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btree.ml
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module type ORDER = sig val order: int end module O32: ORDER = struct let order = 32 end module Map0 (Order: ORDER) (Key: Interfaces.SORTABLE) = struct include Order let odd_order: bool = assert (3 <= order); order / 2 * 2 < order let max_keys: int = order - 1 let min_keys: int = if odd_order then (order - 1) / 2 else order / 2 - 1 type key = Key.t type 'a pairs = (Key.t * 'a) array type 'a t = | Leaf of 'a pairs | Node of 'a pairs * 'a t array (* General functions. *) let is_empty (map: 'a t): bool = match map with | Leaf keys -> Array.length keys = 0 | Node _ -> false let rec cardinal (map: 'a t): int = match map with | Leaf keys -> Array.length keys | Node (keys, children) -> Array.fold_left (fun n child -> n + cardinal child) (Array.length keys) children let empty: 'a t = Leaf [||] let fold_left (f: 'a -> Key.t -> 'b -> 'a) (start: 'a) (map: 'b t): 'a = let rec fold accu map = match map with | Leaf pairs -> Array.fold_left (fun a (key,value) -> f a key value) accu pairs | Node (pairs, children) -> let n = Array.length pairs in assert (n + 1 = Array.length children); let rec fold_interior accu i = assert (i < Array.length children); if i = n then fold accu children.(i) else fold_interior (f (fold accu children.(i)) (fst pairs.(i)) (snd pairs.(i))) (i + 1) in fold_interior accu 0 in fold start map let fold_right (f: 'a -> Key.t -> 'b -> 'a) (start: 'a) (map: 'b t): 'a = let rec fold accu map = match map with | Leaf pairs -> Array.fold_right (fun (key,value) a -> f a key value) pairs accu | Node (pairs, children) -> let n = Array.length pairs in assert (n + 1 = Array.length children); let rec fold_interior accu i = assert (0 <= i); if i = 0 then accu else let i = i - 1 in assert (0 <= i); assert (i < Array.length children); fold_interior (fold (f accu (fst pairs.(i)) (snd pairs.(i))) children.(i)) i in fold_interior (fold accu children.(n)) n in fold start map let bindings (map: 'a t): (Key.t * 'a) list = fold_right (fun lst key value -> (key,value) :: lst) [] map let keys (map: 'a t): Key.t list = fold_right (fun lst key _ -> key :: lst) [] map (* Searching *) let bsearch (key: Key.t) (arr: 'a pairs): int * bool = Array.binsearch Key.compare fst key arr let rec find_opt (key: Key.t) (map: 'a t): 'a option = match map with | Leaf pairs -> let i, exact = bsearch key pairs in if exact then Some (snd pairs.(i)) else None | Node (pairs, children) -> let i, exact = bsearch key pairs in if exact then Some (snd pairs.(i)) else find_opt key children.(i) (* Insertion * ========= *) type 'a insert = | Normal_insert of 'a t | Split_insert of 'a t * (Key.t * 'a) * 'a t let subarray (arr: 'a array) (start: int) (beyond: int): 'a array = (* The subarray of [arr] starting at [start] and ending one before [beyond]. *) assert (0 <= start); assert (start <= beyond); assert (beyond <= Array.length arr); Array.sub arr start (beyond - start) let insert_subarray (arr: 'a array) (i: int) (x: 'a) (start: int) (beyond: int) : 'a array = (* The subarray of [arr] starting at [start] and ending one before [beyond] with [x] inserted at position [i]. *) assert (0 <= start); assert (start <= i); assert (i <= beyond); assert (beyond <= Array.length arr); let arr2 = Array.make (beyond - start + 1) x in Array.blit arr start arr2 0 (i - start); Array.blit arr i arr2 (i - start + 1) (beyond - i); arr2 let split_subarray (arr: 'a array) (i: int) (x: 'a) (y: 'a) (start: int) (beyond: int) : 'a array = (* The subarray of [arr] starting at [start] and ending one before [beyond] with [x] inserted at position [i] and the original value at position [i] replaced by [y]. *) assert (i < beyond); let arr = insert_subarray arr i x start beyond in arr.(i - start + 1) <- y; arr let add_in_leaf (key: Key.t) (value: 'a) (pairs: 'a pairs): 'a insert = let len = Array.length pairs in let i, exact = bsearch key pairs in if exact then Normal_insert (Leaf (Array.replace i (key, value) pairs)) else if len < max_keys then (* Leaf is not full. *) Normal_insert (Leaf (Array.insert i (key, value) pairs)) else (* Leaf is full *) let insert_subarray = insert_subarray pairs i (key, value) and k = order / 2 in if odd_order then if i = k then let left = subarray pairs 0 k and right = subarray pairs k len in Split_insert (Leaf left, (key, value), Leaf right) else if i < k then let left = insert_subarray 0 (k - 1) and right = subarray pairs k len in Split_insert (Leaf left, pairs.(k - 1), Leaf right) else let left = subarray pairs 0 k and right = insert_subarray (k + 1) len in Split_insert (Leaf left, pairs.(k), Leaf right) else begin (* even order *) if i < k then let left = insert_subarray 0 (k - 1) and right = subarray pairs k len in Split_insert (Leaf left, pairs.(k - 1), Leaf right) else let left = subarray pairs 0 (k - 1) and right = insert_subarray k len in Split_insert (Leaf left, pairs.(k - 1), Leaf right) end let add_in_node (i: int) (left: 'a t) (pair: Key.t * 'a) (right: 'a t) (pairs: 'a pairs) (children: 'a t array) : 'a insert = let len = Array.length pairs in if len < max_keys then let pairs = Array.insert i pair pairs and children = Array.insert i left children in assert (Array.valid_index (i + 1) children); children.(i + 1) <- right; Normal_insert (Node (pairs, children)) else (* Node is full. *) let k = order / 2 and insert_subarray = insert_subarray pairs i pair and split_subarray start beyond = split_subarray children i left right start beyond in if odd_order then if i = k then let left_pairs = subarray pairs 0 k and left_children = subarray children 0 (k + 1) and right_pairs = subarray pairs k len and right_children = subarray children k (len + 1) in assert (Array.valid_index k left_children); assert (Array.valid_index 0 right_children); left_children.(k) <- left; right_children.(0) <- right; Split_insert ( Node (left_pairs, left_children), pair, Node (right_pairs, right_children)) else if i < k then let left_pairs = insert_subarray 0 (k - 1) and left_children = split_subarray 0 k and right_pairs = subarray pairs k len and right_children = subarray children k (len + 1) in assert (Array.valid_index (k - 1) pairs); Split_insert ( Node (left_pairs, left_children), pairs.(k - 1), Node (right_pairs, right_children)) else begin let left_pairs = subarray pairs 0 k and left_children = subarray children 0 (k + 1) and right_pairs = insert_subarray (k + 1) len and right_children = split_subarray (k + 1) (len + 1) in assert (Array.valid_index k pairs); Split_insert ( Node (left_pairs, left_children), pairs.(k), Node (right_pairs, right_children)) end else begin (* even order *) if i < k then let left_pairs = insert_subarray 0 (k - 1) and left_children = split_subarray 0 k and right_pairs = subarray pairs k len and right_children = subarray children k (len + 1) in assert (Array.valid_index (k - 1) pairs); Split_insert ( Node (left_pairs, left_children), pairs.(k - 1), Node (right_pairs, right_children)) else let left_pairs = subarray pairs 0 (k - 1) and left_children = subarray children 0 k and right_pairs = insert_subarray k len and right_children = split_subarray k (len + 1) in assert (Array.valid_index (k - 1) pairs); Split_insert ( Node (left_pairs, left_children), pairs.(k - 1), Node (right_pairs, right_children)) end let rec add_aux (key: Key.t) (value: 'a) (map: 'a t): 'a insert = match map with | Leaf pairs -> add_in_leaf key value pairs | Node (pairs, children) -> let i, exact = bsearch key pairs in if exact then (* An exact match has been found. Therefore update the value. *) let pairs = Array.replace i (key,value) pairs in Normal_insert (Node (pairs, children)) else begin (** Add the key value pair into the [i]th child. *) assert (Array.valid_index i children); match add_aux key value children.(i) with | Normal_insert child -> let children = Array.replace i child children in Normal_insert (Node (pairs, children)) | Split_insert (u, y, v) -> add_in_node i u y v pairs children end let add (key: Key.t) (value: 'a) (map: 'a t): 'a t = match add_aux key value map with | Normal_insert map -> map | Split_insert (left, pair, right) -> (* tree grows at the root *) Node ([|pair|], [|left; right|]) (* Deletion * ======== *) type 'a delete = { tree: 'a t; (* The tree with the deleted key value pair. *) pair: Key.t * 'a; (* The deleted key value pair. *) underflow: bool; (* one key less than the minimal number *) } let not_minimal (pairs: 'a pairs): bool = min_keys < Array.length pairs let replace2 (i: int) (left: 'a t) (right: 'a t) (children: 'a t array) : 'a t array = let children = Array.copy children in assert (Array.valid_index i children); assert (Array.valid_index (i + 1) children); children.(i) <- left; children.(i + 1) <- right; children let rotate_keys (to_left: bool) (i: int) (left: 'a pairs) (parent: 'a pairs) (right: 'a pairs) : 'a pairs * 'a pairs * 'a pairs = let open Array in assert (valid_index i parent); if to_left then push parent.(i) left, replace i (first right) parent, remove_first right else remove_last left, replace i (last left) parent, push_front parent.(i) right let rotate_children (to_left: bool) (left: 'a t array) (right: 'a t array) : 'a t array * 'a t array = let open Array in if to_left then push (first right) left, remove_first right else remove_last left, push_front (last left) right let merge_keys (i: int) (left: 'a pairs) (parent: 'a pairs) (right: 'a pairs) : 'a pairs * 'a pairs = assert (Array.valid_index i parent); let len_left = Array.length left and len_right = Array.length right in let merged = Array.make (len_left + 1 + len_right) parent.(i) and parent = Array.remove i parent in Array.blit left 0 merged 0 len_left; Array.blit right 0 merged (len_left + 1) len_right; merged, parent let merge_leaves (i: int) (pair: Key.t * 'a) (pairs1: 'a pairs) (pairs2: 'a pairs) (pairs: 'a pairs) (children: 'a t array) : 'a delete = assert (i + 1 < Array.length children); let merged, pairs = merge_keys i pairs1 pairs pairs2 and children = Array.remove i children and underflow = Array.length pairs <= min_keys in children.(i) <- Leaf merged; {tree = Node (pairs, children); pair; underflow} let merge_nodes (i: int) (pair: Key.t * 'a) (pairs1: 'a pairs) (children1: 'a t array) (pairs2: 'a pairs) (children2: 'a t array) (pairs: 'a pairs) (children: 'a t array) : 'a delete = assert (i + 1 < Array.length children); let pairs_new, pairs = merge_keys i pairs1 pairs pairs2 and children = Array.remove i children and underflow = Array.length pairs <= min_keys and children_new = Array.append children1 children2 in children.(i) <- Node (pairs_new, children_new); {tree = Node (pairs, children); pair; underflow} let handle_underflow (i: int) (* Index of the child where the deletion occurred. *) (underflow_left: bool) (* Underflow happend in the left child? *) (left_child: 'a t) (right_child: 'a t) (pair: Key.t * 'a) (* The deleted key value pair. *) (pairs: 'a pairs) (* The key value pairs of the parent. *) (children: 'a t array) (* The children of the parent. *) : 'a delete = let not_minimal pairs1 pairs2 = if underflow_left then not_minimal pairs2 else not_minimal pairs1 in match left_child, right_child with | Leaf pairs1, Leaf pairs2 when not_minimal pairs1 pairs2 -> (* Right sibling is not minimal, rotate *) let pairs1, pairs, pairs2 = rotate_keys underflow_left i pairs1 pairs pairs2 in let children = replace2 i (Leaf pairs1) (Leaf pairs2) children in {tree = Node (pairs, children); pair; underflow = false} | Leaf pairs1, Leaf pairs2 -> (* Sibling is minimal, merge *) merge_leaves i pair pairs1 pairs2 pairs children | Node (pairs1, children1), Node (pairs2, children2) when not_minimal pairs1 pairs2 -> (* Sibling is not minimal, rotate *) let pairs1, pairs, pairs2 = rotate_keys underflow_left i pairs1 pairs pairs2 and children1, children2 = rotate_children underflow_left children1 children2 in let children = replace2 i (Node (pairs1, children1)) (Node (pairs2, children2)) children in {tree = Node (pairs, children); pair; underflow = false} | Node (pairs1, children1), Node (pairs2, children2) -> (* Sibling is minimal, merge *) merge_nodes i pair pairs1 children1 pairs2 children2 pairs children | _, _ -> assert false (* Cannot happen, tree is balanced. *) let handle_delete (i: int) (* Index of the child where the deletion occurred. *) (pair: Key.t * 'a) (* The deleted key value pair. *) (d: 'a delete) (* The new tree with the key value pair deleted. *) (pairs: 'a pairs) (* The key value pairs of the parent. *) (children: 'a t array) (* The children of the parent. *) : 'a delete = if not d.underflow then { tree = Node (pairs, Array.replace i d.tree children); pair; underflow = false } else let len = Array.length pairs in if i < len then handle_underflow i true d.tree children.(i + 1) pair pairs children else let i = i - 1 in handle_underflow i false children.(i) d.tree pair pairs children let rec remove_last (map: 'a t): 'a delete = match map with | Leaf pairs -> let len = Array.length pairs in assert (0 < len); let pair = Array.last pairs and pairs = Array.remove_last pairs and underflow = Array.length pairs <= min_keys in { tree = Leaf pairs; pair; underflow } | Node (pairs, children) -> let len = Array.length pairs in assert (len + 1 = Array.length children); let d = remove_last children.(len) in handle_delete len d.pair d pairs children let rec remove_aux (key: Key.t) (map: 'a t): 'a delete option = match map with | Leaf pairs -> let i, exact = bsearch key pairs in if exact then let pair = pairs.(i) and pairs = Array.remove i pairs and underflow = Array.length pairs <= min_keys in Some { tree = Leaf pairs; pair; underflow } else None | Node (pairs, children) -> let i, exact = bsearch key pairs in if exact then let d = remove_last children.(i) in let pair = pairs.(i) and pairs = Array.replace i d.pair pairs in Some (handle_delete i pair d pairs children) else Option.map (fun d -> handle_delete i d.pair d pairs children) (remove_aux key children.(i)) let remove (key: Key.t) (map: 'a t): 'a t = match remove_aux key map with | None -> map | Some d -> match d.tree with | Node (pairs, children) when Array.is_empty pairs -> (* tree shrinks at the root *) children.(0) | _ -> d.tree (* Update * ====== *) type 'a update = | Insert of 'a insert | Delete of 'a delete let rec update_aux (key: Key.t) (f: 'a option -> 'a option) (map: 'a t) : 'a update = match map with | Leaf pairs -> let i, exact = bsearch key pairs in if exact then match f (Some (snd pairs.(i))) with | None -> let pairs = Array.remove i pairs and pair = pairs.(i) and underflow = min_keys = Array.length pairs in Delete { tree = Leaf pairs; pair; underflow} | Some value -> Insert (Normal_insert (Leaf (Array.replace i (key,value) pairs))) else begin match f None with | None -> Insert (Normal_insert map) | Some value -> Insert (add_in_leaf key value pairs) end | Node (pairs, children) -> let i, exact = bsearch key pairs in if exact then match f (Some (snd pairs.(i))) with | None -> let d = remove_last children.(i) in let pair = pairs.(i) and pairs = Array.replace i d.pair pairs in Delete (handle_delete i pair d pairs children) | Some value -> Insert (Normal_insert (Node ( Array.replace i (key, value) pairs, children ))) else match update_aux key f children.(i) with | Insert (Normal_insert child) -> Insert (Normal_insert (Node ( pairs, Array.replace i child children ))) | Insert (Split_insert (u, y, v)) -> Insert (add_in_node i u y v pairs children) | Delete d -> Delete (handle_delete i d.pair d pairs children) let update (key: Key.t) (f: 'a option -> 'a option) (map: 'a t): 'a t = match update_aux key f map with | Insert (Normal_insert map) -> map | Insert (Split_insert (u, y, v)) -> Node ( [| y |], [| u; v |] ) | Delete d -> match d.tree with | Node (pairs, children) when Array.length pairs = 0 -> (* tree shrinks at the root *) children.(0) | _ -> d.tree (* Stream of key value pairs * ========================= *) type 'a entry = 'a pairs * 'a t array * int type 'a source = { top: 'a t * int; (* node/leaf and position within the node/leaf *) stack: 'a entry list; } let has_more (source: 'a source): bool = match source.top with | Leaf pairs, i -> i < Array.length pairs | Node (pairs, _ ), i -> i < Array.length pairs let peek (source: 'a source): Key.t * 'a = assert (has_more source); match source.top with | Leaf pairs, i -> pairs.(i) | Node (pairs, _ ), i -> pairs.(i) let rec down (tree: 'a t) (stack: 'a entry list): 'a source = (* Search for the first key value pair of [tree]. *) match tree with | Leaf pairs -> (* We are already on a leaf. The next item is the first key value * pair. *) {top = Leaf pairs, 0; stack} | Node (pairs, children) -> (* Search the first key value pair in the first child. Push the * first key value pair of the node onto the stack. *) down children.(0) ((pairs, children, 0) :: stack) let rec up (stack: 'a entry list): 'a source = (* Search the stack for a node which is positioned on a key value pair. * *) match stack with | [] -> {top = empty, 0; stack = []} | (pairs, children, i) :: stack -> if i < Array.length pairs then {top = Node (pairs, children), i; stack} else up stack let advance (source: 'a source): 'a source = assert (has_more source); match source.top with | Leaf pairs, i -> if i + 1 < Array.length pairs then {source with top = Leaf pairs, i + 1} else up source.stack | Node (pairs, children), i -> assert (i < Array.length pairs); down children.(i + 1) ((pairs, children, i + 1) :: source.stack) let make_source (tree: 'a t): 'a source = down tree [] module Source (Value: Interfaces.ANY) = struct type 'a map = 'a t type item = Key.t * Value.t type t = Value.t source let has_more = has_more let peek = peek let advance = advance let make = make_source end end module Set0 (Order: ORDER) (Key: Interfaces.SORTABLE) = struct module Map = Map0 (Order) (Key) type item = Key.t type t = unit Map.t let is_empty = Map.is_empty let cardinal = Map.cardinal let empty = Map.empty let fold_left (f: 'a -> Key.t -> 'a) (start: 'a) (set: t): 'a = Map.fold_left (fun a key _ -> f a key) start set let fold_right (f: 'a -> Key.t -> 'a) (start: 'a) (set: t): 'a = Map.fold_right (fun a key _ -> f a key) start set let mem (key: Key.t) (set: t): bool = match Map.find_opt key set with | None -> false | Some _ -> true let add (key: Key.t) (set: t): t = Map.add key () set let remove (key: Key.t) (set: t): t = Map.remove key set let elements (set: t): Key.t list = Map.keys set module Source = struct type set = t module M = Map.Source (Unit) type item = Key.t type t = M.t let has_more = M.has_more let peek (source: t): Key.t = M.peek source |> fst let advance = M.advance let make = M.make end end module Map (Key: Interfaces.SORTABLE) = struct include Map0 (O32) (Key) end module Set (Key: Interfaces.SORTABLE) = struct include Set0 (O32) (Key) end (* ================================================================ * Unit Tests * * with sets * ================================================================ *) module Set_order (Order: ORDER) = struct include Set0 (Order) (Int) let do_upward (f: int -> t -> t) (start: int) (beyond: int) (set: t): t = assert (start <= beyond); let rec action i set = if i = beyond then set else action (i + 1) (f i set) in action start set let do_downward (f: int -> t -> t) (start: int) (beyond: int) (set: t): t = assert (start <= beyond); let rec action i set = if i = start then set else let i = i - 1 in action i (f i set) in action beyond set let add_upward (start: int) (beyond: int) (set: t): t = do_upward add start beyond set let add_downward (start: int) (beyond: int) (set: t): t = do_downward add start beyond set let remove_upward (start: int) (beyond: int) (set: t): t = do_upward remove start beyond set let remove_downward (start: int) (beyond: int) (set: t): t = do_downward remove start beyond set let check_range (start: int) (beyond: int) (set: t): bool = let n, ok = fold_left (fun (i, ok) key -> i + 1, ok && key = i) (start, true) set in n = beyond && ok module Source = struct include Source let to_list (source: t): int list = let rec to_list source accu = if has_more source then to_list (advance source) (peek source :: accu) else List.rev accu in to_list source [] end end module Set3 = Set_order (struct let order = 3 end) module Set4 = Set_order (struct let order = 4 end) let string_of (lst: int list): string = "[" ^ String.concat ", " (List.map string_of_int lst) ^ "]" let _ = string_of let%test _ = let module Map = Map (Int) in Map.(cardinal empty) = 0 (* Insertion *) let%test _ = let set = Set4.(add_upward 100 200 empty) in Set4.check_range 100 200 set let%test _ = let set = Set4.(add_downward 0 100 empty) in Set4.check_range 0 100 set let%test _ = let set = Set3.(add_upward 100 200 empty) in Set3.check_range 100 200 set let%test _ = let set = Set3.(add_downward 0 100 empty) in Set3.check_range 0 100 set (* Deletion *) let%test _ = let set = Set3.(add_upward 0 200 empty |> remove_upward 0 100) in Set3.check_range 100 200 set let%test _ = let set = Set3.(add_upward 0 200 empty |> remove_downward 0 100) in Set3.check_range 100 200 set let%test _ = let set = Set4.(add_upward 0 200 empty |> remove_upward 0 100) in Set4.check_range 100 200 set let%test _ = let set = Set4.(add_upward 0 200 empty |> remove_downward 0 100) in Set4.check_range 100 200 set (* ================================================================ * Unit Tests * * with maps * ================================================================ *) module Map_order (Order: ORDER) = struct include Map0 (Order) (Int) let do_upward (f: int -> 'a t -> 'a t) (start: int) (beyond: int) (map: 'a t) : 'a t = assert (start <= beyond); let rec action i map = if i = beyond then map else action (i + 1) (f i map) in action start map let do_downward (f: int -> 'a t -> 'a t) (start: int) (beyond: int) (map: 'a t) : 'a t = assert (start <= beyond); let rec action i map = if i = start then map else let i = i - 1 in action i (f i map) in action beyond map let add_upward (start: int) (beyond: int) (f: int -> 'a) (map: 'a t): 'a t = do_upward (fun i map -> add i (f i) map) start beyond map let add_downward (start: int) (beyond: int) (f: int -> 'a) (map: 'a t): 'a t = do_downward (fun i map -> add i (f i) map) start beyond map let update_upward (start: int) (beyond: int) (f: int -> 'a option -> 'a option) (map: 'a t) : 'a t = do_upward (fun i map -> update i (f i) map) start beyond map let check_range (start: int) (beyond: int) (f: int -> 'a) (map: 'a t): bool = let n, ok = fold_left (fun (i, ok) key value -> i + 1, ok && f key = value) (start, true) map in n = beyond && ok end module Map3 = Map_order (struct let order = 3 end) (* Insertion *) let%test _ = let open Map3 in let map = add_upward 0 100 Fun.id empty in check_range 0 100 Fun.id map let%test _ = let open Map3 in let map = add_downward 0 100 Fun.id empty in check_range 0 100 Fun.id map (* Update *) let%test _ = let open Map3 in let map = update_upward 0 100 (fun i _ -> Some i) empty in check_range 0 100 Fun.id map let%test _ = let open Map3 in let map = add_upward 0 100 Fun.id empty |> update_upward 0 100 (fun _ -> Option.map (fun i -> 2 * i)) in check_range 0 100 (fun i -> 2 * i) map let%test _ = let open Map3 in let map = add_upward 0 100 Fun.id empty |> update_upward 0 100 (fun _ _ -> None) in is_empty map let%test _ = let open Map3 in let f i = if i / 2 * 2 = i then i else 2 * i in let map = add_upward 0 100 Fun.id empty |> add_upward 200 300 Fun.id |> update_upward 0 100 (fun _ -> Option.map f) |> update_upward 100 200 (fun i _ -> Some (f i)) |> update_upward 0 300 (fun i -> if i < 200 then Option.map Fun.id else fun _ -> None) in check_range 0 200 f map (* ================================================================ * Unit Tests * * with streams * ================================================================ *) let%test _ = let open Set3 in let module Source = Set3.Source in let set = add_upward 0 20 empty in Source.(make set |> to_list) = [0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]