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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(*s Heaps *)
module type Ordered = sig
type t
val compare : t -> t -> int
end
module type S =sig
(* Type of functional heaps *)
type t
(* Type of elements *)
type elt
(* The empty heap *)
val empty : t
(* [add x h] returns a new heap containing the elements of [h], plus [x];
complexity $O(log(n))$ *)
val add : elt -> t -> t
(* [maximum h] returns the maximum element of [h]; raises [EmptyHeap]
when [h] is empty; complexity $O(1)$ *)
val maximum : t -> elt
(* [remove h] returns a new heap containing the elements of [h], except
the maximum of [h]; raises [EmptyHeap] when [h] is empty;
complexity $O(log(n))$ *)
val remove : t -> t
(* usual iterators and combinators; elements are presented in
arbitrary order *)
val iter : (elt -> unit) -> t -> unit
val fold : (elt -> 'a -> 'a) -> t -> 'a -> 'a
end
exception EmptyHeap
(*s Functional implementation *)
module Functional(X : Ordered) = struct
(* Heaps are encoded as Braun trees, that are binary trees
where size r <= size l <= size r + 1 for each node Node (l, x, r) *)
type t =
| Leaf
| Node of t * X.t * t
type elt = X.t
let empty = Leaf
let rec add x = function
| Leaf ->
Node (Leaf, x, Leaf)
| Node (l, y, r) ->
if X.compare x y >= 0 then
Node (add y r, x, l)
else
Node (add x r, y, l)
let rec extract = function
| Leaf ->
assert false
| Node (Leaf, y, r) ->
assert (r = Leaf);
y, Leaf
| Node (l, y, r) ->
let x, l = extract l in
x, Node (r, y, l)
let is_above x = function
| Leaf -> true
| Node (_, y, _) -> X.compare x y >= 0
let rec replace_min x = function
| Node (l, _, r) when is_above x l && is_above x r ->
Node (l, x, r)
| Node ((Node (_, lx, _) as l), _, r) when is_above lx r ->
(* lx <= x, rx necessarily *)
Node (replace_min x l, lx, r)
| Node (l, _, (Node (_, rx, _) as r)) ->
(* rx <= x, lx necessarily *)
Node (l, rx, replace_min x r)
| Leaf | Node (Leaf, _, _) | Node (_, _, Leaf) ->
assert false
(* merges two Braun trees [l] and [r],
with the assumption that [size r <= size l <= size r + 1] *)
let rec merge l r = match l, r with
| _, Leaf ->
l
| Node (ll, lx, lr), Node (_, ly, _) ->
if X.compare lx ly >= 0 then
Node (r, lx, merge ll lr)
else
let x, l = extract l in
Node (replace_min x r, ly, l)
| Leaf, _ ->
assert false (* contradicts the assumption *)
let maximum = function
| Leaf -> raise EmptyHeap
| Node (_, x, _) -> x
let remove = function
| Leaf ->
raise EmptyHeap
| Node (l, _, r) ->
merge l r
let rec iter f = function
| Leaf -> ()
| Node (l, x, r) -> iter f l; f x; iter f r
let rec fold f h x0 = match h with
| Leaf ->
x0
| Node (l, x, r) ->
fold f l (fold f r (f x x0))
end