[Haskell-cafe] Help with Bird problem 4.5.6: sequence of successive maxima

[Daniel Fischer]

Am Sonntag, 15. März 2009 21:09 schrieb R J:
> This Bird problem vexes me, in the first instance because it doesn't seem
> to specify a unique solution:
>
> Given a list xs = [x_1, x_2, . . . , x_n], the sequence of successive
> maxima "ssm xs" is the longest subsequence [x_j1, x_j2, x_j3..x_jk] such
> that j_1 = 1 and j_m < j_n => x_jm < x_jn. For example, xs = [3, 1, 3, 4,
> 9, 2, 10, 7] => ssm xs = [3, 4, 9, 10].  Define "ssm" in terms of "foldl".
>
> From this specification, I infer:
>
> ssm []           = []
> ssm [1]          = [1]
> ssm [1, 2, 3]    = [1, 2, 3]
> ssm [1, 0, 3, 2] = [1, 3]
>
> However, what is ssm [1,0,100,2,3,4,5]?  Is it [1, 100] or [1, 2, 3, 4, 5]?
>  I think the latter, but am not certain.  Whichever it is, what's the
> solution?
>
> Thanks.
>

Not particularly efficient, but

module SSM where

import Data.List (maximumBy)
import Data.Ord

ssm :: Ord a => [a] -> [a]
ssm = reverse . maximumBy (comparing length) . foldl comb [[]]
      where
        comb [[]] a = [[a]]
        comb lists a = do
                xs@(h:_) <- lists
                if h < a then [xs,a:xs] else [xs]

I think it is impossible to implement ssm as

foldl f z

without any post-processing and since foldl can't foresee what comes in the 
remainder of the list, you must keep several candidates around.
You can probably make it more efficient by removing all lists 
lst@(h:_) where there's a longer list with head <= h or an equally long list 
with head < h in the store (but doing that efficiently is not trivial).