double stream feed to prevent unneeded memoization?

Question

Normally, definition of primes stream in Richard Bird's formulation of the sieve of Eratosthenes is self-referential:

import Data.List.Ordered (minus, union, unionAll)

ps = 2 : minus [3..]            -- `:` is "cons"
          (foldr (\p r -> p*p : union [p*p+p, p*p+2*p..] r) [] ps)

The primes ps produced by this definition are used as the input to it. To prevent a vicious circle the definition is primed with the initial value, 2. This corresponds to the mathematical definition of the sieve of Eratosthenes as finding primes in the gaps between the composites, enumerated for each prime p by counting up in steps of p,

P = { 2 } ∪ ( { 3,4,5,... } \ ⋃_{_p in P} { p², p²+p, p²+2p, ... } )

The produced stream is used as the input in its own definition. This causes the retention of the whole stream of primes in memory (or most of it anyway). The fixpoint here is sharing, corecursive:

fix f  = xs where xs = f xs                 -- a sharing fixpoint combinator
ps     = fix ((2:) . minus [3..] . foldr (...) [])
    -- = xs where xs = 2 : minus [3..] (foldr (...) [] xs)

The idea (due to Melissa O'Neill) is, then, to separate this into two streams, with an inner loop feeding into the second stream of primes "above" it:

fix2 f  = f xs where xs = f xs          -- double-staged fixpoint combinator
ps2     = fix2 ((2:) . minus [3..] . foldr (...) [])
     -- = 2 : minus [3..] (foldr (...) [] xs) where
     --                        xs = 2 : minus [3..] (foldr (...) [] xs)

Thus, when ps2 produces some prime p, its inner stream xs of "core" primes needs only be instantiated up to about sqrt p, and any primes which are produced by ps2 can get discarded and garbage-collected by the system immediately afterwards:

    \
     \
      <- ps2 <-.
                \
                 \
                  <- xs <-.
                 /         \ 
                 \_________/

Primes produced by the inner loop xs can not be immediately discarded, because they are needed for xs stream itself. When xs has produced a prime q, only its part below sqrt q can be discarded, just after it has been consumed by the foldr part of the computation. In other words this sequence maintains back pointer into itself down to the sqrt of its biggest produced value (as it is being consumed by its consumer, like print).

So with one feed loop (with fix) almost the whole sequence would have to be retained in memory, while with the double feed (with fix2) only the inner loop needs to be mostly retained that only reaches up to the square root of the current value produced by the main stream. Thus the overall space complexity is reduced from about O(N) to about O(sqrt(N)) -- a drastic reduction.

For this to work the code must be compiled with optimizations, i.e. with the -O2 switch, and run standalone. You may also have to use -fno-cse switch. And there must be only one reference to ps2 in the testing code:

main = getLine >>= (read >>> (+(-1)) >>> (`drop` ps2) >>> print . take 5)

In fact, when tested at Ideone, it does show a practically constant memory consumption.

And it's the Sieve of Eratosthenes, not Euler's sieve.

The initial definitions are:

eratos (x:xs) = x : eratos (minus xs $ map (*x) [x..] )    -- ps = eratos [2..]
eulers (x:xs) = x : eulers (minus xs $ map (*x) (x:xs))    -- ps = eulers [2..]

Both are very inefficient because of the premature handling of the multiples. It is easy to remedy the first definition by fusing the map and the enumeration into one enumeration moved further away (from x to x*x, i.e. [x*x, x*x+x..]), so that its handling can be postponed -- because here each prime's multiples are generated independently (enumerated at fixed intervals):

eratos (p:ps) xs | (h,t) <- span (< p*p) xs =                 -- ps = 2 : eratos ps [2..]
                    h ++ eratos ps (minus t [p*p, p*p+p..])   -- "postponed sieve"

which is the same as Bird's sieve at the top of this post, segment-wise:

ps = 2 : [n | (r:q:_, px) <- (zip . tails . (2:) . map (^2) <*> inits) ps,
              n           <- [r+1..q-1] `minus` foldr union [] 
                               [[s+p, s+2*p..q-1] | p <- px, let s = r`div`p*p]]

((f <*> g) x = f x (g x) is used here as a pointfree shorthand.)

There is no easy fix for the second definition, i.e. eulers.

addition: you can see the same idea implemented with Python generators, for comparison, here.

In fact, that Python code employs a telescopic, multistage recursive production of ephemeral primes streams; in Haskell we can arrange for it with the non-sharing, multi-staged fixpoint combinator _Y:

primes = 2 : _Y ((3:) . sieve 5 . unionAll . map (\p -> [p*p, p*p+2*p..]))
  where
    _Y g = g (_Y g)                                   -- == g . g . g . g . ....

    sieve k s@(x:xs) | k < x = k : sieve (k+2) s      -- == [k,k+2..] \\ s,
                     | True  =     sieve (k+2) xs     --    when s ⊂ [k,k+2..]