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(* *)
(* OCaml *)
(* *)
(* Xavier Leroy, projet Cristal, INRIA Rocquencourt *)
(* *)
(* Copyright 1996 Institut National de Recherche en Informatique et *)
(* en Automatique. All rights reserved. This file is distributed *)
(* under the terms of the GNU Library General Public License, with *)
(* the special exception on linking described in file ../LICENSE. *)
(* *)
(***********************************************************************)
(* Sets over ordered types *)
module type OrderedType =
sig
type t
val compare: t -> t -> int
end
module type S =
sig
type elt
type t
val empty: t
val is_empty: t -> bool
val mem: elt -> t -> bool
val add: elt -> t -> t
val singleton: elt -> t
val remove: elt -> t -> t
val union: t -> t -> t
val inter: t -> t -> t
val diff: t -> t -> t
val compare: t -> t -> int
val equal: t -> t -> bool
val subset: t -> t -> bool
val iter: (elt -> unit) -> t -> unit
val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a
val for_all: (elt -> bool) -> t -> bool
val exists: (elt -> bool) -> t -> bool
val filter: (elt -> bool) -> t -> t
val partition: (elt -> bool) -> t -> t * t
val cardinal: t -> int
val elements: t -> elt list
val min_elt: t -> elt
val max_elt: t -> elt
val choose: t -> elt
val split: elt -> t -> t * bool * t
val find: elt -> t -> elt
end
module Make(Ord: OrderedType) =
struct
type elt = Ord.t
type t = Empty | Node of t * elt * t * int
(* Sets are represented by balanced binary trees (the heights of the
children differ by at most 2 *)
let height = function
Empty -> 0
| Node(_, _, _, h) -> h
(* Creates a new node with left son l, value v and right son r.
We must have all elements of l < v < all elements of r.
l and r must be balanced and | height l - height r | <= 2.
Inline expansion of height for better speed. *)
let create l v r =
let hl = match l with Empty -> 0 | Node(_,_,_,h) -> h in
let hr = match r with Empty -> 0 | Node(_,_,_,h) -> h in
Node(l, v, r, (if hl >= hr then hl + 1 else hr + 1))
(* Same as create, but performs one step of rebalancing if necessary.
Assumes l and r balanced and | height l - height r | <= 3.
Inline expansion of create for better speed in the most frequent case
where no rebalancing is required. *)
let bal l v r =
let hl = match l with Empty -> 0 | Node(_,_,_,h) -> h in
let hr = match r with Empty -> 0 | Node(_,_,_,h) -> h in
if hl > hr + 2 then begin
match l with
Empty -> invalid_arg "Set.bal"
| Node(ll, lv, lr, _) ->
if height ll >= height lr then
create ll lv (create lr v r)
else begin
match lr with
Empty -> invalid_arg "Set.bal"
| Node(lrl, lrv, lrr, _)->
create (create ll lv lrl) lrv (create lrr v r)
end
end else if hr > hl + 2 then begin
match r with
Empty -> invalid_arg "Set.bal"
| Node(rl, rv, rr, _) ->
if height rr >= height rl then
create (create l v rl) rv rr
else begin
match rl with
Empty -> invalid_arg "Set.bal"
| Node(rll, rlv, rlr, _) ->
create (create l v rll) rlv (create rlr rv rr)
end
end else
Node(l, v, r, (if hl >= hr then hl + 1 else hr + 1))
(* Insertion of one element *)
let rec add x = function
Empty -> Node(Empty, x, Empty, 1)
| Node(l, v, r, _) as t ->
let c = Ord.compare x v in
if c = 0 then t else
if c < 0 then bal (add x l) v r else bal l v (add x r)
let singleton x = Node(Empty, x, Empty, 1)
(* Beware: those two functions assume that the added v is *strictly*
smaller (or bigger) than all the present elements in the tree; it
does not test for equality with the current min (or max) element.
Indeed, they are only used during the "join" operation which
respects this precondition.
*)
let rec add_min_element v = function
| Empty -> singleton v
| Node (l, x, r, h) ->
bal (add_min_element v l) x r
let rec add_max_element v = function
| Empty -> singleton v
| Node (l, x, r, h) ->
bal l x (add_max_element v r)
(* Same as create and bal, but no assumptions are made on the
relative heights of l and r. *)
let rec join l v r =
match (l, r) with
(Empty, _) -> add_min_element v r
| (_, Empty) -> add_max_element v l
| (Node(ll, lv, lr, lh), Node(rl, rv, rr, rh)) ->
if lh > rh + 2 then bal ll lv (join lr v r) else
if rh > lh + 2 then bal (join l v rl) rv rr else
create l v r
(* Smallest and greatest element of a set *)
let rec min_elt = function
Empty -> raise Not_found
| Node(Empty, v, r, _) -> v
| Node(l, v, r, _) -> min_elt l
let rec max_elt = function
Empty -> raise Not_found
| Node(l, v, Empty, _) -> v
| Node(l, v, r, _) -> max_elt r
(* Remove the smallest element of the given set *)
let rec remove_min_elt = function
Empty -> invalid_arg "Set.remove_min_elt"
| Node(Empty, v, r, _) -> r
| Node(l, v, r, _) -> bal (remove_min_elt l) v r
(* Merge two trees l and r into one.
All elements of l must precede the elements of r.
Assume | height l - height r | <= 2. *)
let merge t1 t2 =
match (t1, t2) with
(Empty, t) -> t
| (t, Empty) -> t
| (_, _) -> bal t1 (min_elt t2) (remove_min_elt t2)
(* Merge two trees l and r into one.
All elements of l must precede the elements of r.
No assumption on the heights of l and r. *)
let concat t1 t2 =
match (t1, t2) with
(Empty, t) -> t
| (t, Empty) -> t
| (_, _) -> join t1 (min_elt t2) (remove_min_elt t2)
(* Splitting. split x s returns a triple (l, present, r) where
- l is the set of elements of s that are < x
- r is the set of elements of s that are > x
- present is false if s contains no element equal to x,
or true if s contains an element equal to x. *)
let rec split x = function
Empty ->
(Empty, false, Empty)
| Node(l, v, r, _) ->
let c = Ord.compare x v in
if c = 0 then (l, true, r)
else if c < 0 then
let (ll, pres, rl) = split x l in (ll, pres, join rl v r)
else
let (lr, pres, rr) = split x r in (join l v lr, pres, rr)
(* Implementation of the set operations *)
let empty = Empty
let is_empty = function Empty -> true | _ -> false
let rec mem x = function
Empty -> false
| Node(l, v, r, _) ->
let c = Ord.compare x v in
c = 0 || mem x (if c < 0 then l else r)
let rec remove x = function
Empty -> Empty
| Node(l, v, r, _) ->
let c = Ord.compare x v in
if c = 0 then merge l r else
if c < 0 then bal (remove x l) v r else bal l v (remove x r)
let rec union s1 s2 =
match (s1, s2) with
(Empty, t2) -> t2
| (t1, Empty) -> t1
| (Node(l1, v1, r1, h1), Node(l2, v2, r2, h2)) ->
if h1 >= h2 then
if h2 = 1 then add v2 s1 else begin
let (l2, _, r2) = split v1 s2 in
join (union l1 l2) v1 (union r1 r2)
end
else
if h1 = 1 then add v1 s2 else begin
let (l1, _, r1) = split v2 s1 in
join (union l1 l2) v2 (union r1 r2)
end
let rec inter s1 s2 =
match (s1, s2) with
(Empty, t2) -> Empty
| (t1, Empty) -> Empty
| (Node(l1, v1, r1, _), t2) ->
match split v1 t2 with
(l2, false, r2) ->
concat (inter l1 l2) (inter r1 r2)
| (l2, true, r2) ->
join (inter l1 l2) v1 (inter r1 r2)
let rec diff s1 s2 =
match (s1, s2) with
(Empty, t2) -> Empty
| (t1, Empty) -> t1
| (Node(l1, v1, r1, _), t2) ->
match split v1 t2 with
(l2, false, r2) ->
join (diff l1 l2) v1 (diff r1 r2)
| (l2, true, r2) ->
concat (diff l1 l2) (diff r1 r2)
type enumeration = End | More of elt * t * enumeration
let rec cons_enum s e =
match s with
Empty -> e
| Node(l, v, r, _) -> cons_enum l (More(v, r, e))
let rec compare_aux e1 e2 =
match (e1, e2) with
(End, End) -> 0
| (End, _) -> -1
| (_, End) -> 1
| (More(v1, r1, e1), More(v2, r2, e2)) ->
let c = Ord.compare v1 v2 in
if c <> 0
then c
else compare_aux (cons_enum r1 e1) (cons_enum r2 e2)
let compare s1 s2 =
compare_aux (cons_enum s1 End) (cons_enum s2 End)
let equal s1 s2 =
compare s1 s2 = 0
let rec subset s1 s2 =
match (s1, s2) with
Empty, _ ->
true
| _, Empty ->
false
| Node (l1, v1, r1, _), (Node (l2, v2, r2, _) as t2) ->
let c = Ord.compare v1 v2 in
if c = 0 then
subset l1 l2 && subset r1 r2
else if c < 0 then
subset (Node (l1, v1, Empty, 0)) l2 && subset r1 t2
else
subset (Node (Empty, v1, r1, 0)) r2 && subset l1 t2
let rec iter f = function
Empty -> ()
| Node(l, v, r, _) -> iter f l; f v; iter f r
let rec fold f s accu =
match s with
Empty -> accu
| Node(l, v, r, _) -> fold f r (f v (fold f l accu))
let rec for_all p = function
Empty -> true
| Node(l, v, r, _) -> p v && for_all p l && for_all p r
let rec exists p = function
Empty -> false
| Node(l, v, r, _) -> p v || exists p l || exists p r
let rec filter p = function
Empty -> Empty
| Node(l, v, r, _) ->
(* call [p] in the expected left-to-right order *)
let l' = filter p l in
let pv = p v in
let r' = filter p r in
if pv then join l' v r' else concat l' r'
let rec partition p = function
Empty -> (Empty, Empty)
| Node(l, v, r, _) ->
(* call [p] in the expected left-to-right order *)
let (lt, lf) = partition p l in
let pv = p v in
let (rt, rf) = partition p r in
if pv
then (join lt v rt, concat lf rf)
else (concat lt rt, join lf v rf)
let rec cardinal = function
Empty -> 0
| Node(l, v, r, _) -> cardinal l + 1 + cardinal r
let rec elements_aux accu = function
Empty -> accu
| Node(l, v, r, _) -> elements_aux (v :: elements_aux accu r) l
let elements s =
elements_aux [] s
let choose = min_elt
let rec find x = function
Empty -> raise Not_found
| Node(l, v, r, _) ->
let c = Ord.compare x v in
if c = 0 then v
else find x (if c < 0 then l else r)
end
|