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rb_array.ml
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module type BITSIZE = sig val bitsize: int end module Make (Bitsize: BITSIZE) = struct include Bitsize let branching: int = assert (0 < bitsize); 1 lsl bitsize let slot (i: int) (l: int): int = (* The slot of index [i] at level [l]. i/ 2^(l * bitsize) *) i lsr (l * bitsize) let offset (i: int) (s: int) (l: int): int = (* The offset of index [i] in slot [s] in a tree at level [l]. i- s * 2^(l * bitsize) *) i - s lsl (l * bitsize) let full_size (l: int): int = (* The size of a full radix balanced array at level [l]. *) assert (0 <= l); 1 lsl ((l + 1) * bitsize) type 'a t = | Leaf of 'a array | Node of { size: int; level: int; nodes: 'a t array} let level: 'a t -> int = function | Leaf _ -> 0 | Node node -> node.level let is_full: 'a t -> bool = function | Leaf arr -> Array.length arr = full_size 0 | Node node -> node.size = full_size node.level let length: 'a t -> int = (* The length of the radix balanced array. *) function | Leaf arr -> Array.length arr | Node node -> node.size let has_some (t: 'a t): bool = 0 < length t let is_empty (t: 'a t): bool = 0 = length t let check_invariant (t: 'a t): bool = let rec check is_root = function | Leaf arr -> let len = Array.length arr in len <= branching && (is_root || 0 < len) | Node node -> let nchildren = Array.length node.nodes in Array.for_all (check false) node.nodes && Array.for_all (fun child -> level child + 1 = node.level) node.nodes && nchildren <= branching && 1 <= nchildren && (not is_root || 2 <= nchildren) && ( node.size = Array.fold_left (fun size child -> size + length child) 0 node.nodes ) in check true t let empty: 'a t = Leaf [| |] (* Folding *) let fold_left (f: 'a -> 'b -> 'a) (start: 'a) (t: 'b t): 'a = let rec fold start = function | Leaf arr -> Array.fold_left f start arr | Node node -> Array.fold_left fold start node.nodes in fold start t let foldi_left (f: 'a -> int -> 'b -> 'a) (start: 'a) (t: 'b t): 'a = fold_left (fun (start,idx) e -> f start idx e, (idx + 1)) (start, 0) t |> fst (* Element Retrieval *) let rec element (i: int) (t: 'a t): 'a = (* The element at index [i] in the radix balanced array [t]. *) assert (0 <= i); assert (i < length t); match t with | Leaf arr -> arr.(i) | Node node -> let s = slot i node.level in let o = offset i s node.level in element o node.nodes.(s) let first (t: 'a t): 'a = (* The first element of the non empty radix balanced array [t]. *) assert (has_some t); let rec fst = function | Leaf arr -> Array.first arr | Node node -> fst (Array.first node.nodes) in fst t let last (t: 'a t): 'a = (* The last element of the non empty radix balanced array [t]. *) assert (has_some t); let rec fst = function | Leaf arr -> Array.last arr | Node node -> fst (Array.last node.nodes) in fst t (* Element Replacement *) let rec replace (i: int) (e: 'a) (t: 'a t): 'a t = (* Replace the element at index [i] by the element [e] within the radix balanced array [t]. *) assert (0 <= i); assert (i < length t); match t with | Leaf arr -> Leaf (Array.replace i e arr) | Node node -> let s = slot i node.level in let o = offset i s node.level in Node {node with nodes = Array.replace s (replace o e node.nodes.(s)) node.nodes } (* Element Insertion at the Rear End *) let rec singleton_tree (lev: int) (e: 'a): 'a t = (* Construct tree at level [lev] with the element [e]. *) if lev = 0 then Leaf [| e |] else Node { size = 1; level = lev; nodes = [| singleton_tree (lev - 1) e |] } let rec push_not_full (e: 'a) (t: 'a t): 'a t = (* Append the element [e] at the rear end of the radix balanced array [t] which is not full. *) assert (not (is_full t)); match t with | Leaf arr -> Leaf (Array.push e arr) | Node node -> let slot = Array.length node.nodes - 1 in assert (0 <= slot); let nodes = if is_full node.nodes.(slot) then Array.push (singleton_tree (node.level - 1) e) node.nodes else Array.replace slot (push_not_full e node.nodes.(slot)) node.nodes in Node {node with nodes; size = node.size + 1} let push (e: 'a) (t: 'a t): 'a t = (* Append the element [e] at the rear end of the radix balanced array [t]. *) let lev = level t and len = length t in if len = full_size lev then Node { size = len + 1; level = lev + 1; nodes = [| t; singleton_tree lev e|] } else push_not_full e t (* Element Removal from the Rear End *) let rec pop_aux (is_root: bool) (t: 'a t): 'a * 'a t = (* Remove the last element from a nonempty tree. *) assert (has_some t); match t with | Leaf arr -> Array.(last arr, Leaf (remove_last arr)) | Node node -> let j = Array.length node.nodes - 1 in assert (0 <= j); let child = node.nodes.(j) in let len = length child in if is_root && j = 1 && len = 1 then (* Last child of the root node has only one element. *) last child, node.nodes.(0) else let e, nodes = if len = 1 then (* Last child has only one element. *) last child, Array.remove_last node.nodes else (* Normal case. *) let e, child = pop_aux false child in e, Array.replace j child node.nodes in e, Node { node with size = node.size - 1; nodes } let pop (t: 'a t): 'a * 'a t = assert (has_some t); pop_aux true t let pop_opt (t: 'a t): ('a * 'a t) option = if is_empty t then None else Some (pop_aux true t) end module Branching2: BITSIZE = struct let bitsize: int = 1 end module Branching32: BITSIZE = struct let bitsize: int = 5 end include Make (Branching32) (* Unit Tests * ********** *) module Rb = Make (Branching2) let fill (start: int) (beyond: int): int Rb.t = assert (start <= beyond); let rec fl start t = if start = beyond then t else fl (start + 1) (Rb.push start t) in fl start Rb.empty let check_fill (start: int) (beyond: int): bool = let rec check start t = if start = beyond then Rb.check_invariant t else Rb.check_invariant t && check (start + 1) (Rb.push start t) in check start Rb.empty let check_fold (start: int) (beyond: int) (t: int Rb.t): bool = start + Rb.length t = beyond && Rb.foldi_left (fun ok idx e -> ok && e = start + idx) true t let check_element (start: int) (beyond: int) (t: int Rb.t): bool = let rec check_from i start = if start = beyond then true else start = Rb.element i t && check_from (i + 1) (start + 1) in check_from 0 start let check_pop (start: int) (beyond: int) (t: int Rb.t): bool = let rec check beyond t = Rb.check_invariant t && ( if beyond = start then Rb.is_empty t else Rb.has_some t && let e, t = Rb.pop t in e + 1 = beyond && check (beyond - 1) t ) in check beyond t let%test _ = let start = 10 and beyond = 100 in check_fill start beyond let%test _ = let start = 10 and beyond = 100 in check_fold start beyond (fill start beyond) let%test _ = let start = 10 and beyond = 100 in check_pop start beyond (fill start beyond) let%test _ = let start = 10 and beyond = 100 in check_element start beyond (fill start beyond) let%test _ = Rb.(check_invariant empty) let%test _ = Rb.check_invariant (fill 0 25)