FSharpx.Collections包括一个功能性的Heap集合https://github.com/fsharp/fsharpx/blob/master/src/FSharpx.Core/Collections/Heap.fsi以及它的PriortityQueue接口https://github.com /fsharp/fsharpx/blob/master/src/FSharpx.Core/Collections/PriorityQueue.fs
令人惊讶的是,除了不再有Pervasives.compare函数和"比较"函数现已合并到基础运算符之外,接受的答案仍然几乎与F#的所有变化一起工作七年以上. Microsoft.FSharp.Core.Operators.compare.
也就是说,引用的博客条目将二项式堆实现为通用堆,而不是优先级队列的特定要求,而不需要优先级的泛型类型,这可以只是整数类型的比较效率,以及它说的但没有实现额外的改进,只保留最小值作为单独的字段来提高效率,只需检查队列中的最高优先级项目.
以下模块代码实现了从该代码派生的二项式堆优先级队列,其效率得到提高,它不使用通用比较进行优先级比较,而更有效的O(1)方法用于检查队列顶部(尽管在插入和删除条目的开销成本较高,尽管它们仍然是O(log n) - n是队列中的条目数).此代码更适合优先级队列的常规应用,其中队列的顶部比插入和/或顶级项删除更频繁地被读取.请注意,当删除顶部元素并将其重新插入队列中时,它不如MinHeap有效,因为完整的"deleteMin"和"插入"必须以更多的计算开销执行.代码如下:
[<RequireQualifiedAccess>]
module BinomialHeapPQ =
// type 'a treeElement = Element of uint32 * 'a
type 'a treeElement = class val k:uint32 val v:'a new(k,v) = { k=k;v=v } end
type 'a tree = Node of uint32 * 'a treeElement * 'a tree list
type 'a heap = 'a tree list
type 'a outerheap = | HeapEmpty | HeapNotEmpty of 'a treeElement * 'a heap
let empty = HeapEmpty
let isEmpty = function | HeapEmpty -> true | _ -> false
let inline private rank (Node(r,_,_)) = r
let inline private root (Node(_,x,_)) = x
exception Empty_Heap
let getMin = function | HeapEmpty -> None
| HeapNotEmpty(min,_) -> Some min
let rec private findMin heap =
match heap with | [] -> raise Empty_Heap //guarded so should never happen
| [node] -> root node,[]
| topnode::heap' ->
let min,subheap = findMin heap' in let rtn = root topnode
match subheap with
| [] -> if rtn.k > min.k then min,[] else rtn,[]
| minnode::heap'' ->
let rmn = root minnode
if rtn.k <= rmn.k then rtn,heap
else rmn,minnode::topnode::heap''
let private mergeTree (Node(r,kv1,ts1) as tree1) (Node (_,kv2,ts2) as tree2) =
if kv1.k > kv2.k then Node(r+1u,kv2,tree1::ts2)
else Node(r+1u,kv1,tree2::ts1)
let rec private insTree (newnode: 'a tree) heap =
match heap with
| [] -> [newnode]
| topnode::heap' -> if (rank newnode) < (rank topnode) then newnode::heap
else insTree (mergeTree newnode topnode) heap'
let insert k v = let kv = treeElement(k,v) in let nn = Node(0u,kv,[])
function | HeapEmpty -> HeapNotEmpty(kv,[nn])
| HeapNotEmpty(min,heap) -> let nmin = if k > min.k then min else kv
HeapNotEmpty(nmin,insTree nn heap)
let rec private merge' heap1 heap2 = //doesn't guaranty minimum tree node as head!!!
match heap1,heap2 with
| _,[] -> heap1
| [],_ -> heap2
| topheap1::heap1',topheap2::heap2' ->
match compare (rank topheap1) (rank topheap2) with
| -1 -> topheap1::merge' heap1' heap2
| 1 -> topheap2::merge' heap1 heap2'
| _ -> insTree (mergeTree topheap1 topheap2) (merge' heap1' heap2')
let merge oheap1 oheap2 = match oheap1,oheap2 with
| _,HeapEmpty -> oheap1
| HeapEmpty,_ -> oheap2
| HeapNotEmpty(min1,heap1),HeapNotEmpty(min2,heap2) ->
let min = if min1.k > min2.k then min2 else min1
HeapNotEmpty(min,merge' heap1 heap2)
let rec private removeMinTree = function
| [] -> raise Empty_Heap // will never happen as already guarded
| [node] -> node,[]
| t::ts -> let t',ts' = removeMinTree ts
if (root t).k <= (root t').k then t,ts else t',t::ts'
let deleteMin =
function | HeapEmpty -> HeapEmpty
| HeapNotEmpty(_,heap) ->
match heap with
| [] -> HeapEmpty // should never occur: non empty heap with no elements
| [Node(_,_,heap')] -> match heap' with
| [] -> HeapEmpty
| _ -> let min,_ = findMin heap'
HeapNotEmpty(min,heap')
| _::_ -> let Node(_,_,ts1),ts2 = removeMinTree heap
let nheap = merge' (List.rev ts1) ts2 in let min,_ = findMin nheap
HeapNotEmpty(min,nheap)
let reinsertMinAs k v pq = insert k v (deleteMin pq)
请注意,"treeElement"类型有两个选项,以适应测试方式.在我关于使用优先级队列筛选素数的答案中提到的应用程序中,上面的代码比MinHeap的功能实现慢了大约80%(非多处理模式,因为上面的二项式堆不适合于 - 场地调整); 这是因为二项式堆的"删除后插入"操作的额外计算复杂性,而不是为MinHeap实现有效地组合这些操作的能力.
因此,MinHeap优先级队列更适合这种类型的应用程序,并且还需要有效的就地调整,而二项式堆优先级队列更适合需要能够有效地将两个队列合并为一个的情况.
编辑：纠正纯函数版本的deleteMin函数中的错误并添加ofSeq函数。
我在关于 F# prime sieves 的答案中实现了两个版本的基于 MinHeap 二进制堆的优先级队列，第一个是纯函数代码（较慢），第二个是基于数组的（ResizeArray，它构建在 DotNet List 上，内部使用存储列表的数组）。非功能版本在某种程度上是合理的，因为在400 多年前Michael Eytzinger发明的基于谱系树的模型之后，MinHeap 通常被实现为可变数组二进制堆。
在该答案中，我没有实现“从队列中删除最高优先级项目”功能，因为算法不需要它，但我确实实现了“在队列中重新插入顶部项目”功能，因为算法确实需要它，并且函数与“deleteMin”函数所需的非常相似；区别在于，无需使用新参数重新插入顶部“最小”项目，而是只需从队列中删除最后一个项目（与插入新项目时的方式类似，但更简单），然后重新插入该项目以替换顶部队列中的（最小）项目（只需调用“reinsertMinAt”函数）。我还实现了一个“调整”函数，它将函数应用于所有队列元素，然后重新堆放最终结果以提高效率，该函数是该答案中埃拉托斯特尼算法分页筛的要求。
在下面的代码中，我实现了上面描述的“deleteMin”函数以及“ofSeq”函数，该函数可用于从使用内部“reheapify”函数的优先级/内容元组对元素序列构建新队列为了效率。
在与优先级“k”值相关的比较中，通过将大于符号更改为小于符号，可以轻松地将根据此代码的 MinHeap 更改为“MaxHeap”，反之亦然。最小/最大堆支持具有相同无符号整数“Key”优先级的多个元素，但不保留具有相同优先级的条目的顺序；换句话说，如果有其他条目具有与我不需要的相同优先级并且当前代码更高效，则不能保证进入队列的第一个元素将是弹出到最小位置的第一个元素。如果需要的话，可以修改代码以保留顺序（继续向下移动新插入，直到超过相同优先级的任何条目）。
最小/最大堆优先队列的优点是，与其他类型的非简单队列相比，它的计算复杂度开销较小，在 O(1) 时间内生成 Min 或 Max（取决于是 MinHeap 还是 MaxHeap 实现），最坏情况下插入和删除的时间为 O(log n)，而调整和构建只需要 O(n) 时间，其中“n”是队列中当前元素的数量。与删除然后插入相比，“resinsertMinAs”函数的优点在于，它将最坏情况下的时间从两倍减少到 O(log n)，并且通常比这更好，因为重新插入通常靠近队列的开头，因此不需要全面扫描。
与具有指向最小值的指针的附加选项以产生 O(1) 查找最小值性能的二项式堆相比，MinHeap 可能稍微简单一些，因此在执行相同的工作时速度更快，尤其是在不需要时二项式堆提供的“合并堆”功能。与使用 MinHeap 相比，使用二项式堆“合并”函数“reinsertMinAs”可能需要更长的时间，因为通常平均需要进行更多的比较。
MinHeap 优先级队列特别适合解决埃拉托色尼增量筛问题，如其他链接答案中所示，并且很可能是 Melissa E. O'Neill 在她的论文中使用的队列，表明特纳素数筛是无论是算法还是性能，都不是真正的埃拉托斯特尼筛法。
以下纯函数代码将“deleteMin”和“ofSeq”函数添加到该代码中：
[<RequireQualifiedAccess>]
module MinHeap =
type MinHeapTreeEntry<'T> = class val k:uint32 val v:'T new(k,v) = { k=k;v=v } end
[<CompilationRepresentation(CompilationRepresentationFlags.UseNullAsTrueValue)>]
[<NoEquality; NoComparison>]
type MinHeapTree<'T> =
| HeapEmpty
| HeapOne of MinHeapTreeEntry<'T>
| HeapNode of MinHeapTreeEntry<'T> * MinHeapTree<'T> * MinHeapTree<'T> * uint32
let empty = HeapEmpty
let getMin pq = match pq with | HeapOne(kv) | HeapNode(kv,_,_,_) -> Some kv | _ -> None
let insert k v pq =
let kv = MinHeapTreeEntry(k,v)
let rec insert' kv msk pq =
match pq with
| HeapEmpty -> HeapOne kv
| HeapOne kvn -> if k < kvn.k then HeapNode(kv,pq,HeapEmpty,2u)
else HeapNode(kvn,HeapOne kv,HeapEmpty,2u)
| HeapNode(kvn,l,r,cnt) ->
let nc = cnt + 1u
let nmsk = if msk <> 0u then msk <<< 1 else
let s = int32 (System.Math.Log (float nc) / System.Math.Log(2.0))
(nc <<< (32 - s)) ||| 1u //never ever zero again with the or'ed 1
if k <= kvn.k then if (nmsk &&& 0x80000000u) = 0u then HeapNode(kv,insert' kvn nmsk l,r,nc)
else HeapNode(kv,l,insert' kvn nmsk r,nc)
else if (nmsk &&& 0x80000000u) = 0u then HeapNode(kvn,insert' kv nmsk l,r,nc)
else HeapNode(kvn,l,insert' kv nmsk r,nc)
insert' kv 0u pq
let private reheapify kv k pq =
let rec reheapify' pq =
match pq with
| HeapEmpty | HeapOne _ -> HeapOne kv
| HeapNode(kvn,l,r,cnt) ->
match r with
| HeapOne kvr when k > kvr.k ->
match l with //never HeapEmpty
| HeapOne kvl when k > kvl.k -> //both qualify, choose least
if kvl.k > kvr.k then HeapNode(kvr,l,HeapOne kv,cnt)
else HeapNode(kvl,HeapOne kv,r,cnt)
| HeapNode(kvl,_,_,_) when k > kvl.k -> //both qualify, choose least
if kvl.k > kvr.k then HeapNode(kvr,l,HeapOne kv,cnt)
else HeapNode(kvl,reheapify' l,r,cnt)
| _ -> HeapNode(kvr,l,HeapOne kv,cnt) //only right qualifies
| HeapNode(kvr,_,_,_) when k > kvr.k -> //need adjusting for left leaf or else left leaf
match l with //never HeapEmpty or HeapOne
| HeapNode(kvl,_,_,_) when k > kvl.k -> //both qualify, choose least
if kvl.k > kvr.k then HeapNode(kvr,l,reheapify' r,cnt)
else HeapNode(kvl,reheapify' l,r,cnt)
| _ -> HeapNode(kvr,l,reheapify' r,cnt) //only right qualifies
| _ -> match l with //r could be HeapEmpty but l never HeapEmpty
| HeapOne(kvl) when k > kvl.k -> HeapNode(kvl,HeapOne kv,r,cnt)
| HeapNode(kvl,_,_,_) when k > kvl.k -> HeapNode(kvl,reheapify' l,r,cnt)
| _ -> HeapNode(kv,l,r,cnt) //just replace the contents of pq node with sub leaves the same
reheapify' pq
let reinsertMinAs k v pq =
let kv = MinHeapTreeEntry(k,v)
reheapify kv k pq
let deleteMin pq =
let rec delete' kv msk pq =
match pq with
| HeapEmpty -> kv,empty //should never get here as should flock off up before an empty is reached
| HeapOne kvn -> kvn,empty
| HeapNode(kvn,l,r,cnt) ->
let nmsk = if msk <> 0u then msk <<< 1 else
let s = int32 (System.Math.Log (float cnt) / System.Math.Log(2.0))
(cnt <<< (32 - s)) ||| 1u //never ever zero again with the or'ed 1
if (nmsk &&& 0x80000000u) = 0u then let kvl,pql = delete' kvn nmsk l
match pql with
| HeapEmpty -> kvl,HeapOne kvn
| HeapOne _ | HeapNode _ -> kvl,HeapNode(kvn,pql,r,cnt - 1u)
else let kvr,pqr = delete' kvn nmsk r
kvr,HeapNode(kvn,l,pqr,cnt - 1u)
match pq with
| HeapEmpty | HeapOne _ -> empty //for the case of deleting from queue either empty or one entry
| HeapNode(kv,_,_,cnt) -> let nkv,npq = delete' kv 0u pq in reinsertMinAs nkv.k nkv.v npq
let adjust f (pq:MinHeapTree<_>) = //adjust all the contents using the function, then rebuild by reheapify
let rec adjust' pq =
match pq with
| HeapEmpty -> pq
| HeapOne kv -> HeapOne(MinHeapTreeEntry(f kv.k kv.v))
| HeapNode (kv,l,r,cnt) -> let nkv = MinHeapTreeEntry(f kv.k kv.v)
reheapify nkv nkv.k (HeapNode(kv,adjust' l,adjust' r,cnt))
adjust' pq
let ofSeq (sq:seq<MinHeapTreeEntry<_>>) =
let cnt = sq |> Seq.length |> uint32 in let hcnt = cnt / 2u in let nmrtr = sq.GetEnumerator()
let rec build' i =
if nmrtr.MoveNext() && i <= cnt then
if i > hcnt then HeapOne(nmrtr.Current)
else let i2 = i + i in HeapNode(nmrtr.Current,build' i2,build' (i2 + 1u),cnt - i)
else HeapEmpty
build' 1u
以下代码将 deleteMin 和 ofSeq 函数添加到基于数组的版本中：
[<RequireQualifiedAccess>]
module MinHeap =
type MinHeapTreeEntry<'T> = class val k:uint32 val v:'T new(k,v) = { k=k;v=v } end
type MinHeapTree<'T> = ResizeArray<MinHeapTreeEntry<'T>>
let empty<'T> = MinHeapTree<MinHeapTreeEntry<'T>>()
let getMin (pq:MinHeapTree<_>) = if pq.Count > 0 then Some pq.[0] else None
let insert k v (pq:MinHeapTree<_>) =
if pq.Count = 0 then pq.Add(MinHeapTreeEntry(0xFFFFFFFFu,v)) //add an extra entry so there's always a right max node
let mutable nxtlvl = pq.Count in let mutable lvl = nxtlvl <<< 1 //1 past index of value added times 2
pq.Add(pq.[nxtlvl - 1]) //copy bottom entry then do bubble up while less than next level up
while ((lvl <- lvl >>> 1); nxtlvl <- nxtlvl >>> 1; nxtlvl <> 0) do
let t = pq.[nxtlvl - 1] in if t.k > k then pq.[lvl - 1] <- t else lvl <- lvl <<< 1; nxtlvl <- 0 //causes loop break
pq.[lvl - 1] <- MinHeapTreeEntry(k,v); pq
let reinsertMinAs k v (pq:MinHeapTree<_>) = //do minify down for value to insert
let mutable nxtlvl = 1 in let mutable lvl = nxtlvl in let cnt = pq.Count
while (nxtlvl <- nxtlvl <<< 1; nxtlvl < cnt) do
let lk = pq.[nxtlvl - 1].k in let rk = pq.[nxtlvl].k in let oldlvl = lvl
let k = if k > lk then lvl <- nxtlvl; lk else k in if k > rk then nxtlvl <- nxtlvl + 1; lvl <- nxtlvl
if lvl <> oldlvl then pq.[oldlvl - 1] <- pq.[lvl - 1] else nxtlvl <- cnt //causes loop break
pq.[lvl - 1] <- MinHeapTreeEntry(k,v); pq
let deleteMin (pq:MinHeapTree<_>) =
if pq.Count <= 2 then empty else //if contains one or less entries, return empty queue
let btmi = pq.Count - 2 in let btm = pq.[btmi] in pq.RemoveAt btmi
reinsertMinAs btm.k btm.v pq
let adjust f (pq:MinHeapTree<_>) = //adjust all the contents using the function, then re-heapify
if pq <> null then
let cnt = pq.Count
if cnt > 1 then
for i = 0 to cnt - 2 do //change contents using function
let e = pq.[i] in let k,v = e.k,e.v in pq.[i] <- MinHeapTreeEntry (f k v)
for i = cnt/2 downto 1 do //rebuild by reheapify
let kv = pq.[i - 1] in let k = kv.k
let mutable nxtlvl = i in let mutable lvl = nxtlvl
while (nxtlvl <- nxtlvl <<< 1; nxtlvl < cnt) do
let lk = pq.[nxtlvl - 1].k in let rk = pq.[nxtlvl].k in let oldlvl = lvl
let k = if k > lk then lvl <- nxtlvl; lk else k in if k > rk then nxtlvl <- nxtlvl + 1; lvl <- nxtlvl
if lvl <> oldlvl then pq.[oldlvl - 1] <- pq.[lvl - 1] else nxtlvl <- cnt //causes loop break
pq.[lvl - 1] <- kv
pq