Magicloud Magiclouds magicloud.magiclouds at gmail.com
Sun May 19 13:32:57 UTC 2019

```Awesome, thanks for the explanation.

On Sun, May 19, 2019 at 2:32 PM William Yager <will.yager at gmail.com> wrote:
>
> I realized that there is a simplification which makes the transformation more obvious. I should have put the base case inside the recursive step, rather than special-casing it:
>
> module Main where
> import Data.Map.Strict as Map
> import Data.Vector as Vec
>
>
> -- The recursive step
>
> rec :: (Int -> [[Int]]) -> Int -> [[Int]]
> rec rec 0 = [[]]
> rec rec n = do
>     this <- [1..n]
>     other <- rec (n - this)
>     return \$ (this : other)
>
> -- Non-dynamic
>
> sumsTo1 :: Int -> [[Int]]
> sumsTo1 = rec sumsTo1
>
> -- Dynamic (corecursive)
>
> sumsTo2 :: Int -> [[Int]]
> sumsTo2 n = lookup Map.! n
>     where
>     lookup = go 0 Map.empty
>     go m acc | m > n = acc
>              | otherwise = go (m + 1) (Map.insert m (rec (acc Map.!) m) acc)
>
> -- Dynamic (lazy)
>
> sumsTo3 :: Int -> [[Int]]
> sumsTo3 n = lookup Vec.! n
>     where
>     lookup = generate (n + 1) \$ rec (lookup Vec.!)
>
> main = do
>     let a = sumsTo1 10
>         b = sumsTo2 10
>         c = sumsTo3 10
>     print (a == b && b == c)
>     print a
>
> Also, to expand on this:
>
> * Corecursive DP is good in cases where you can figure out which order to generate things in, especially if you can drop no-longer-relevant data as you go
> * Lazy DP (using Vector) is good and fast in the case where the data is dense in the (n-dimensional) integers. Also very elegant!
> * If your DP dependency graph doesn't have any nice properties (not trivially dense in the integers, not easily predictable dependencies), you can implement your algorithm using e.g. a State monad over a map of cached values. However, I think this requires the recursive step to be written in terms of a monad rather than a non-monadic function (so that you can interrupt the control flow of the recursive step).
>
>
> On Sat, May 18, 2019 at 11:49 PM Magicloud Magiclouds <magicloud.magiclouds at gmail.com> wrote:
>>
>> Cool. Thanks.
>>
>> On Sat, May 18, 2019 at 10:18 PM William Yager <will.yager at gmail.com> wrote:
>> >
>> > Here are two mechanical strategies for implementing DP in haskell:
>> >
>> > module Main where
>> > import Data.Map.Strict as Map
>> > import Data.Vector as Vec
>> >
>> >
>> > -- The recursive step
>> >
>> > rec :: (Int -> [[Int]]) -> Int -> [[Int]]
>> > rec rec n = do
>> >     this <- [1..n]
>> >     other <- rec (n - this)
>> >     return \$ (this : other)
>> >
>> > -- Non-dynamic
>> >
>> > sumsTo1 :: Int -> [[Int]]
>> > sumsTo1 0 = [[]]
>> > sumsTo1 n = rec sumsTo1 n
>> >
>> > -- Dynamic (corecursive)
>> >
>> > sumsTo2 :: Int -> [[Int]]
>> > sumsTo2 n = lookup Map.! n
>> >     where
>> >     lookup = go 1 (Map.singleton 0 [[]])
>> >     go m acc | m > n = acc
>> >              | otherwise = go (m + 1) (Map.insert m (rec (acc Map.!) m) acc)
>> >
>> > -- Dynamic (lazy)
>> >
>> > sumsTo3 :: Int -> [[Int]]
>> > sumsTo3 n = lookup Vec.! n
>> >     where
>> >     lookup = generate (n + 1) \$ \m ->
>> >         if m == 0
>> >         then [[]]
>> >         else rec (lookup Vec.!) m
>> >
>> > main = do
>> >     let a = sumsTo1 10
>> >         b = sumsTo2 10
>> >         c = sumsTo3 10
>> >     print (a == b && b == c)
>> >     print a
>> >
>> > In case the formatting is messed up, see
>> > https://gist.github.com/wyager/7daebb351d802bbb2a624b71c0f343d3
>> >
>> > On Sat, May 18, 2019 at 10:15 PM Magicloud Magiclouds <magicloud.magiclouds at gmail.com> wrote:
>> >>
>> >> Thanks. This is kind like my original (did not get through) thought.
>> >>
>> >> On Sat, May 18, 2019 at 8:25 PM Thorkil Naur <naur at post11.tele.dk> wrote:
>> >> >
>> >> > Hello,
>> >> >
>> >> > On Sat, May 18, 2019 at 12:33:00PM +0800, Magicloud Magiclouds wrote:
>> >> > > ...
>> >> > > 1 - 9, nine numbers. Show all the possible combinations that sum up to
>> >> > > 10. Different orders are counted as the same.
>> >> > >
>> >> > > For example, [1, 4, 5].
>> >> >
>> >> > With
>> >> >
>> >> >   sumIs n [] = if n == 0 then [[]] else []
>> >> >   sumIs n (x:xs)
>> >> >     = (if n < x then
>> >> >         []
>> >> >       else
>> >> >         map (x:) \$ sumIs (n-x) xs
>> >> >       )
>> >> >       ++ sumIs n xs
>> >> >
>> >> > we can do:
>> >> >
>> >> > Prelude Main> sumIs 10 [1..9]
>> >> > [[1,2,3,4],[1,2,7],[1,3,6],[1,4,5],[1,9],[2,3,5],[2,8],[3,7],[4,6]]
>> >> >
>> >> > > ...
>> >> >
>> >> > Best
>> >> > Thorkil
>> >> _______________________________________________