Haskell: effective computation of recursively-defined function values

Question

I want to compute the recursively-defined function values r(i,j), which are defined by

r i j  | i<0 || j<0   = 0
       | i==0 && j==0 = 1
       | otherwise    = (i-1) * r (i-2) j + r (i-1) (j-1)

Obviously, the NxN table of these coefficients can be computed in O(N^2). Unfortunately, the straightforward evaluation, like

[[r i j | j <-[0..50]]| i <- [0..50]]

is performed in screamingly ineffective way (exponential complexity). Apparently, Haskell builds the entire recursion tree for each r i j and ignores the values of previously computed r (i-1) (j-1) etc.

What is the elegant and effective way to compute such a table?

... I don't know haskell but it seems you want dynamic programming. (googleit) — flight
– flight, Commented May 17, 2012 at 9:08
You want memoization, see haskell.org/haskellwiki/Memoization — fuz
– fuz, Commented May 17, 2012 at 9:23
if you want just a single value, rather than the whole table you might want to calculate a generating function for the recursive series. I'm no expert on this but I once heard about that in a lecture about discrete math/combinatorics. maybe the fellas at math.stackexchange.com are able to help you in that regard. Words I associate with that kind of math is "umbral calculus", "gosper algorithm" and "Zeilberger algorithm". Maybe it is helpful. — epsilonhalbe
– epsilonhalbe, Commented May 17, 2012 at 12:11

dave4420 · Accepted Answer · 2012-05-17 09:48:16Z

3

As FUZxxl says, this is a memoisation question.

r i j | i < 0 || j < 0 = 0
      | otherwise      = rss !! i !! j

rss = [[r' i j | j <- [0..]] | i <- [0..]]
  where r' 0 0 = 1
        r' i j = (i-1) * r (i-2) j + r (i-1) (j-1)

If you need an explicit table of precisely the values up to 50, you can use take 51 (map (take 51) rss), or [[r i j | j <-[0..50]]| i <- [0..50]] as you mentioned. Otherwise you can just call r or reference rss directly.

edited May 17, 2012 at 9:48

answered May 17, 2012 at 9:41

dave4420

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2 Comments

Jedai Over a year ago

This is still slower than it could be though, since access will be O(i+j). If you limit yourself to a given size, an array would be more appropriate (lazy array, not unboxed) for O(1) access, and if you want unlimited, a trie of trie (IntMap) should still be faster than a list of list.

Thomas M. DuBuisson Over a year ago

And if you are tasteful enough to hate the array library API then use vector instead.

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