segtree_2d.hpp

// 点の重複があっても別の点として set などがされる
template <typename Monoid, typename XY, bool SMALL_X = false>
struct SegTree_2D {
  using MX = Monoid;
  using S = typename MX::value_type;
  static_assert(MX::commute);
  int N;
  // X to idx
  vc<XY> keyX;
  int minX;
  // top node の点列
  vc<XY> all_Y;
  vc<int> pos;
  // segtree data
  int NX, log, size;
  vc<int> indptr;
  vc<S> dat;
  // fractional cascading
  vc<int> to_left;

  SegTree_2D(vc<XY>& X, vc<XY>& Y)
      : SegTree_2D(len(X), [&](int i) -> tuple<XY, XY, S> {
          return {X[i], Y[i], MX::unit()};
        }) {}

  SegTree_2D(vc<XY>& X, vc<XY>& Y, vc<S>& vals)
      : SegTree_2D(len(X), [&](int i) -> tuple<XY, XY, S> {
          return {X[i], Y[i], vals[i]};
        }) {}

  // f(i) = (x,y,val)
  template <typename F>
  SegTree_2D(int N, F f) {
    vc<XY> X(N), Y(N);
    vc<S> wt(N);
    FOR(i, N) {
      auto [a, b, c] = f(i);
      X[i] = a, Y[i] = b, wt[i] = c;
    }
    if (!SMALL_X) {
      keyX = X;
      UNIQUE(keyX);
      NX = len(keyX);
    } else {
      minX = (X.empty() ? 0 : MIN(X));
      NX = (X.empty() ? 1 : MAX(X) - minX + 1);
    }

    log = 0;
    while ((1 << log) < NX) ++log;
    size = (1 << log);

    vc<int> IX(N);
    FOR(i, N) IX[i] = xtoi(X[i]);
    indptr.assign(2 * size, 0);
    for (auto i: IX) {
      i += size;
      while (i) indptr[i]++, i /= 2;
    }
    indptr = cumsum<int>(indptr);
    dat.assign(2 * indptr.back(), MX::unit());
    to_left.assign(indptr[size], 0);

    vc<int> ptr = indptr;
    vc<int> I = argsort(Y);
    pos.resize(N);
    FOR(i, N) pos[I[i]] = i;
    for (auto raw_idx: I) {
      int i = IX[raw_idx] + size;
      int j = -1;
      while (i) {
        int p = ptr[i];
        ptr[i]++;
        dat[indptr[i + 1] + p] = wt[raw_idx];
        if (j != -1) { to_left[p] = (j % 2 == 0); }
        j = i, i /= 2;
      }
    }
    to_left = cumsum<int>(to_left);

    FOR(i, 2 * size) {
      int off = 2 * indptr[i], n = indptr[i + 1] - indptr[i];
      FOR_R(j, 1, n) {
        dat[off + j] = MX::op(dat[off + 2 * j + 0], dat[off + 2 * j + 1]);
      }
    }
    all_Y = Y;
    sort(all(all_Y));
  }

  // 最初に与えた点群の index
  void multiply(int raw_idx, S val) {
    int i = 1, p = pos[raw_idx];
    while (1) {
      multiply_i(i, p - indptr[i], val);
      if (i >= size) break;
      int lc = to_left[p] - to_left[indptr[i]];
      int rc = (p - indptr[i]) - lc;
      if (to_left[p + 1] - to_left[p]) {
        p = indptr[2 * i + 0] + lc;
        i = 2 * i + 0;
      } else {
        p = indptr[2 * i + 1] + rc;
        i = 2 * i + 1;
      }
    }
  }

  // 最初に与えた点群の index
  void set(int raw_idx, S val) {
    int i = 1, p = pos[raw_idx];
    while (1) {
      set_i(i, p - indptr[i], val);
      if (i >= size) break;
      int lc = to_left[p] - to_left[indptr[i]];
      int rc = (p - indptr[i]) - lc;
      if (to_left[p + 1] - to_left[p]) {
        p = indptr[2 * i + 0] + lc;
        i = 2 * i + 0;
      } else {
        p = indptr[2 * i + 1] + rc;
        i = 2 * i + 1;
      }
    }
  }

  S prod(XY lx, XY rx, XY ly, XY ry) {
    assert(lx <= rx && ly <= ry);
    int L = xtoi(lx), R = xtoi(rx);
    S res = MX::unit();
    auto dfs = [&](auto& dfs, int i, int l, int r, int a, int b) -> void {
      if (a == b || R <= l || r <= L) return;
      if (L <= l && r <= R) {
        res = MX::op(res, prod_i(i, a, b));
        return;
      }
      int la = to_left[indptr[i] + a] - to_left[indptr[i]];
      int ra = a - la;
      int lb = to_left[indptr[i] + b] - to_left[indptr[i]];
      int rb = b - lb;
      int m = (l + r) / 2;
      dfs(dfs, 2 * i + 0, l, m, la, lb);
      dfs(dfs, 2 * i + 1, m, r, ra, rb);
    };
    dfs(dfs, 1, 0, size, LB(all_Y, ly), LB(all_Y, ry));
    return res;
  }

  // 矩形内の全点を数える, NlogN
  int count(XY lx, XY rx, XY ly, XY ry) {
    assert(lx <= rx && ly <= ry);
    int L = xtoi(lx), R = xtoi(rx);
    int res = 0;
    auto dfs = [&](auto& dfs, int i, int l, int r, int a, int b) -> void {
      if (a == b || R <= l || r <= L) return;
      if (L <= l && r <= R) {
        res += b - a;
        return;
      }
      int la = to_left[indptr[i] + a] - to_left[indptr[i]];
      int ra = a - la;
      int lb = to_left[indptr[i] + b] - to_left[indptr[i]];
      int rb = b - lb;
      int m = (l + r) / 2;
      dfs(dfs, 2 * i + 0, l, m, la, lb);
      dfs(dfs, 2 * i + 1, m, r, ra, rb);
    };
    dfs(dfs, 1, 0, size, LB(all_Y, ly), LB(all_Y, ry));
    return res;
  }

private:
  inline int xtoi(XY x) {
    if constexpr (SMALL_X) return clamp<XY>(x - minX, 0, NX);
    return LB(keyX, x);
  }

  S prod_i(int i, int a, int b) {
    int LID = indptr[i], n = indptr[i + 1] - indptr[i];
    int off = 2 * LID;
    int L = n + a, R = n + b;
    S val = MX::unit();
    while (L < R) {
      if (L & 1) val = MX::op(val, dat[off + (L++)]);
      if (R & 1) val = MX::op(dat[off + (--R)], val);
      L >>= 1, R >>= 1;
    }
    return val;
  }
  void multiply_i(int i, int j, S val) {
    int LID = indptr[i], n = indptr[i + 1] - indptr[i];
    int off = 2 * LID;
    j += n;
    while (j) {
      dat[off + j] = MX::op(dat[off + j], val);
      j >>= 1;
    }
  }
  void set_i(int i, int j, S val) {
    int LID = indptr[i], n = indptr[i + 1] - indptr[i];
    int off = 2 * LID;
    j += n;
    dat[off + j] = val;
    while (j > 1) {
      j /= 2;
      dat[off + j] = MX::op(dat[off + 2 * j + 0], dat[off + 2 * j + 1]);
    }
  }
};