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

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• Last update: 2024-08-10 03:11:07+09:00
• Include: #include "structure/others/permutation-tree.hpp"
• Link: https://codeforces.com/blog/entry/78898

順列 $A$ に対応する順列木(Permutation Tree, 析合树) を構築します。

$ \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ます。

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ます。定義を以下に示します。

• 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。

コンストラクタ

PermutationTree()

順列木を初期化します。

build

static NP build(vector<int> &A)

順列 A に対応する順列木を返します。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 が格納されます。

制約

• A は $[0, n)$ からなる順列

計算量

• $O(n \log n)$

Depends on

• structure/class/acted-monoid.hpp
• structure/class/range-add-range-min.hpp
• 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 "../class/range-add-range-min.hpp"
#include "../segment-tree/lazy-segment-tree.hpp"

/**
 * @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;
    LazySegmentTree seg{RangeAddRangeMin<int>(), vector<int>(n)};
    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/class/range-add-range-min.hpp"
template <typename T>
struct RangeAddRangeMin {
  using S = T;
  using F = T;
  static constexpr S op(const S &a, const S &b) { return min(a, b); }
  static constexpr S e() { return numeric_limits<T>::max(); }
  static constexpr F mapping(const S &x, const F &f) { return x + f; }
  static constexpr F composition(const F &f, const F &g) { return f + g; }
  static constexpr F id() { return {0}; }
};
#line 2 "structure/class/acted-monoid.hpp"

template <typename S2, typename Op, typename E, typename F2, typename Mapping,
          typename Composition, typename Id>
struct LambdaActedMonoid {
  using S = S2;
  using F = F2;

  S op(const S &a, const S &b) const { return _op(a, b); }

  S e() const { return _e(); }

  S mapping(const S &x, const F &f) const { return _mapping(x, f); }

  F composition(const F &f, const F &g) const { return _composition(f, g); }

  F id() const { return _id(); }

  LambdaActedMonoid(Op _op, E _e, Mapping _mapping, Composition _composition,
                    Id _id)
      : _op(_op),
        _e(_e),
        _mapping(_mapping),
        _composition(_composition),
        _id(_id) {}

 private:
  Op _op;

  E _e;

  Mapping _mapping;

  Composition _composition;

  Id _id;
};

template <typename Op, typename E, typename Mapping, typename Composition,
          typename Id>
LambdaActedMonoid(Op _op, E _e, Mapping _mapping, Composition _composition,
                  Id _id)
    -> LambdaActedMonoid<decltype(_e()), Op, E, decltype(_id()), Mapping,
                         Composition, Id>;

/*
struct ActedMonoid {
  using S = ?;
  using F = ?;
  static constexpr S op(const S& a, const S& b) {}
  static constexpr S e() {}
  static constexpr S mapping(const S &x, const F &f) {}
  static constexpr F composition(const F &f, const F &g) {}
  static constexpr F id() {}
};
*/
#line 2 "structure/segment-tree/lazy-segment-tree.hpp"

template <typename ActedMonoid>
struct LazySegmentTree {
  using S = typename ActedMonoid::S;
  using F = typename ActedMonoid::F;

 private:
  ActedMonoid m;

  int n{}, sz{}, height{};

  vector<S> data;

  vector<F> lazy;

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

  inline void all_apply(int k, const F &x) {
    data[k] = m.mapping(data[k], x);
    if (k < sz) lazy[k] = m.composition(lazy[k], x);
  }

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

 public:
  LazySegmentTree() = default;

  explicit LazySegmentTree(ActedMonoid m, int n) : m(m), n(n) {
    sz = 1;
    height = 0;
    while (sz < n) sz <<= 1, height++;
    data.assign(2 * sz, m.e());
    lazy.assign(2 * sz, m.id());
  }

  explicit LazySegmentTree(ActedMonoid m, const vector<S> &v)
      : LazySegmentTree(m, v.size()) {
    build(v);
  }

  void build(const vector<S> &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 S &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);
  }

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

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

  S prod(int l, int r) {
    if (l >= r) return m.e();
    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);
    }
    S L = m.e(), R = m.e();
    for (; l < r; l >>= 1, r >>= 1) {
      if (l & 1) L = m.op(L, data[l++]);
      if (r & 1) R = m.op(data[--r], R);
    }
    return m.op(L, R);
  }

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

  void apply(int k, const F &f) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    data[k] = m.mapping(data[k], f);
    for (int i = 1; i <= height; i++) update(k >> i);
  }

  void apply(int l, int r, const F &f) {
    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++, f);
        if (r & 1) all_apply(--r, f);
      }
      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);
    S sum = m.e();
    do {
      while ((l & 1) == 0) l >>= 1;
      if (check(m.op(sum, data[l]))) {
        while (l < sz) {
          propagate(l);
          l <<= 1;
          auto nxt = m.op(sum, data[l]);
          if (not check(nxt)) {
            sum = nxt;
            l++;
          }
        }
        return l + 1 - sz;
      }
      sum = m.op(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);
    S sum = m.e();
    do {
      r--;
      while (r > 1 and (r & 1)) r >>= 1;
      if (check(m.op(data[r], sum))) {
        while (r < sz) {
          propagate(r);
          r = (r << 1) + 1;
          auto nxt = m.op(data[r], sum);
          if (not check(nxt)) {
            sum = nxt;
            r--;
          }
        }
        return r - sz;
      }
      sum = m.op(data[r], sum);
    } while ((r & -r) != r);
    return -1;
  }
};
#line 3 "structure/others/permutation-tree.hpp"

/**
 * @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;
    LazySegmentTree seg{RangeAddRangeMin<int>(), vector<int>(n)};
    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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