平衡二分探索木を使ったsetとmultisetの実装(C#)
C++のset
set
順序付けされたデータを重複を排除して保持するもの。C#のSortedSet
multiset
順序付けされたデータを重複を許容しながら保持するもの。C#に類似のものはない。set
lower_bound()は該当値以上になる最小Indexを、upper_bound()は該当値より大きくなる最小Indexを返す関数で、いずれもO(logN)になる。
どちらも平衡二分探索木で実現できる。平衡二分探索木は二分探索木を、木構造がなるべく偏らないように工夫したものである(偏ると効率が悪くなる=O(N)に近くなる)。ここではRandomized Binary Search Tree(RBST)と呼ばれる、値を追加するときに、なるべく木の高さの偏りがなくなるよう、追加候補の位置に重みづけをしてランダム選択する方法で実装してみる。
詳細はこちらが参考になった
プログラミングコンテストでのデータ構造 2 ~平衡二分探索木編~
/// <summary> /// Self-Balancing Binary Search Tree /// (using Randamized BST) /// </summary> public class SB_BinarySearchTree<T> where T : IComparable { public class Node { public T Value; public Node LChild; public Node RChild; public int Count; //size of the sub tree public Node(T v) { Value = v; Count = 1; } } static Random _rnd = new Random(); public static int Count(Node t) { return t == null ? 0 : t.Count; } static Node Update(Node t) { t.Count = Count(t.LChild) + Count(t.RChild) + 1; return t; } public static Node Merge(Node l, Node r) { if (l == null || r == null) return l == null ? r : l; if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble()) { l.RChild = Merge(l.RChild, r); return Update(l); } else { r.LChild = Merge(l, r.LChild); return Update(r); } } /// <summary> /// split as [0, k), [k, n) /// </summary> public static Tuple<Node, Node> Split(Node t, int k) { if (t == null) return new Tuple<Node, Node>(null, null); if (k <= Count(t.LChild)) { var s = Split(t.LChild, k); t.LChild = s.Item2; return new Tuple<Node, Node>(s.Item1, Update(t)); } else { var s = Split(t.RChild, k - Count(t.LChild) - 1); t.RChild = s.Item1; return new Tuple<Node, Node>(Update(t), s.Item2); } } public static Node Remove(Node t, T v) { if (Find(t, v) == null) return t; return RemoveAt(t, LowerBound(t, v)); } public static Node RemoveAt(Node t, int k) { var s = Split(t, k); var s2 = Split(s.Item2, 1); return Merge(s.Item1, s2.Item2); } public static bool Contains(Node t, T v) { return Find(t, v) != null; } public static Node Find(Node t, T v) { while (t != null) { var cmp = t.Value.CompareTo(v); if (cmp > 0) t = t.LChild; else if (cmp < 0) t = t.RChild; else break; } return t; } public static Node FindByIndex(Node t, int idx) { if (t == null) return null; var currentIdx = Count(t) - Count(t.RChild) - 1; while (t != null) { if (currentIdx == idx) return t; if (currentIdx > idx) { t = t.LChild; currentIdx -= (Count(t == null ? null : t.RChild) + 1); } else { t = t.RChild; currentIdx += (Count(t == null ? null : t.LChild) + 1); } } return null; } public static int UpperBound(Node t, T v) { var torg = t; if (t == null) return -1; var ret = Int32.MaxValue; var idx = Count(t) - Count(t.RChild) - 1; while (t != null) { var cmp = t.Value.CompareTo(v); if (cmp > 0) { ret = Math.Min(ret, idx); t = t.LChild; idx -= (Count(t == null ? null : t.RChild) + 1); } else if (cmp <= 0) { t = t.RChild; idx += (Count(t == null ? null : t.LChild) + 1); } } return ret == Int32.MaxValue ? Count(torg) : ret; } public static int LowerBound(Node t, T v) { var torg = t; if (t == null) return -1; var idx = Count(t) - Count(t.RChild) - 1; var ret = Int32.MaxValue; while (t != null) { var cmp = t.Value.CompareTo(v); if (cmp >= 0) { if (cmp == 0) ret = Math.Min(ret, idx); t = t.LChild; if (t == null) ret = Math.Min(ret, idx); idx -= t == null ? 0 : (Count(t.RChild) + 1); } else if (cmp < 0) { t = t.RChild; idx += (Count(t == null ? null : t.LChild) + 1); if (t == null) return idx; } } return ret == Int32.MaxValue ? Count(torg) : ret; } public static Node Insert(Node t, T v) { var ub = LowerBound(t, v); return InsertByIdx(t, ub, v); } static Node InsertByIdx(Node t, int k, T v) { var s = Split(t, k); return Merge(Merge(s.Item1, new Node(v)), s.Item2); } public static IEnumerable<T> Enumerate(Node t) { var ret = new List<T>(); Enumerate(t, ret); return ret; } static void Enumerate(Node t, List<T> ret) { if (t == null) return; Enumerate(t.LChild, ret); ret.Add(t.Value); Enumerate(t.RChild, ret); } }
平衡二分探索木ができてしまえば、set
/// <summary> /// C-like set /// </summary> public class Set<T> where T : IComparable { protected SB_BinarySearchTree<T>.Node _root; public T this[int idx]{ get { return ElementAt(idx); } } public int Count() { return SB_BinarySearchTree<T>.Count(_root); } public virtual void Insert(T v) { if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v); else { if (SB_BinarySearchTree<T>.Find(_root, v) != null) return; _root = SB_BinarySearchTree<T>.Insert(_root, v); } } public void Clear() { _root = null; } public void Remove(T v) { _root = SB_BinarySearchTree<T>.Remove(_root, v); } public bool Contains(T v) { return SB_BinarySearchTree<T>.Contains(_root, v); } public T ElementAt(int k) { var node = SB_BinarySearchTree<T>.FindByIndex(_root, k); if (node == null) throw new IndexOutOfRangeException(); return node.Value; } public int Count(T v) { return SB_BinarySearchTree<T>.UpperBound(_root, v) - SB_BinarySearchTree<T>.LowerBound(_root, v); } public int LowerBound(T v) { return SB_BinarySearchTree<T>.LowerBound(_root, v); } public int UpperBound(T v) { return SB_BinarySearchTree<T>.UpperBound(_root, v); } public Tuple<int, int> EqualRange(T v) { if (!Contains(v)) return new Tuple<int, int>(-1, -1); return new Tuple<int, int>(SB_BinarySearchTree<T>.LowerBound(_root, v), SB_BinarySearchTree<T>.UpperBound(_root, v) - 1); } public List<T> ToList() { return new List<T>(SB_BinarySearchTree<T>.Enumerate(_root)); } }
set
/// <summary> /// C-like multiset /// </summary> public class MultiSet<T> : Set<T> where T : IComparable { public override void Insert(T v) { if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v); else _root = SB_BinarySearchTree<T>.Insert(_root, v); } }