module Make = functor (X : Set.OrderedType) -> struct type elt = X.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 x and right son r. l and r must be balanced and | height l - height r | <= 2. Inline expansion of height for better speed. *) let create l x 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, x, 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. Inline expansion of create for better speed in the most frequent case where no rebalancing is required. *) let bal l x 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 x r) else begin match lr with Empty -> invalid_arg "Set.bal" | Node(lrl, lrv, lrr, _)-> create (create ll lv lrl) lrv (create lrr x 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 x rl) rv rr else begin match rl with Empty -> invalid_arg "Set.bal" | Node(rll, rlv, rlr, _) -> create (create l x rll) rlv (create rlr rv rr) end end else Node(l, x, r, (if hl >= hr then hl + 1 else hr + 1)) (* Same as bal, but repeat rebalancing until the final result is balanced. *) let rec join l x r = match bal l x r with Empty -> invalid_arg "Set.join" | Node(l', x', r', _) as t' -> let d = height l' - height r' in if d < -2 or d > 2 then join l' x' r' else t' (* Merge two trees l and r into one. All elements of l must precede the elements of r. Assumes | height l - height r | <= 2. *) let rec merge t1 t2 = match (t1, t2) with (Empty, t) -> t | (t, Empty) -> t | (Node(l1, v1, r1, h1), Node(l2, v2, r2, h2)) -> bal l1 v1 (bal (merge r1 l2) v2 r2) (* Same as merge, but does not assume anything about l and r. *) let rec concat t1 t2 = match (t1, t2) with (Empty, t) -> t | (t, Empty) -> t | (Node(l1, v1, r1, h1), Node(l2, v2, r2, h2)) -> join l1 v1 (join (concat r1 l2) v2 r2) (* Splitting *) let rec split x = function Empty -> (Empty, None, Empty) | Node(l, v, r, _) -> let c = X.compare x v in if c = 0 then (l, Some v, r) else if c < 0 then let (ll, vl, rl) = split x l in (ll, vl, join rl v r) else let (lr, vr, rr) = split x r in (join l v lr, vr, 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 = X.compare x v in c = 0 || mem x (if c < 0 then l else r) let rec add x = function Empty -> Node(Empty, x, Empty, 1) | Node(l, v, r, _) as t -> let c = X.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) let rec remove x = function Empty -> Empty | Node(l, v, r, _) -> let c = X.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, None, r2) -> concat (inter l1 l2) (inter r1 r2) | (l2, Some _, 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, None, r2) -> join (diff l1 l2) v1 (diff r1 r2) | (l2, Some _, r2) -> concat (diff l1 l2) (diff r1 r2) let rec compare_aux l1 l2 = match (l1, l2) with ([], []) -> 0 | ([], _) -> -1 | (_, []) -> 1 | (Empty :: t1, Empty :: t2) -> compare_aux t1 t2 | (Node(Empty, v1, r1, _) :: t1, Node(Empty, v2, r2, _) :: t2) -> let c = compare v1 v2 in if c <> 0 then c else compare_aux (r1::t1) (r2::t2) | (Node(l1, v1, r1, _) :: t1, t2) -> compare_aux (l1 :: Node(Empty, v1, r1, 0) :: t1) t2 | (t1, Node(l2, v2, r2, _) :: t2) -> compare_aux t1 (l2 :: Node(Empty, v2, r2, 0) :: t2) let compare s1 s2 = compare_aux [s1] [s2] 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 = X.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 l (f v (fold f r accu)) 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 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 let choose = min_elt end