函数式编程——在f# /OCaml中实现类似快速排序函数的尾部递归版本

Question

是否有可能实现快速排序算法的尾部递归版本(通过延续模式)?如果是，又该如何实现呢?

正常(未优化)版本:

let rec quicksort list =
 match list with
 | [] -> []
 | element::[] -> [element]
 | pivot::rest -> let ``elements smaller than pivot``, ``elements larger or equal to pivot``= 
                    rest |> List.partition(fun element -> element < pivot)
                  quicksort ``elements smaller than pivot`` @ [pivot] @ quicksort ``elements larger or equal to pivot``

Answer 1

直接样式:

let rec quicksort list =
    match list with
    | [] -> []
    | [element] -> [element]
    | pivot::rest ->
        let left, right = List.partition (fun element -> element < pivot) rest in
        let sorted_left = quicksort left in
        let sorted_right = quicksort right in
        sorted_left @ [pivot] @ sorted_right

我的第一个简单的翻译与Laurent的版本非常相似，除了有点奇怪地缩进，以表明带有延续的调用实际上是一种绑定:

let rec quicksort list cont =
    match list with
    | [] -> cont []
    | element::[] -> cont [element]
    | pivot::rest ->
        let left, right = List.partition (fun element -> element < pivot) rest in
        quicksort left (fun sorted_left ->
        quicksort right (fun sorted_right ->
        cont (sorted_left @ [pivot] @ sorted_right)))
let qsort li = quicksort li (fun x -> x)

与Laurent相反，我发现很容易检查cont没有被遗忘:从直接样式翻译的CPS函数具有线性使用延续的属性，在每个分支的尾部位置一次且仅一次。很容易检查没有忘记这样的调用。

但实际上，对于大多数快速排序运行(假设您得到一个大致的对数行为，因为您不是不幸的，或者您先打乱了输入)，调用堆栈不是问题，因为它只以对数方式增长。更令人担忧的是对@的频繁调用，它的左参数是线性的。一种常见的优化技术是将函数定义为"向累加器列表添加输入"而不是返回列表:

let rec quicksort list accu =
    match list with
    | [] -> accu
    | element::[] -> element::accu
    | pivot::rest ->
        let left, right = List.partition (fun element -> element < pivot) rest in
        let sorted_right = quicksort right accu in
        quicksort left (pivot :: sorted_right)
let qsort li = quicksort li []

当然这也可以变成CPS:

let rec quicksort list accu cont =
    match list with
    | [] -> cont accu
    | element::[] -> cont (element::accu)
    | pivot::rest ->
        let left, right = List.partition (fun element -> element < pivot) rest in
        quicksort right accu (fun sorted_right ->
        quicksort left (pivot :: sorted_right) cont)
let qsort li = quicksort li [] (fun x -> x)    

现在，最后一个技巧是通过将延续转换为数据结构来"去功能化"它们(假设数据结构的分配比闭包的分配更有效):

type 'a cont =
  | Left of 'a list * 'a * 'a cont
  | Return
let rec quicksort list accu cont =
    match list with
    | [] -> eval_cont cont accu
    | element::[] -> eval_cont cont (element::accu)
    | pivot::rest ->
        let left, right = List.partition (fun element -> element < pivot) rest in
        quicksort right accu (Left (left, pivot, cont))
and eval_cont = function
  | Left (left, pivot, cont) ->
    (fun sorted_right -> quicksort left (pivot :: sorted_right) cont)
  | Return -> (fun x -> x)
let qsort li = quicksort li [] Return

最后，我为eval_cont选择了function .. fun风格，以表明这些只是来自CPS版本的代码片段，但下面的版本可能通过提高字体进行了更好的优化:

and eval_cont cont accu = match cont with
  | Left (left, pivot, cont) ->
    quicksort left (pivot :: accu) cont
  | Return -> accu

Answer 2

快速尝试，似乎有效:

let rec quicksort list cont =
    match list with
    | [] -> cont []
    | element::[] -> cont [element]
    | pivot::rest ->
        let ``elements smaller than pivot``, ``elements larger or equal to pivot`` =
            rest |> List.partition (fun element -> element < pivot)
        quicksort ``elements smaller than pivot``
            (fun x -> quicksort ``elements larger or equal to pivot`` (fun y -> cont (x @ [pivot] @ y)))
> quicksort [2; 6; 3; 8; 5; 1; 9; 4] id;;
val it : int list = [1; 2; 3; 4; 5; 6; 8; 9]

编辑:

当然，这段代码效率非常低。我希望没有人会在实际代码中使用它。代码不难编写，但是延续可能很难阅读并且容易出错(很容易忘记对cont的调用)。如果你想玩更多，你可以写一个延续单子(Brian写了一篇关于它的博客文章)。

Answer 3

Continuation monad(从这里偷来的)也可以使用(通常使代码更具可读性):

type ContinuationMonad() =
    // ma -> (a -> mb) -> mb
    member this.Bind (m, f) = fun c -> m (fun a -> f a c)
    // a -> ma
    member this.Return x = fun k -> k x
    // ma -> ma
    member this.ReturnFrom m = m
let cont = ContinuationMonad()
// Monadic definition of QuickSort
// it's shame F# doesn't allow us to use generic monad code
// (we need to use 'cont' monad here)
// otherwise we could run the same code as Identity monad, for instance
// producing direct (non-cont) behavior
let rec qsm = function
     |[]    -> cont.Return []
     |x::xs -> cont {
        let l,r = List.partition ((>=)x) xs
        let! ls = qsm l 
        let! rs = qsm r
        return (ls @ x :: rs) }
// Here we run our cont with id
let qs xs = qsm xs id     
printf "%A" (qs [2;6;3;8;5;1;9;4])