seq/inversion.hpp

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• Last update: 2026-09-13 11:46:00+09:00
• Include: #include "seq/inversion.hpp"

Depends on

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

Required by

• other/sliding_puzzle_solver.hpp

Verified with

• test/3_yukicoder/1838.test.cpp
• test/3_yukicoder/694.test.cpp
• test/4_aoj/ALDS1_2_A.test.cpp
• test/4_aoj/ALDS1_5.test.cpp

Code

#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(const 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;
}

template <typename T>
ll inversion_between(vc<T> A, vc<T> B) {
  int N = len(A);
  map<T, vc<int>> MP;
  FOR(i, N) MP[B[i]].eb(i);
  vc<int> TO(N);
  FOR_R(i, N) {
    auto& I = MP[A[i]];
    if (I.empty()) return -1;
    TO[i] = POP(I);
  }
  return inversion(TO);
}

#line 1 "ds/fenwicktree/fenwicktree_01.hpp"

#line 1 "other/bit.hpp"

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_sgn(int x) { return (__builtin_parity(unsigned(x)) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T kth_bit(int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return T(1) << k;
}
template <typename T>
bool has_kth_bit(T x, int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return x >> k & 1;
}

template <typename UINT>
struct all_bit {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_bit(UINT s) : s(s) {}
  struct iter {
    UINT s;
    int operator*() const { return lowbit(s); }
    void operator++() { s &= s - 1; }
    bool operator!=(nullptr_t) const { return s; }
  };
  iter begin() const { return {s}; }
  nullptr_t end() const { return nullptr; }
};

template <typename UINT>
struct all_subset {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_subset(UINT s) : s(s) {}
  struct iter {
    UINT s, t;
    bool done = false;
    UINT operator*() const { return t; }
    void operator++() {
      done = (t == 0);
      t = (t - 1) & s;
    }
    bool operator!=(nullptr_t) const { return !done; }
  };
  iter begin() const { return {s, s}; }
  nullptr_t end() const { return nullptr; }
};

constexpr u64 full_mask(int n) {
  assert(0 <= n && n <= 64);
  return n == 64 ? -1ULL : (1ULL << n) - 1;
}

u64 bit_reverse(u64 x) {
  x = ((x & 0x5555555555555555ULL) << 1) | ((x >> 1) & 0x5555555555555555ULL);
  x = ((x & 0x3333333333333333ULL) << 2) | ((x >> 2) & 0x3333333333333333ULL);
  x = ((x & 0x0f0f0f0f0f0f0f0fULL) << 4) | ((x >> 4) & 0x0f0f0f0f0f0f0f0fULL);
  x = ((x & 0x00ff00ff00ff00ffULL) << 8) | ((x >> 8) & 0x00ff00ff00ff00ffULL);
  x = ((x & 0x0000ffff0000ffffULL) << 16) | ((x >> 16) & 0x0000ffff0000ffffULL);
  x = (x << 32) | (x >> 32);
  return x;
}
#line 1 "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 id() { return X(0); }
  static constexpr bool commute = true;
};
#line 3 "ds/fenwicktree/fenwicktree.hpp"

template <typename Monoid>
struct FenwickTree {
  using G = Monoid;
  using MX = 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::id());
    total = G::id();
  }
  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::id();
    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() const { return total; }
  E sum_all() const { return total; }
  E sum(int k) const { return prefix_sum(k); }
  E prod(int k) const { return prefix_prod(k); }
  E prefix_sum(int k) const { return prefix_prod(k); }
  E prefix_prod(int k) const {
    chmin(k, n);
    E ret = G::id();
    for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
    return ret;
  }
  E sum(int L, int R) const { return prod(L, R); }
  E prod(int L, int R) const {
    chmax(L, 0), chmin(R, n);
    if (L == 0) return prefix_prod(R);
    assert(0 <= L && L <= R && R <= n);
    E pos = G::id(), neg = G::id();
    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() const {
    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);
    assert(0 <= k && k < n);
    total = G::op(total, x);
    for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
  }
  void set(int k, E x) { add(k, G::op(G::inverse(prod(k, k + 1)), x)); }

  template <class F>
  int max_right(const F check, int L = 0) const {
    assert(check(G::id()));
    E s = G::id();
    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 (i + (1 << k) <= L || 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) const {
    assert(check(L, G::id()));
    E s = G::id();
    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 (i + (1 << k) <= L || check(i + (1 << k), t)) {
          i += (1 << k), s = t;
        }
      }
    }
    return i;
  }

  template <class F>
  int min_left(const F check, int R) const {
    assert(check(G::id()));
    E s = G::id();
    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) const {
    return max_right([&k](E x) -> bool { return x <= k; }, L);
  }
};
#line 3 "ds/fenwicktree/fenwicktree_01.hpp"

struct FenwickTree_01 {
  using MX = Monoid_Add<int>;
  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);
  }
  void build(vc<int> dat) {
    build(len(dat), [&](int i) -> int { return dat[i]; });
  }

  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) {
      u64 x = f(i);
      assert(x == 0 || x == 1);
      dat[i / 64] |= x << (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) {
    assert(0 <= k && k <= N);
    int ans = bit.sum(k / 64);
    ans += popcnt(dat[k / 64] & ((u64(1) << (k % 64)) - 1));
    return ans;
  }
  int sum(int L, int R) {
    assert(0 <= L && L <= R && R <= N);
    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;
  }
  int get(int i) {
    assert(0 <= i && i < N);
    return sum(i, i + 1);
  }
  int prod(int L, int R) { return sum(L, R); }

  void add(int k, int x) {
    assert(0 <= k && k < N);
    if (x == 1) {
      assert(sum(k, k + 1) == 0);
      dat[k / 64] |= u64(1) << (k % 64);
      bit.add(k / 64, 1);
    }
    elif (x == -1) {
      assert(sum(k, k + 1) == 1);
      dat[k / 64] &= ~(u64(1) << (k % 64));
      bit.add(k / 64, -1);
    }
    else assert(0);
  }
  void multiply(int k, int x) { add(k, x); }

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

  int kth(int k, int L = 0) {
    assert(0 <= L && L <= N);
    assert(0 <= k && k < sum_all());
    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) {
    assert(0 <= k && k <= N);
    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) {
    assert(0 <= k && k <= N);
    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]);
  }

  string to_string() {
    string out;
    FOR(i, N) out += '0' + (dat[i / 64] >> (i & 63) & 1);
    return out;
  }
};
#line 2 "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(const 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;
}

template <typename T>
ll inversion_between(vc<T> A, vc<T> B) {
  int N = len(A);
  map<T, vc<int>> MP;
  FOR(i, N) MP[B[i]].eb(i);
  vc<int> TO(N);
  FOR_R(i, N) {
    auto& I = MP[A[i]];
    if (I.empty()) return -1;
    TO[i] = POP(I);
  }
  return inversion(TO);
}

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