Permutation Tree(順列木) (structure/others/permutation-tree.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "structure/others/permutation-tree.hpp"
• Link: https://codeforces.com/blog/entry/78898

概要

順列 $A$ に対応する順列木(Permutation Tree, 析合树) を構築する.

$ \max\limits_{l \leq i \leq r} A_i - \min\limits_{l \leq i \leq r} A_i = r-l$ が成立するような $[l, r]$ を良い連続部分列と定義する. この良い連続部分列を効率的に求める.

順列木のそれぞれのノードはいくつかのタイプに分類される. 定義を以下に示す.

• LEAF: 葉ノード. 数列のある $1$ 要素に対応する.
• JOIN: そのノードが持つ子の列から、連続する $2$ つ以上の子を選んだときに全てが良い連続部分列となる.
• CUT: そのノードが持つ子の列から、どの連続する $2$ つ以上の子を選んでも良い連続部分列にはならない.

使い方

• build(A): 順列 A に対応する順列木を返す. A には $0$ 以上 $n - 1$ 以下の値が $1$ つずる含まれることを要求する. Node には、ノードのタイプ type, そのノードに対応する順列の index の区間 $[l, r)$, ノードに含まれる要素の値 [min_v, max_v), 子 ch が格納される.

計算量

• $O(n \log n)$

Depends on

• Lazy-Segment-Tree(遅延伝搬セグメント木) (structure/segment-tree/lazy-segment-tree.hpp)

Verified with

• test/verify/yukicoder-1720.test.cpp

Code

#include "../segment-tree/lazy-segment-tree.hpp"

/**
 * @brief Permutation Tree(順列木)
 *
 * @see https://codeforces.com/blog/entry/78898
 */
struct PermutationTree {
 public:
  enum NodeType { JOIN_ASC, JOIN_DESC, LEAF, CUT };

  struct Node {
    NodeType type;
    int l, r;          // [l, r)
    int min_v, max_v;  // [min_v, max_v)
    vector<Node *> ch;

    size_t size() const { return r - l; }

    bool is_join() const { return type == JOIN_ASC or type == JOIN_DESC; };

    bool is_leaf() const { return type == LEAF; }

    bool is_cut() const { return type == CUT; }
  };

  using NP = Node *;

  PermutationTree() = default;

 private:
  static void add_child(NP t, NP c) {
    t->ch.emplace_back(c);
    t->l = min(t->l, c->l);
    t->r = max(t->r, c->r);
    t->min_v = min(t->min_v, c->min_v);
    t->max_v = max(t->max_v, c->max_v);
  }

 public:
  static NP build(vector<int> &A) {
    int n = (int)A.size();

    vector<int> desc{-1};
    vector<int> asc{-1};
    vector<NP> st;

    auto f = [](int a, int b) { return min(a, b); };
    auto g = [](int a, int b) { return a + b; };
    constexpr int lim = (1 << 30) - 1;
    auto seg = get_lazy_segment_tree(vector<int>(n), f, g, g, lim, 0);

    for (int i = 0; i < n; i++) {
      while (~desc.back() and A[i] > A[desc.back()]) {
        seg.apply(desc[desc.size() - 2] + 1, desc.back() + 1,
                  A[i] - A[desc.back()]);
        desc.pop_back();
      }
      while (~asc.back() and A[i] < A[asc.back()]) {
        seg.apply(asc[asc.size() - 2] + 1, asc.back() + 1,
                  A[asc.back()] - A[i]);
        asc.pop_back();
      }
      desc.emplace_back(i);
      asc.emplace_back(i);

      NP t = new Node{LEAF, i, i + 1, A[i], A[i] + 1, {}};
      for (;;) {
        NodeType type = CUT;
        if (not st.empty()) {
          if (st.back()->max_v == t->min_v) {
            type = JOIN_ASC;
          } else if (t->max_v == st.back()->min_v) {
            type = JOIN_DESC;
          }
        }
        if (type != CUT) {
          NP r = st.back();
          if (type != r->type) {
            r = new Node{type, r->l, r->r, r->min_v, r->max_v, {r}};
          }
          add_child(r, t);
          st.pop_back();
          t = r;
        } else if (seg.prod(0, i + 1 - (int)t->size()) == 0) {
          t = new Node{CUT, t->l, t->r, t->min_v, t->max_v, {t}};
          do {
            add_child(t, st.back());
            st.pop_back();
          } while (t->max_v - t->min_v != t->size());
          reverse(begin(t->ch), end(t->ch));
        } else {
          break;
        }
      }
      st.emplace_back(t);
      seg.apply(0, i + 1, -1);
    }
    return st[0];
  }
};

#line 1 "structure/segment-tree/lazy-segment-tree.hpp"
/**
 * @brief Lazy-Segment-Tree(遅延伝搬セグメント木)
 *
 */
template <typename T, typename E, typename F, typename G, typename H>
struct LazySegmentTree {
 private:
  int n{}, sz{}, height{};
  vector<T> data;
  vector<E> lazy;
  const F f;
  const G g;
  const H h;
  const T ti;
  const E ei;

  inline void update(int k) { data[k] = f(data[2 * k + 0], data[2 * k + 1]); }

  inline void all_apply(int k, const E &x) {
    data[k] = g(data[k], x);
    if (k < sz) lazy[k] = h(lazy[k], x);
  }

  inline void propagate(int k) {
    if (lazy[k] != ei) {
      all_apply(2 * k + 0, lazy[k]);
      all_apply(2 * k + 1, lazy[k]);
      lazy[k] = ei;
    }
  }

 public:
  LazySegmentTree() = default;

  explicit LazySegmentTree(int n, const F f, const G g, const H h, const T &ti,
                           const E &ei)
      : n(n), f(f), g(g), h(h), ti(ti), ei(ei) {
    sz = 1;
    height = 0;
    while (sz < n) sz <<= 1, height++;
    data.assign(2 * sz, ti);
    lazy.assign(2 * sz, ei);
  }

  explicit LazySegmentTree(const vector<T> &v, const F f, const G g, const H h,
                           const T &ti, const E &ei)
      : LazySegmentTree(v.size(), f, g, h, ti, ei) {
    build(v);
  }

  void build(const vector<T> &v) {
    assert(n == (int)v.size());
    for (int k = 0; k < n; k++) data[k + sz] = v[k];
    for (int k = sz - 1; k > 0; k--) update(k);
  }

  void set(int k, const T &x) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    data[k] = x;
    for (int i = 1; i <= height; i++) update(k >> i);
  }

  T get(int k) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    return data[k];
  }

  T operator[](int k) { return get(k); }

  T prod(int l, int r) {
    if (l >= r) return ti;
    l += sz;
    r += sz;
    for (int i = height; i > 0; i--) {
      if (((l >> i) << i) != l) propagate(l >> i);
      if (((r >> i) << i) != r) propagate((r - 1) >> i);
    }
    T L = ti, R = ti;
    for (; l < r; l >>= 1, r >>= 1) {
      if (l & 1) L = f(L, data[l++]);
      if (r & 1) R = f(data[--r], R);
    }
    return f(L, R);
  }

  T all_prod() const { return data[1]; }

  void apply(int k, const E &x) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    data[k] = g(data[k], x);
    for (int i = 1; i <= height; i++) update(k >> i);
  }

  void apply(int l, int r, const E &x) {
    if (l >= r) return;
    l += sz;
    r += sz;
    for (int i = height; i > 0; i--) {
      if (((l >> i) << i) != l) propagate(l >> i);
      if (((r >> i) << i) != r) propagate((r - 1) >> i);
    }
    {
      int l2 = l, r2 = r;
      for (; l < r; l >>= 1, r >>= 1) {
        if (l & 1) all_apply(l++, x);
        if (r & 1) all_apply(--r, x);
      }
      l = l2, r = r2;
    }
    for (int i = 1; i <= height; i++) {
      if (((l >> i) << i) != l) update(l >> i);
      if (((r >> i) << i) != r) update((r - 1) >> i);
    }
  }

  template <typename C>
  int find_first(int l, const C &check) {
    if (l >= n) return n;
    l += sz;
    for (int i = height; i > 0; i--) propagate(l >> i);
    T sum = ti;
    do {
      while ((l & 1) == 0) l >>= 1;
      if (check(f(sum, data[l]))) {
        while (l < sz) {
          propagate(l);
          l <<= 1;
          auto nxt = f(sum, data[l]);
          if (not check(nxt)) {
            sum = nxt;
            l++;
          }
        }
        return l + 1 - sz;
      }
      sum = f(sum, data[l++]);
    } while ((l & -l) != l);
    return n;
  }

  template <typename C>
  int find_last(int r, const C &check) {
    if (r <= 0) return -1;
    r += sz;
    for (int i = height; i > 0; i--) propagate((r - 1) >> i);
    T sum = ti;
    do {
      r--;
      while (r > 1 and (r & 1)) r >>= 1;
      if (check(f(data[r], sum))) {
        while (r < sz) {
          propagate(r);
          r = (r << 1) + 1;
          auto nxt = f(data[r], sum);
          if (not check(nxt)) {
            sum = nxt;
            r--;
          }
        }
        return r - sz;
      }
      sum = f(data[r], sum);
    } while ((r & -r) != r);
    return -1;
  }
};

template <typename T, typename E, typename F, typename G, typename H>
LazySegmentTree<T, E, F, G, H> get_lazy_segment_tree(int N, const F &f,
                                                     const G &g, const H &h,
                                                     const T &ti, const E &ei) {
  return LazySegmentTree{N, f, g, h, ti, ei};
}

template <typename T, typename E, typename F, typename G, typename H>
LazySegmentTree<T, E, F, G, H> get_lazy_segment_tree(const vector<T> &v,
                                                     const F &f, const G &g,
                                                     const H &h, const T &ti,
                                                     const E &ei) {
  return LazySegmentTree{v, f, g, h, ti, ei};
}
#line 2 "structure/others/permutation-tree.hpp"

/**
 * @brief Permutation Tree(順列木)
 *
 * @see https://codeforces.com/blog/entry/78898
 */
struct PermutationTree {
 public:
  enum NodeType { JOIN_ASC, JOIN_DESC, LEAF, CUT };

  struct Node {
    NodeType type;
    int l, r;          // [l, r)
    int min_v, max_v;  // [min_v, max_v)
    vector<Node *> ch;

    size_t size() const { return r - l; }

    bool is_join() const { return type == JOIN_ASC or type == JOIN_DESC; };

    bool is_leaf() const { return type == LEAF; }

    bool is_cut() const { return type == CUT; }
  };

  using NP = Node *;

  PermutationTree() = default;

 private:
  static void add_child(NP t, NP c) {
    t->ch.emplace_back(c);
    t->l = min(t->l, c->l);
    t->r = max(t->r, c->r);
    t->min_v = min(t->min_v, c->min_v);
    t->max_v = max(t->max_v, c->max_v);
  }

 public:
  static NP build(vector<int> &A) {
    int n = (int)A.size();

    vector<int> desc{-1};
    vector<int> asc{-1};
    vector<NP> st;

    auto f = [](int a, int b) { return min(a, b); };
    auto g = [](int a, int b) { return a + b; };
    constexpr int lim = (1 << 30) - 1;
    auto seg = get_lazy_segment_tree(vector<int>(n), f, g, g, lim, 0);

    for (int i = 0; i < n; i++) {
      while (~desc.back() and A[i] > A[desc.back()]) {
        seg.apply(desc[desc.size() - 2] + 1, desc.back() + 1,
                  A[i] - A[desc.back()]);
        desc.pop_back();
      }
      while (~asc.back() and A[i] < A[asc.back()]) {
        seg.apply(asc[asc.size() - 2] + 1, asc.back() + 1,
                  A[asc.back()] - A[i]);
        asc.pop_back();
      }
      desc.emplace_back(i);
      asc.emplace_back(i);

      NP t = new Node{LEAF, i, i + 1, A[i], A[i] + 1, {}};
      for (;;) {
        NodeType type = CUT;
        if (not st.empty()) {
          if (st.back()->max_v == t->min_v) {
            type = JOIN_ASC;
          } else if (t->max_v == st.back()->min_v) {
            type = JOIN_DESC;
          }
        }
        if (type != CUT) {
          NP r = st.back();
          if (type != r->type) {
            r = new Node{type, r->l, r->r, r->min_v, r->max_v, {r}};
          }
          add_child(r, t);
          st.pop_back();
          t = r;
        } else if (seg.prod(0, i + 1 - (int)t->size()) == 0) {
          t = new Node{CUT, t->l, t->r, t->min_v, t->max_v, {t}};
          do {
            add_child(t, st.back());
            st.pop_back();
          } while (t->max_v - t->min_v != t->size());
          reverse(begin(t->ch), end(t->ch));
        } else {
          break;
        }
      }
      st.emplace_back(t);
      seg.apply(0, i + 1, -1);
    }
    return st[0];
  }
};

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