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:heavy_check_mark: ds/wavelet_matrix_with_fenwick.hpp

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Code

#pragma once
#include "ds/fenwick.hpp"
#include "ds/bit_vector.hpp"

template <class Key, Key U, class M>
class wavelet_matrix_with_fenwick {
 public:
  using key_type = Key;
  using value_type = typename M::type;

  wavelet_matrix_with_fenwick(vector<Key> a, M m = M())
      : wavelet_matrix_with_fenwick(a, a, m) {}
  wavelet_matrix_with_fenwick(vector<Key> a, vector<value_type> b, M m = M())
      : m_(m), size_(a.size()) {
    vector<bool> bit(size_);
    vector<key_type> nxt_a(size_);
    vector<value_type> nxt_b(size_);
    repr(t, B) {
      int l = 0, r = size_;
      rep(i, size_) {
        bit[i] = a[i] >> t & 1;
        (bit[i] ? nxt_a[r - 1] : nxt_a[l]) = a[i];
        (bit[i] ? nxt_b[--r] : nxt_b[l++]) = b[i];
      }
      mat[t] = bit_vector(all(bit));
      zeros[t] = l;
      fwk[t + 1] = fenwick_tree<M>(b, m);
      reverse(nxt_a.begin() + r, nxt_a.end());
      reverse(nxt_b.begin() + r, nxt_b.end());
      swap(a, nxt_a);
      swap(b, nxt_b);
    }
    fwk[0] = fenwick_tree<M>(b, m);
  }

  int size() const { return size_; }
  key_type operator[](int i) const { return access(i); }
  key_type access(int i) const {
    key_type res = 0;
    repr(t, B) {
      if (mat[t][i]) {
        res |= key_type(1) << t;
        i = zeros[t] + mat[t].rank1(i);
      } else {
        i = mat[t].rank0(i);
      }
    }
    return res;
  }
  // #occurences of x on [l, r)
  int rank(key_type x, int l, int r) const {
    tie(l, r) = range(x, l, r);
    return r - l;
  }
  // -1 if #occurences <= k
  int select(key_type x, int l, int k) const {
    int r;
    tie(l, r) = range(x, l, size());
    l += k;
    if (l >= r) return -1;
    rep(t, B) {
      if (x >> t & 1)
        l = mat[t].select1(l - zeros[t]);
      else
        l = mat[t].select0(l);
    }
    return l;
  }
  // k-th greatest on [l, r)
  key_type quantile(int l, int r, int k) const {
    key_type res = 0;
    repr(t, B) {
      int r1 = mat[t].rank1(l, r);
      if (r1 > k) {
        res |= key_type(1) << t;
        l = zeros[t] + mat[t].rank1(l);
        r = l + r1;
      } else {
        k -= r1;
        int r0 = r - l - r1;
        l = mat[t].rank0(l);
        r = l + r0;
      }
    }
    return res;
  }
  // k-th smallest on [l, r)
  key_type rquantile(int l, int r, int k) const {
    return quantile(l, r, r - l - k - 1);
  }
  int rangefreq(int l, int r, key_type ub) const {
    int res = 0;
    repr(t, B) {
      if (ub >> t & 1) {
        int r1l = mat[t].rank1(l), r1r = mat[t].rank1(r);
        int r0lr = (r - l) - (r1r - r1l);
        res += r0lr;
        l = zeros[t] + r1l, r = zeros[t] + r1r;
      } else {
        l = mat[t].rank0(l), r = mat[t].rank0(r);
      }
    }
    return res;
  }
  value_type range_reduce(int l, int r, key_type ub) const {
    value_type res = m_.unit();
    repr(t, B) {
      if (ub >> t & 1) {
        int r1l = mat[t].rank1(l), r1r = mat[t].rank1(r);
        res = m_.op(res, fwk[t + 1].sum(l, r));
        l = zeros[t] + r1l, r = zeros[t] + r1r;
        res = m_.op(res, m_.inv(fwk[t].sum(l, r)));
      } else {
        l = mat[t].rank0(l), r = mat[t].rank0(r);
      }
    }
    return res;
  }
  // int rangefreq(int l, int r, key_type lb, key_type ub) const {
  //   return rangefreq(B - 1, l, r, lb, ub - 1);
  // }
  int rangefreq(int l, int r, key_type lb, key_type ub) const {
    return rangefreq(l, r, ub) - rangefreq(l, r, lb);
  }
  value_type range_reduce(int l, int r, key_type lb, key_type ub) const {
    return m_.op(range_reduce(l, r, ub), m_.inv(range_reduce(l, r, lb)));
  }
  // -1 if no such elt
  key_type succ(int l, int r, key_type x) const {
    int k = rangefreq(l, r, x);
    return k == r - l ? -1 : rquantile(l, r, k);
  }
  // -1 if no such elt
  key_type pred(int l, int r, key_type x) const {
    int k = rangefreq(l, r, x);
    return k ? rquantile(l, r, k - 1) : -1;
  }

 private:
  static constexpr int calc_b(key_type u) {
    int res = 0;
    for (key_type x = u - 1; x; x /= 2) res++;
    return res;
  }
  static constexpr int B = calc_b(U);
  static_assert(B <= sizeof(key_type) * CHAR_BIT);
  M m_;
  int size_;
  array<int, B> zeros;
  array<bit_vector, B> mat;
  array<fenwick_tree<M>, B + 1> fwk;

  pair<int, int> range(key_type x, int l, int r) const {
    repr(t, B) {
      if (x >> t & 1) {
        l = zeros[t] + mat[t].rank1(l);
        r = zeros[t] + mat[t].rank1(r);
      } else {
        l = mat[t].rank0(l);
        r = mat[t].rank0(r);
      }
    }
    return make_pair(l, r);
  }
  // // inclusive
  // int rangefreq(int t, int l, int r, key_type x, key_type lb, key_type
  // ub) const {
  //   if (t == -1 || ()) return r - l;
  //   if (lb >> t & 1) {
  //     l = zeros[t] + mat[t].rank1(l);
  //     r = zeros[t] + mat[t].rank1(r);
  //     return rangefreq(t - 1, l, r, lb, ub);
  //   } else if (~ub >> t & 1) {
  //     l = mat[t].rank0(l);
  //     r = mat[t].rank0(r);
  //     return rangefreq(t - 1, l, r, lb, ub);
  //   } else {
  //     l = mat[t].rank0(l);
  //     r = zeros[t] + mat[t].rank1(r);
  //     return rangefreq(t - 1, l, zeros[t], lb, ~0) +
  //            rangefreq(t - 1, zeros[t], r, 0, ub);
  //   }
  // }
};

#line 2 "prelude.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using vi = vector<int>;
using vvi = vector<vector<int>>;
using vll = vector<ll>;
using vvll = vector<vector<ll>>;
using vc = vector<char>;
#define rep2(i, m, n) for (auto i = (m); i < (n); i++)
#define rep(i, n) rep2(i, 0, n)
#define repr2(i, m, n) for (auto i = (n); i-- > (m);)
#define repr(i, n) repr2(i, 0, n)
#define all(x) begin(x), end(x)
auto ndvec(int n, auto e) { return vector(n, e); }
auto ndvec(int n, auto ...e) { return vector(n, ndvec(e...)); }
auto comp_key(auto&& f) { return [&](auto&& a, auto&& b) { return f(a) < f(b); }; }
auto& max(const auto& a, const auto& b) { return a < b ? b : a; }
auto& min(const auto& a, const auto& b) { return b < a ? b : a; }
#if __cpp_lib_ranges
namespace R = std::ranges;
namespace V = std::views;
#endif
#line 3 "algebra.hpp"

#define CONST(val) [=] { return val; }
#define WRAP_FN(func) \
  [](auto&&... args) { return func(forward<decltype(args)>(args)...); }

template <class Unit, class Op>
struct monoid : private Unit, private Op {
  using type = decltype(declval<Unit>()());
  monoid(Unit unit, Op op) : Unit(unit), Op(op) {}
  type unit() const { return Unit::operator()(); }
  type op(type a, type b) const { return Op::operator()(a, b); }
};

template <class Unit, class Op, class Inv>
struct group : monoid<Unit, Op>, private Inv {
  using type = typename monoid<Unit, Op>::type;
  group(Unit unit, Op op, Inv inv) : monoid<Unit, Op>(unit, op), Inv(inv) {}
  type inv(type a) const { return Inv::operator()(a); }
};

template <class T>
struct addition {
  using type = T;
  type unit() const { return 0; }
  type op(type a, type b) const { return a + b; }
  type inv(type a) const { return -a; }
};

template <class T>
struct maximum {
  using type = T;
  type unit() const { return numeric_limits<T>::min(); }
  type op(type a, type b) const { return a > b ? a : b; }
};

template <class T>
struct minimum {
  using type = T;
  type unit() const { return numeric_limits<T>::max(); }
  type op(type a, type b) const { return a > b ? b : a; }
};

template <class T, T nul = -1>
struct assign {
  using type = T;
  type unit() const { return nul; }
  type op(type a, type b) const { return b == nul ? a : b; }
};
#line 3 "bit/clz.hpp"

#pragma GCC target("lzcnt")

template <class T>
int clz(T x) {
    if (!x) return sizeof(T) * 8;
    if constexpr (sizeof(T) <= sizeof(unsigned)) {
        return __builtin_clz((unsigned)x);
    } else if constexpr (sizeof(T) <= sizeof(unsigned long long)) {
        return __builtin_clzll((unsigned long long)x);
    } else if constexpr (sizeof(T) <= sizeof(unsigned long long) * 2) {
        int l = clz((unsigned long long)(x >> sizeof(unsigned long long) * 8));
        return l != sizeof(unsigned long long) * 8 ? l : l + clz((unsigned long long)x);
    }
}
#line 4 "bit/ilog2.hpp"

template <class T>
__attribute__((pure)) int ilog2(T x) { assert(x != 0); return sizeof(T) * 8 - 1 - clz(x); }

template <class T>
__attribute__((pure)) int ilog2_ceil(T x) { return x == 0 || x == 1 ? 0 : ilog2(x - 1) + 1; }

template <class T, enable_if_t<is_signed_v<T>>* = nullptr>
__attribute__((pure)) T bit_floor(T x) { return T(1) << ilog2(x); }

template <class T, enable_if_t<is_signed_v<T>>* = nullptr>
__attribute__((pure)) T bit_ceil(T x) { return T(1) << ilog2_ceil(x); }
#line 4 "ds/fenwick.hpp"

template <class M>
class fenwick_tree {
 public:
  using value_type = typename M::type;
  fenwick_tree() = default;
  fenwick_tree(vector<value_type> v, M m = M()) : m(m), data(move(v)) {
    data.insert(data.cbegin(), m.unit());
    for (int i = 1; i < data.size(); i++) {
      if (i + lsb(i) < data.size())
        data[i + lsb(i)] = m.op(data[i + lsb(i)], data[i]);
    }
  }
  template <class Iter>
  fenwick_tree(Iter f, Iter l, M m = M())
      : fenwick_tree(vector<value_type>(f, l), m) {}
  fenwick_tree(int n, M m = M()) : m(m), data(n + 1, m.unit()) {}
  int size() const { return data.size() - 1; }
  void clear() { fill(data.begin(), data.end(), m.unit()); }
  void add(int i, value_type v) {
    for (i++; i < data.size(); i += lsb(i)) data[i] = m.op(data[i], v);
  }
  void sub(int i, value_type v) { add(i, m.inv(v)); }
  void assign(int i, value_type v) { add(i, m.op(v, m.inv(sum(i, i + 1)))); }
  value_type sum(int r) const {
    value_type res = m.unit();
    for (; r; r -= lsb(r)) res = m.op(res, data[r]);
    return res;
  }
  value_type sum(int l, int r) const { return m.op(m.inv(sum(l)), sum(r)); }
  template <class F>
  int partition_point(F pred = F()) const {
    int i = 0;
    value_type s = m.unit();
    if (!pred(s)) return i;
    for (int w = bit_floor(data.size()); w; w >>= 1) {
      if (i + w < data.size()) {
        value_type s2 = m.op(s, data[i + w]);
        if (pred(s2)) i += w, s = s2;
      }
    }
    return i + 1;
  }
  // min i s.t. sum(i) >= x
  template <class Comp = less<value_type>>
  int lower_bound(value_type x, Comp comp = Comp()) const {
    return partition_point([&](value_type s) { return comp(s, x); });
  }

 private:
  M m;
  vector<value_type> data;
  static int lsb(int a) { return a & -a; }
};
#line 3 "bit/popcnt.hpp"

#pragma GCC target("popcnt")

template <class T>
int popcnt(T a) {
  if constexpr (sizeof(T) <= sizeof(unsigned)) {
    return __builtin_popcount((unsigned)a);
  } else if constexpr (sizeof(T) <= sizeof(unsigned long long)) {
    return __builtin_popcountll((unsigned long long)a);
  } else if constexpr (sizeof(T) <= sizeof(unsigned long long) * 2) {
    return popcnt((unsigned long long)a) +
           popcnt((unsigned long long)(a >> sizeof(unsigned long long) * 8));
  }
}
#line 4 "ds/bit_vector.hpp"

class bit_vector {
 public:
  bit_vector(int n = 0) : bit(n / 8 + 1), sum(n / 8 + 2) {}
  template <class It>
  bit_vector(It a, It last) : bit_vector(last - a) {
    int n = last - a;
    rep(i, n) bit[i / 64] |= uint64_t(a[i] != 0) << (i % 64);
    rep(i, bit.size()) sum[i + 1] = sum[i] + popcnt(bit[i]);
  }
  int size() const { return bit.size() * 64; }
  bool operator[](int i) const { return bit[i / 64] >> (i % 64) & 1; }
  int rank0(int r) const { return r - rank1(r); }
  int rank0(int l, int r) const { return rank0(r) - rank0(l); }
  int rank1(int r) const {
    return sum[r / 64] + popcnt(bit[r / 64] & ~(~uint64_t(0) << (r % 64)));
  }
  int rank1(int l, int r) const { return rank1(r) - rank1(l); }
  int rank(bool b, int r) const { return b ? rank1(r) : rank0(r); }
  int rank(bool b, int l, int r) const { return b ? rank1(l, r) : rank0(l, r); }
  int select0(int l, int k) const {
    int r = bit.size() * 8;
    while (l + 1 < r) {
      int m = (l + r) / 2;
      (rank0(m) <= k ? l : r) = m;
    }
    return l;
  }
  int select0(int k) const { return select0(0, k); }
  int select1(int l, int k) const {
    int r = bit.size() * 8;
    while (l + 1 < r) {
      int m = (l + r) / 2;
      (rank1(m) <= k ? l : r) = m;
    }
    return l;
  }
  int select1(int k) const { return select1(0, k); }
  int select(bool v, int k) const { return v ? select1(k) : select0(k); }
  int select(bool v, int l, int k) const {
    return v ? select1(l, k) : select0(l, k);
  }

 private:
  vector<uint64_t> bit;
  vector<int> sum;
};
#line 4 "ds/wavelet_matrix_with_fenwick.hpp"

template <class Key, Key U, class M>
class wavelet_matrix_with_fenwick {
 public:
  using key_type = Key;
  using value_type = typename M::type;

  wavelet_matrix_with_fenwick(vector<Key> a, M m = M())
      : wavelet_matrix_with_fenwick(a, a, m) {}
  wavelet_matrix_with_fenwick(vector<Key> a, vector<value_type> b, M m = M())
      : m_(m), size_(a.size()) {
    vector<bool> bit(size_);
    vector<key_type> nxt_a(size_);
    vector<value_type> nxt_b(size_);
    repr(t, B) {
      int l = 0, r = size_;
      rep(i, size_) {
        bit[i] = a[i] >> t & 1;
        (bit[i] ? nxt_a[r - 1] : nxt_a[l]) = a[i];
        (bit[i] ? nxt_b[--r] : nxt_b[l++]) = b[i];
      }
      mat[t] = bit_vector(all(bit));
      zeros[t] = l;
      fwk[t + 1] = fenwick_tree<M>(b, m);
      reverse(nxt_a.begin() + r, nxt_a.end());
      reverse(nxt_b.begin() + r, nxt_b.end());
      swap(a, nxt_a);
      swap(b, nxt_b);
    }
    fwk[0] = fenwick_tree<M>(b, m);
  }

  int size() const { return size_; }
  key_type operator[](int i) const { return access(i); }
  key_type access(int i) const {
    key_type res = 0;
    repr(t, B) {
      if (mat[t][i]) {
        res |= key_type(1) << t;
        i = zeros[t] + mat[t].rank1(i);
      } else {
        i = mat[t].rank0(i);
      }
    }
    return res;
  }
  // #occurences of x on [l, r)
  int rank(key_type x, int l, int r) const {
    tie(l, r) = range(x, l, r);
    return r - l;
  }
  // -1 if #occurences <= k
  int select(key_type x, int l, int k) const {
    int r;
    tie(l, r) = range(x, l, size());
    l += k;
    if (l >= r) return -1;
    rep(t, B) {
      if (x >> t & 1)
        l = mat[t].select1(l - zeros[t]);
      else
        l = mat[t].select0(l);
    }
    return l;
  }
  // k-th greatest on [l, r)
  key_type quantile(int l, int r, int k) const {
    key_type res = 0;
    repr(t, B) {
      int r1 = mat[t].rank1(l, r);
      if (r1 > k) {
        res |= key_type(1) << t;
        l = zeros[t] + mat[t].rank1(l);
        r = l + r1;
      } else {
        k -= r1;
        int r0 = r - l - r1;
        l = mat[t].rank0(l);
        r = l + r0;
      }
    }
    return res;
  }
  // k-th smallest on [l, r)
  key_type rquantile(int l, int r, int k) const {
    return quantile(l, r, r - l - k - 1);
  }
  int rangefreq(int l, int r, key_type ub) const {
    int res = 0;
    repr(t, B) {
      if (ub >> t & 1) {
        int r1l = mat[t].rank1(l), r1r = mat[t].rank1(r);
        int r0lr = (r - l) - (r1r - r1l);
        res += r0lr;
        l = zeros[t] + r1l, r = zeros[t] + r1r;
      } else {
        l = mat[t].rank0(l), r = mat[t].rank0(r);
      }
    }
    return res;
  }
  value_type range_reduce(int l, int r, key_type ub) const {
    value_type res = m_.unit();
    repr(t, B) {
      if (ub >> t & 1) {
        int r1l = mat[t].rank1(l), r1r = mat[t].rank1(r);
        res = m_.op(res, fwk[t + 1].sum(l, r));
        l = zeros[t] + r1l, r = zeros[t] + r1r;
        res = m_.op(res, m_.inv(fwk[t].sum(l, r)));
      } else {
        l = mat[t].rank0(l), r = mat[t].rank0(r);
      }
    }
    return res;
  }
  // int rangefreq(int l, int r, key_type lb, key_type ub) const {
  //   return rangefreq(B - 1, l, r, lb, ub - 1);
  // }
  int rangefreq(int l, int r, key_type lb, key_type ub) const {
    return rangefreq(l, r, ub) - rangefreq(l, r, lb);
  }
  value_type range_reduce(int l, int r, key_type lb, key_type ub) const {
    return m_.op(range_reduce(l, r, ub), m_.inv(range_reduce(l, r, lb)));
  }
  // -1 if no such elt
  key_type succ(int l, int r, key_type x) const {
    int k = rangefreq(l, r, x);
    return k == r - l ? -1 : rquantile(l, r, k);
  }
  // -1 if no such elt
  key_type pred(int l, int r, key_type x) const {
    int k = rangefreq(l, r, x);
    return k ? rquantile(l, r, k - 1) : -1;
  }

 private:
  static constexpr int calc_b(key_type u) {
    int res = 0;
    for (key_type x = u - 1; x; x /= 2) res++;
    return res;
  }
  static constexpr int B = calc_b(U);
  static_assert(B <= sizeof(key_type) * CHAR_BIT);
  M m_;
  int size_;
  array<int, B> zeros;
  array<bit_vector, B> mat;
  array<fenwick_tree<M>, B + 1> fwk;

  pair<int, int> range(key_type x, int l, int r) const {
    repr(t, B) {
      if (x >> t & 1) {
        l = zeros[t] + mat[t].rank1(l);
        r = zeros[t] + mat[t].rank1(r);
      } else {
        l = mat[t].rank0(l);
        r = mat[t].rank0(r);
      }
    }
    return make_pair(l, r);
  }
  // // inclusive
  // int rangefreq(int t, int l, int r, key_type x, key_type lb, key_type
  // ub) const {
  //   if (t == -1 || ()) return r - l;
  //   if (lb >> t & 1) {
  //     l = zeros[t] + mat[t].rank1(l);
  //     r = zeros[t] + mat[t].rank1(r);
  //     return rangefreq(t - 1, l, r, lb, ub);
  //   } else if (~ub >> t & 1) {
  //     l = mat[t].rank0(l);
  //     r = mat[t].rank0(r);
  //     return rangefreq(t - 1, l, r, lb, ub);
  //   } else {
  //     l = mat[t].rank0(l);
  //     r = zeros[t] + mat[t].rank1(r);
  //     return rangefreq(t - 1, l, zeros[t], lb, ~0) +
  //            rangefreq(t - 1, zeros[t], r, 0, ub);
  //   }
  // }
};
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