最小的自由数 - 分而治之的算法

Question

I'm reading a book Pearls of Functional Algorithm Design . Tried implementing the divide and conquer solution for smallest free number problem.

minfree xs = minfrom 0 (length xs) xs

minfrom a 0 _  = a
minfrom a n xs = if m == b - a
                    then minfrom b (n-m) vs
                    else minfrom a (m)   us
    where b = a + 1 + (n `div` 2)
          (us,vs) = partition (<b) xs
          m = length us

But this one works no faster than the solution that one might call "naive" solution. Which is

import Data.List ((\\))
minfree' = head . (\\) [0..]

I don't know why this is like this, what's wrong with the divide and conquer algorithm and how to improve it.

Tried using BangPatterns , implementing the version of partition that also returns first list's length in the tuple, so it eliminates additional traversal for m =length us . None of them made improvement.

First one takes more than 5 seconds, whereas second one does it almost instantly in ghci on input [0..9999999] .

Answer 1

You have pathological input on which head . (\\\\) [0..] head . (\\\\) [0..] performs in O(N) time. \\\\ is defined as follows :

(\\) =  foldl (flip delete)

delete x xs is an O(N) operation that removes the first x from xs . foldl (flip delete) xs ys deletes all elements of ys from xs one by one.

In [0..] \\\\ [0..9999999] , we always find the next element to be deleted at the head of the list, so the result can be evaluated in linear time.

If you instead type minfree' (reverse [0..9999999]) into GHCi, that takes quadratic time and you find that it pretty much never finishes.

The divide-and-conquer algorithm on the other hand would not slow down on the reversed input.