Re: Insertion Sort from Permutation Sort

From: Julio Mariño <jmarino_at_fi.upm.es>
Date: Wed, 16 Oct 2013 11:32:17 +0200

Permutation sort was one of the standard benchmarks of the Babel FL
language in the early nineties. The fact that its lazy implementation was
equivalent to insertion sort was well known, and we actually used the
performance figures to impress the Prolog people ;)

Julio


On Wed, Oct 16, 2013 at 9:05 AM, Sebastian Fischer <
sebf_at_informatik.uni-kiel.de> wrote:

> Hello!
>
> A folklore example of our community is "permutation sort", where a
> list of numbers is sorted by "first" permuting all its elements
> nondeterministically and "then" checking whether the result is sorted.
>
> An important aspect of this example is pruning thanks to laziness.
> Although the program looks like permuting first and then checking for
> sortedness, large parts of the search space are pruned by lazy
> evaluation, interleaving the generator and the test.
>
> That said, the pruned parts are not large enough to make the resulting
> program efficient. Experimental results suggest that it is still
> exponential in the worst case.
>
> It just occurred to me that permutation sort can be expressed in a way
> that allows significantly more pruning. The essential idea is the
> following observation:
>
> isSorted (insert x xs) ==> isSorted xs
>
> or equivalently
>
> not (isSorted xs) ==> not (isSorted (insert x xs))
>
> where `isSorted` checks whether a given list is sorted and `insert`
> inserts the first argument into the second at an arbitrary position
> nondeterministically.
>
> Thanks to this observation, we can discard subsequent insertions if we
> detect that intermediate results are already out of order.
>
> One way to implement a `permute` operation is to use the `foldl`
> function like this:
>
> permute :: [a] -> [a]
> permute = foldl (flip insert) []
>
> In order to collect intermediate results, we can use `scanl` from the
> `List` module instead of `foldl`
>
> insertions :: [a] -> [[a]]
> insertions = scanl (flip insert) []
>
> For example, `insertions [1,2]` has two possible results,
> `[[],[1],[1,2]]` or `[[],[1],[2,1]]`. The last element of `insertions
> l` is always a permutation of `l`.
>
> In analogy to permutation sort defined as
>
> psort :: [Int] -> [Int]
> psort xs | isSorted p = p
> where
> p = permute xs
>
> we can define a function isort that uses `insertions` rather than
> `permute`.
>
> isort :: [Int] -> [Int]
> isort xs | all isSorted ps = last ps
> where
> ps = insertions xs
>
> Thanks to the above observation (and laziness of `all`), this
> definition detects early if the final permutation is unsorted based on
> unsortedness of an intermediate insertion.
>
> It seems that unsorted insertions are discarded immediately, just as
> if the `isSorted` check would have been incorporated into the
> definition of `insert`. Experiments suggest that indeed, the resulting
> code has quadratic worst case complexity, just like insertion sort.
>
> The complete code is attached to this email.
>
> Best regards,
> Sebastian
>
> P.S. My observation is inspired by a paper [1] shown to me by Keisuke
> Nakano after I gave a Curry tutorial at NII Tokyo. Although
> `insertions` does not seem to compute an "improving sequence" in the
> sense of that paper, the technique seems certainly related.
>
> [1] Pruning with improving sequences in lazy functional programs
> Hideya Iwasaki,Takeshi Morimoto,Yasunao Takano
> http://link.springer.com/article/10.1007%2Fs10990-012-9086-3
>
> _______________________________________________
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> curry_at_lists.RWTH-Aachen.DE
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>
>


-- 
Julio Mariño
Babel research group
Universidad Politecnica de Madrid


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Received on Wed Oct 16 2013 - 13:01:00 CEST

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