带有 2 个递归调用的 F# 尾调用优化？

Question

当我编写这个函数时，我知道我不会得到尾调用优化。我还没有想出处理这个问题的好方法，希望其他人可以提供建议。

我有一棵树：

type Heap<'a> =
| E
| T of int * 'a * Heap<'a> * Heap<'a> 

我想计算其中有多少个节点：

let count h =
    let rec count' h acc =
        match h with 
        | E -> 0 + acc
        | T(_, value, leftChild, rightChild) ->
            let acc = 1 + acc
            (count' leftChild acc) + (count' rightChild acc)

    count' h 0

这不是因为添加了子节点的计数而没有优化。如果树有 100 万个节点，你知道如何让这样的事情工作吗？

谢谢，德里克

---

这里是使用 CPS 实现计数。尽管如此，它还是把堆栈搞砸了。

let count h =
    let rec count' h acc cont =
        match h with
        | E -> cont (1 + acc)
        | T(_,_,left,right) ->
            let f = (fun lc -> count' right lc cont)
            count' left acc f

    count' h 0 (fun (x: int) -> x)

也许我可以想出一些方法来将树分成足够多的块，这样我就可以在不破坏堆栈的情况下进行计数？

有人询问生成树的代码。它在下面。

member this.ParallelHeaps threads =
    let rand = new Random()
    let maxVal = 1000000

    let rec heaper i h =
        if i < 1 then
            h
        else
            let heap = LeftistHeap.insert (rand.Next(100,2 * maxVal)) h
            heaper (i - 1) heap

    let heaps = Array.create threads E
    printfn "Creating heap of %d elements, with %d threads" maxVal threads
    let startTime = DateTime.Now
    seq { for i in 0 .. (threads - 1) ->
          async { Array.set heaps i (heaper (maxVal / threads) E) }}
    |> Async.Parallel
    |> Async.RunSynchronously 
    |> ignore

    printfn "Creating %d sub-heaps took %f milliseconds" threads (DateTime.Now - startTime).TotalMilliseconds
    let startTime = DateTime.Now

    Array.length heaps |> should_ equal threads <| "The size of the heaps array should match the number of threads to process the heaps"

    let rec reMerge i h =
        match i with 
        | -1 -> h
        | _  -> 
            printfn "heap[%d].count = %d" i (LeftistHeap.count heaps.[i])
            LeftistHeap.merge heaps.[i] (reMerge (i-1) h)

    let heap = reMerge (threads-1) E
    printfn "Merging %d heaps took %f milliseconds" threads (DateTime.Now - startTime).TotalMilliseconds
    printfn "heap min: %d" (LeftistHeap.findMin heap)

    LeftistHeap.count heap |> should_ equal maxVal <| "The count of the reMerged heap should equal maxVal"

Answer 1

您可以使用延续传递样式 (CPS) 来解决该问题。请参阅 Matthew Podwysocki 的 Recursing on Recursion - Continuation Passing。

let tree_size_cont tree = 
  let rec size_acc tree acc cont = 
    match tree with 
    | Leaf _ -> cont (1 + acc) 
    | Node(_, left, right) -> 
         size_acc left acc (fun left_size -> 
         size_acc right left_size cont) 

  size_acc tree 0 (fun x -> x)

另请注意，在调试版本中，尾调用优化被禁用。如果不想在 Release 模式下运行，可以在 Visual Studio 的项目属性中启用优化。

Answer 2

CPS 是一个很好的通用解决方案，但您可能还想考虑显式使用堆栈，因为它会更快并且可以说更简单：

let count heap =
  let stack = System.Collections.Generic.Stack[heap]
  let mutable n = 0
  while stack.Count > 0 do
    match stack.Pop() with
    | E -> ()
    | T(_, _, heap1, heap2) ->
        n <- n + 1
        stack.Push heap1
        stack.Push heap2
  n