seq/inversion.hpp

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• Last update: 2024-04-09 15:17:41+09:00
• Include: #include "seq/inversion.hpp"

Depends on

• alg/monoid/add.hpp
• ds/fenwicktree/fenwicktree.hpp
• ds/fenwicktree/fenwicktree_01.hpp

Verified with

• test/aoj/ALDS1_2_A.test.cpp
• test/aoj/ALDS1_5.test.cpp
• test/yukicoder/1838.test.cpp
• test/yukicoder/694.test.cpp
• test_atcoder/abc190f.test.cpp

Code

#pragma once
#include "ds/fenwicktree/fenwicktree_01.hpp"

template <typename T>
ll inversion(vc<T> A) {
  int N = len(A);
  if (A.empty()) return 0;
  ll ANS = 0;
  FenwickTree_01 bit(N);
  auto I = argsort(A);
  for (auto& i: I) {
    ANS += bit.sum_all() - bit.sum(i);
    bit.add(i, 1);
  }
  return ANS;
}

// i 番目：A_i が先頭になるように rotate したときの転倒数
template <typename T, bool SMALL = false>
vi inversion_rotate(vc<T>& A) {
  const int N = len(A);
  if (!SMALL) {
    auto key = A;
    UNIQUE(key);
    for (auto&& x: A) x = LB(key, x);
  }
  ll K = MAX(A) + 1;
  ll ANS = 0;
  FenwickTree<Monoid_Add<int>> bit(K);
  for (auto&& x: A) {
    ANS += bit.sum(x + 1, K);
    bit.add(x, 1);
  }
  vi res(N);
  FOR(i, N) {
    res[i] = ANS;
    ll x = A[i];
    ANS = ANS + bit.sum(x + 1, K) - bit.prefix_sum(x);
  }
  return res;
}

// inv[i][j] = inversion A[i:j) であるような (N+1, N+1) テーブル
template <typename T>
vvc<int> all_range_inversion(vc<T>& A) {
  int N = len(A);
  vv(int, dp, N + 1, N + 1);
  FOR_R(L, N + 1) FOR(R, L + 2, N + 1) {
    dp[L][R] = dp[L][R - 1] + dp[L + 1][R] - dp[L + 1][R - 1];
    if (A[L] > A[R - 1]) ++dp[L][R];
  }
  return dp;
}

#line 2 "ds/fenwicktree/fenwicktree_01.hpp"

#line 2 "alg/monoid/add.hpp"

template <typename E>
struct Monoid_Add {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
  static constexpr X inverse(const X &x) noexcept { return -x; }
  static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
  static constexpr X unit() { return X(0); }
  static constexpr bool commute = true;
};
#line 3 "ds/fenwicktree/fenwicktree.hpp"

template <typename Monoid>
struct FenwickTree {
  using G = Monoid;
  using E = typename G::value_type;
  int n;
  vector<E> dat;
  E total;

  FenwickTree() {}
  FenwickTree(int n) { build(n); }
  template <typename F>
  FenwickTree(int n, F f) {
    build(n, f);
  }
  FenwickTree(const vc<E>& v) { build(v); }

  void build(int m) {
    n = m;
    dat.assign(m, G::unit());
    total = G::unit();
  }
  void build(const vc<E>& v) {
    build(len(v), [&](int i) -> E { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m;
    dat.clear();
    dat.reserve(n);
    total = G::unit();
    FOR(i, n) { dat.eb(f(i)); }
    for (int i = 1; i <= n; ++i) {
      int j = i + (i & -i);
      if (j <= n) dat[j - 1] = G::op(dat[i - 1], dat[j - 1]);
    }
    total = prefix_sum(m);
  }

  E prod_all() { return total; }
  E sum_all() { return total; }
  E sum(int k) { return prefix_sum(k); }
  E prod(int k) { return prefix_prod(k); }
  E prefix_sum(int k) { return prefix_prod(k); }
  E prefix_prod(int k) {
    chmin(k, n);
    E ret = G::unit();
    for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
    return ret;
  }
  E sum(int L, int R) { return prod(L, R); }
  E prod(int L, int R) {
    chmax(L, 0), chmin(R, n);
    if (L == 0) return prefix_prod(R);
    assert(0 <= L && L <= R && R <= n);
    E pos = G::unit(), neg = G::unit();
    while (L < R) { pos = G::op(pos, dat[R - 1]), R -= R & -R; }
    while (R < L) { neg = G::op(neg, dat[L - 1]), L -= L & -L; }
    return G::op(pos, G::inverse(neg));
  }

  vc<E> get_all() {
    vc<E> res(n);
    FOR(i, n) res[i] = prod(i, i + 1);
    return res;
  }

  void add(int k, E x) { multiply(k, x); }
  void multiply(int k, E x) {
    static_assert(G::commute);
    total = G::op(total, x);
    for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
  }

  template <class F>
  int max_right(const F check, int L = 0) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  // check(i, x)
  template <class F>
  int max_right_with_index(const F check, int L = 0) {
    assert(check(L, G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(i + (1 << k), t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(i + (1 << k), t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  template <class F>
  int min_left(const F check, int R) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = R;
    // false になるところまで戻る
    int k = 0;
    while (i > 0 && check(s)) {
      s = G::op(s, dat[i - 1]);
      k = lowbit(i);
      i -= i & -i;
    }
    if (check(s)) {
      assert(i == 0);
      return 0;
    }
    // 2^k 進むと ok になる
    // false を維持して進む
    while (k) {
      --k;
      E t = G::op(s, G::inverse(dat[i + (1 << k) - 1]));
      if (!check(t)) { i += (1 << k), s = t; }
    }
    return i + 1;
  }

  int kth(E k, int L = 0) {
    return max_right([&k](E x) -> bool { return x <= k; }, L);
  }
};
#line 4 "ds/fenwicktree/fenwicktree_01.hpp"

struct FenwickTree_01 {
  int N, n;
  vc<u64> dat;
  FenwickTree<Monoid_Add<int>> bit;
  FenwickTree_01() {}
  FenwickTree_01(int n) { build(n); }
  template <typename F>
  FenwickTree_01(int n, F f) {
    build(n, f);
  }

  void build(int m) {
    N = m;
    n = ceil<int>(N + 1, 64);
    dat.assign(n, u64(0));
    bit.build(n);
  }

  template <typename F>
  void build(int m, F f) {
    N = m;
    n = ceil<int>(N + 1, 64);
    dat.assign(n, u64(0));
    FOR(i, N) { dat[i / 64] |= u64(f(i)) << (i % 64); }
    bit.build(n, [&](int i) -> int { return popcnt(dat[i]); });
  }

  int sum_all() { return bit.sum_all(); }
  int sum(int k) { return prefix_sum(k); }
  int prefix_sum(int k) {
    int ans = bit.sum(k / 64);
    ans += popcnt(dat[k / 64] & ((u64(1) << (k % 64)) - 1));
    return ans;
  }
  int sum(int L, int R) {
    if (L == 0) return prefix_sum(R);
    int ans = 0;
    ans -= popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
    ans += popcnt(dat[R / 64] & ((u64(1) << (R % 64)) - 1));
    ans += bit.sum(L / 64, R / 64);
    return ans;
  }

  void add(int k, int x) {
    if (x == 1) add(k);
    if (x == -1) remove(k);
  }

  void add(int k) {
    dat[k / 64] |= u64(1) << (k % 64);
    bit.add(k / 64, 1);
  }
  void remove(int k) {
    dat[k / 64] &= ~(u64(1) << (k % 64));
    bit.add(k / 64, -1);
  }

  int kth(int k, int L = 0) {
    if (k >= sum_all()) return N;
    k += popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
    L /= 64;
    int mid = 0;
    auto check = [&](auto e) -> bool {
      if (e <= k) chmax(mid, e);
      return e <= k;
    };
    int idx = bit.max_right(check, L);
    if (idx == n) return N;
    k -= mid;
    u64 x = dat[idx];
    int p = popcnt(x);
    if (p <= k) return N;
    k = binary_search([&](int n) -> bool { return (p - popcnt(x >> n)) <= k; },
                      0, 64, 0);
    return 64 * idx + k;
  }

  int next(int k) {
    int idx = k / 64;
    k %= 64;
    u64 x = dat[idx] & ~((u64(1) << k) - 1);
    if (x) return 64 * idx + lowbit(x);
    idx = bit.kth(0, idx + 1);
    if (idx == n || !dat[idx]) return N;
    return 64 * idx + lowbit(dat[idx]);
  }

  int prev(int k) {
    if (k == N) --k;
    int idx = k / 64;
    k %= 64;
    u64 x = dat[idx];
    if (k < 63) x &= (u64(1) << (k + 1)) - 1;
    if (x) return 64 * idx + topbit(x);
    idx = bit.min_left([&](auto e) -> bool { return e <= 0; }, idx) - 1;
    if (idx == -1) return -1;
    return 64 * idx + topbit(dat[idx]);
  }
};
#line 3 "seq/inversion.hpp"

template <typename T>
ll inversion(vc<T> A) {
  int N = len(A);
  if (A.empty()) return 0;
  ll ANS = 0;
  FenwickTree_01 bit(N);
  auto I = argsort(A);
  for (auto& i: I) {
    ANS += bit.sum_all() - bit.sum(i);
    bit.add(i, 1);
  }
  return ANS;
}

// i 番目：A_i が先頭になるように rotate したときの転倒数
template <typename T, bool SMALL = false>
vi inversion_rotate(vc<T>& A) {
  const int N = len(A);
  if (!SMALL) {
    auto key = A;
    UNIQUE(key);
    for (auto&& x: A) x = LB(key, x);
  }
  ll K = MAX(A) + 1;
  ll ANS = 0;
  FenwickTree<Monoid_Add<int>> bit(K);
  for (auto&& x: A) {
    ANS += bit.sum(x + 1, K);
    bit.add(x, 1);
  }
  vi res(N);
  FOR(i, N) {
    res[i] = ANS;
    ll x = A[i];
    ANS = ANS + bit.sum(x + 1, K) - bit.prefix_sum(x);
  }
  return res;
}

// inv[i][j] = inversion A[i:j) であるような (N+1, N+1) テーブル
template <typename T>
vvc<int> all_range_inversion(vc<T>& A) {
  int N = len(A);
  vv(int, dp, N + 1, N + 1);
  FOR_R(L, N + 1) FOR(R, L + 2, N + 1) {
    dp[L][R] = dp[L][R - 1] + dp[L + 1][R] - dp[L + 1][R - 1];
    if (A[L] > A[R - 1]) ++dp[L][R];
  }
  return dp;
}

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