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平衡二分探索木(RBST)

説明

RBST(Randomized Binary Search Tree)は平衡二分探索木の一種。ランダムなノードを根にして期待値的に木の高さをO(logN) に抑える。

遅延評価が不要な場合は引数が 3 つのコンストラクタを呼び出す。左からノードの大きさ, 二項演算, 単位元。

計算量

O(logN)

実装例

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template< class Monoid, class OperatorMonoid = Monoid >
struct RandomizedBinarySearchTree
{
  using F = function< Monoid(Monoid, Monoid) >;
  using G = function< Monoid(Monoid, OperatorMonoid) >;
  using H = function< OperatorMonoid(OperatorMonoid, OperatorMonoid) >;
  using P = function< OperatorMonoid(OperatorMonoid, int) >;

  inline int xor128()
  {
    static int x = 123456789;
    static int y = 362436069;
    static int z = 521288629;
    static int w = 88675123;
    int t;

    t = x ^ (x << 11);
    x = y;
    y = z;
    z = w;
    return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8));
  }

  struct Node
  {
    Node *l, *r;
    int cnt;
    Monoid key, sum;
    OperatorMonoid lazy;

    Node() {}

    Node(const Monoid &k, const OperatorMonoid &p) : cnt(1), key(k), sum(k), lazy(p), l(nullptr), r(nullptr) {}
  };

  vector< Node > pool;
  int ptr;

  const Monoid M1;
  const OperatorMonoid OM0;
  const F f;
  const G g;
  const H h;
  const P p;

  RandomizedBinarySearchTree(int sz, const F &f, const Monoid &M1) :
      pool(sz), ptr(0), f(f), g(G()), h(H()), p(P()), M1(M1), OM0(OperatorMonoid()) {}

  RandomizedBinarySearchTree(int sz, const F &f, const G &g, const H &h, const P &p,
                             const Monoid &M1, const OperatorMonoid &OM0) :
      pool(sz), ptr(0), f(f), g(g), h(h), p(p), M1(M1), OM0(OM0) {}

  inline Node *alloc(const Monoid &key) { return &(pool[ptr++] = Node(key, OM0)); }

  virtual Node *clone(Node *t) { return t; }

  inline int count(const Node *t) { return t ? t->cnt : 0; }

  inline Monoid sum(const Node *t) { return t ? t->sum : M1; }

  inline Node *update(Node *t)
  {
    t->cnt = count(t->l) + count(t->r) + 1;
    t->sum = f(f(sum(t->l), sum(t->r)), t->key);
    return t;
  }

  Node *propagete(Node *t)
  {
    t = clone(t);
    if(t->lazy != OM0) {
      t->key = g(t->key, t->lazy);
      if(t->l) {
        t->l = clone(t->l);
        t->l->lazy = h(t->l->lazy, t->lazy);
        t->l->sum = f(t->l->sum, p(t->lazy, count(t->l)));
      }
      if(t->r) {
        t->r = clone(t->r);
        t->r->lazy = h(t->r->lazy, t->lazy);
        t->r->sum = f(t->r->sum, p(t->lazy, count(t->r)));
      }
      t->lazy = OM0;
    }
    return update(t);
  }

  Node *merge(Node *l, Node *r)
  {
    if(!l || !r) return l ? l : r;
    if(xor128() % (l->cnt + r->cnt) < l->cnt) {
      l = propagete(l);
      l->r = merge(l->r, r);
      return update(l);
    } else {
      r = propagete(r);
      r->l = merge(l, r->l);
      return update(r);
    }
  }

  pair< Node *, Node * > split(Node *t, int k)
  {
    if(!t) return {t, t};
    t = propagete(t);
    if(k <= count(t->l)) {
      auto s = split(t->l, k);
      t->l = s.second;
      return {s.first, update(t)};
    } else {
      auto s = split(t->r, k - count(t->l) - 1);
      t->r = s.first;
      return {update(t), s.second};
    }
  }

  Node *build(int l, int r, const vector< Monoid > &v)
  {
    if(l + 1 >= r) return alloc(v[l]);
    return merge(build(l, (l + r) >> 1, v), build((l + r) >> 1, r, v));
  }

  Node *build(const vector< Monoid > &v)
  {
    ptr = 0;
    return build(0, (int) v.size(), v);
  }

  void dump(Node *r, typename vector< Monoid >::iterator &it)
  {
    if(!r) return;
    r = propagete(r);
    dump(r->l, it);
    *it = r->key;
    dump(r->r, ++it);
  }

  vector< Monoid > dump(Node *r)
  {
    vector< Monoid > v((size_t) count(r));
    auto it = begin(v);
    dump(r, it);
    return v;
  }

  string to_string(Node *r)
  {
    auto s = dump(r);
    string ret;
    for(int i = 0; i < s.size(); i++) ret += ", ";
    return (ret);
  }

  void insert(Node *&t, int k, const Monoid &v)
  {
    auto x = split(t, k);
    t = merge(merge(x.first, alloc(v)), x.second);
  }

  void erase(Node *&t, int k)
  {
    auto x = split(t, k);
    t = merge(x.first, split(x.second, 1).second);
  }

  Monoid query(Node *&t, int a, int b)
  {
    auto x = split(t, a);
    auto y = split(x.second, b - a);
    auto ret = sum(y.first);
    t = merge(x.first, merge(y.first, y.second));
    return ret;
  }

  void set_propagate(Node *&t, int a, int b, const OperatorMonoid &p)
  {
    auto x = split(t, a);
    auto y = split(x.second, b - a);
    y.first->lazy = h(y.first->lazy, p);
    t = merge(x.first, merge(propagete(y.first), y.second));
  }

  void set_element(Node *&t, int k, const Monoid &x)
  {
    t = propagete(t);
    if(k < count(t->l)) set_element(t->l, k, x);
    else if(k == count(t->l)) t->key = t->sum = x;
    else set_element(t->r, k - count(t->l) - 1, x);
    t = update(t);
  }


  int size(Node *t)
  {
    return count(t);
  }

  bool empty(Node *t)
  {
    return !t;
  }

  Node *makeset()
  {
    return (nullptr);
  }
};

応用 1: multiset, set

RBSTを単に multiset や set として使うことも可能。k 番目に小さい値を O(logn) で取得できる機能を追加で持つ。

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template< class T >
struct OrderedMultiSet : RandomizedBinarySearchTree< T >
{
  using RBST = RandomizedBinarySearchTree< T >;
  using Node = typename RBST::Node;

  OrderedMultiSet(int sz) : RBST(sz, [&](T x, T y) { return x; }, T()) {}

  T kth_element(Node *t, int k)
  {
    if(k < RBST::count(t->l)) return kth_element(t->l, k);
    if(k == RBST::count(t->l)) return t->key;
    return kth_element(t->r, k - RBST::count(t->l) - 1);
  }

  virtual void insert_key(Node *&t, const T &x)
  {
    RBST::insert(t, lower_bound(t, x), x);
  }

  void erase_key(Node *&t, const T &x)
  {
    if(!count(t, x)) return;
    RBST::erase(t, lower_bound(t, x));
  }

  int count(Node *t, const T &x)
  {
    return upper_bound(t, x) - lower_bound(t, x);
  }

  int lower_bound(Node *t, const T &x)
  {
    if(!t) return 0;
    if(x <= t->key) return lower_bound(t->l, x);
    return lower_bound(t->r, x) + RBST::count(t->l) + 1;
  }

  int upper_bound(Node *t, const T &x)
  {
    if(!t) return 0;
    if(x < t->key) return upper_bound(t->l, x);
    return upper_bound(t->r, x) + RBST::count(t->l) + 1;
  }
};
Copy
template< class T >
struct OrderedSet : OrderedMultiSet< T >
{
  using SET = OrderedMultiSet< T >;
  using RBST = typename SET::RBST;
  using Node = typename RBST::Node;

  OrderedSet(int sz) : OrderedMultiSet< T >(sz) {}

  void insert_key(Node *&t, const T &x) override
  {
    if(SET::count(t, x)) return;
    RBST::insert(t, SET::lower_bound(t, x), x);
  }
};

応用 2: 完全永続

永続をします（ア。コンストラクタに与える pool の大きさに十分な余裕を持つこと。

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template< class Monoid, class OperatorMonoid = Monoid >
struct PersistentRandomizedBinarySearchTree : RandomizedBinarySearchTree< Monoid, OperatorMonoid >
{
  using RBST = RandomizedBinarySearchTree< Monoid, OperatorMonoid >;
  using Node = typename RBST::Node;
  using F = typename RBST::F;
  using G = typename RBST::G;
  using H = typename RBST::H;
  using P = typename RBST::P;

  PersistentRandomizedBinarySearchTree(int sz, const F &f, const Monoid &M1) :
      RBST(sz, f, M1) {}

  PersistentRandomizedBinarySearchTree(int sz, const F &f, const G &g, const H &h, const P &p,
                                      const Monoid &M1, const OperatorMonoid &OM0) :
      RBST(sz, f, g, h, p, M1, OM0) {}

  Node *clone(Node *t) override { return &(RBST::pool[RBST::ptr++] = *t); }

  Node *rebuild(Node *r) { return RBST::build(RBST::dump(r)); }
};

問題例