
    union (x:xs) (y:ys) = case (compare x y) of 
           LT -> x : union  xs  (y:ys)
           EQ -> x : union  xs     ys 
           GT -> y : union (x:xs)  ys
    union  a      b     = a ++ b

(This turned out to be quite a project, with some quite subtle points.) It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the square root of b and a. To wit,

> primesR 10100 10200                                            -- Sol.3
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(5,428 reductions, 11,310 cells)
 
> takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME   -- TME of Q.31
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(140,313 reductions, 381,058 cells)
 
> takeWhile (<= 10200) $ dropWhile (< 10100) $ sieve [2..]       -- Sol.2
     where sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ]
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(54,893,566 reductions, 79,935,263 cells, 6 garbage collections)
 
> filter isPrime [10100..10200]                                  -- Sol.1
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(15,750 reductions, 29,292 cells)                        -- isPrime: Q.31

(testing with Hugs of Nov 2002).

This solution is faster but not immediate. It has a certain preprocessing stage but then goes on fast to produce the whole range. To illustrate, to produce the 49 primes in 1000-wide range above 120200300100 it takes about 18 seconds on my oldish notebook for the 1st version, with the first number produced almost immediately (~ 0.4 sec); but this version spews up all 49 primes in one go after just under 1 sec.

Solution 4.

For very wide ranges, specifically when a < \sqrt{b}, we're better off just using the primes sequence itself, without any post-processing:

primes :: Integral a => [a]
primes = primesTME          -- of Q.31
 
primesR :: Integral a => a -> a -> [a]
primesR a b = takeWhile (<= b) . dropWhile (< a) $ primes