rectangle_add_point_get.hpp¶
对多个轴对齐矩形整体加权，并查询指定点累计受到的贡献；适合矩形覆盖、点询问模型。
Complexity: Time: \(O(r \log (r + 1))\) preprocessing and \(O(\log^{2} (r + 1))\) per rectangle addition or point query; the batch helper is \(O((R + Q) \log (R + Q))\) with the default Fenwick backend (custom \(C\) follows its operation costs). Space: \(O(r \log r)\) for the dynamic structure and \(O(R + Q)\) for the helper.
AC 记录：rectangle_add_point_get。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(r log(r+1)) preprocessing and O(log^2(r+1)) per rectangle
/// addition or point query; the batch helper is O((R + Q) log(R + Q))
/// with the default Fenwick backend (custom C follows its operation costs).
/// Space: O(r log r) for the dynamic structure and O(R + Q) for the helper.
#include "noya/ds/point_add_range_sum.hpp"
#include "noya/ds/sparse_fenwick_2d.hpp"
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Rectangle-add point-get data structure on sparse coordinates.
/// A half-open rectangle update is its four-corner two-dimensional difference;
/// the value at a point is the inclusive prefix sum of those differences.
/// Register all rectangles before build(), including rectangles added later.
template <class T, class Coordinate = int> struct rectangle_add_point_get_tree {
sparse_fenwick_2d<T, Coordinate> bit;
void reserve_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up) {
if (L == R || dn == up) {
return;
}
bit.reserve(L, dn);
bit.reserve(L, up);
bit.reserve(R, dn);
bit.reserve(R, up);
}
void build() { bit.build(); }
void add_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up, const T &dif) {
if (L == R || dn == up) {
return;
}
bit.add(L, dn, dif);
bit.add(L, up, -dif);
bit.add(R, dn, -dif);
bit.add(R, up, dif);
}
T point_get(Coordinate x, Coordinate y) const {
return bit.prefix_sum_inclusive(x, y);
}
};
/// @brief Offline rectangle-add point-get queries.
/// Add weight w to [l, r) x [d, u), then return the value at every point.
/// Sweep the two vertical edges of each rectangle, maintaining y differences.
/// Coordinates may be negative; weights and their negations are computed in T.
template <class T, class C = noya::fenwick<T>>
std::vector<T>
rectangle_add_point_get(std::vector<std::array<int, 5>> rs,
std::vector<std::array<int, 2>> pt) {
std::vector<T> ans(pt.size());
if (pt.empty()) {
return ans;
}
std::vector<int> ys;
ys.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
ys.push_back(d);
ys.push_back(u);
}
if (ys.empty()) {
return ans;
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
// Right edges swap the y boundaries instead of negating an int weight.
std::vector<std::array<int, 4>> es;
es.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
int a = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
int b = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
es.push_back({l, a, b, w});
es.push_back({r, b, a, w});
}
std::sort(es.begin(), es.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<std::array<int, 3>> qs;
qs.reserve(pt.size());
for (int i = 0; i < int(pt.size()); ++i) {
qs.push_back({pt[i][0], pt[i][1], i});
}
std::sort(qs.begin(), qs.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t ei = 0;
for (auto [x, y, i] : qs) {
while (ei < es.size() && es[ei][0] <= x) {
auto [edge_x, a, b, w] = es[ei++];
T delta(w);
bit.add(a, delta);
bit.add(b, -delta);
}
int end = int(std::upper_bound(ys.begin(), ys.end(), y) - ys.begin());
ans[i] = bit.prod(0, end);
}
return ans;
}
} // namespace noya
// NOYA_RECTANGLE_ADD_POINT_GET_HPP
#ifndef NOYA_RECTANGLE_ADD_POINT_GET_HPP
#define NOYA_RECTANGLE_ADD_POINT_GET_HPP 1
/// @complexity Time: O(r log(r+1)) preprocessing and O(log^2(r+1)) per rectangle
/// addition or point query; the batch helper is O((R + Q) log(R + Q))
/// with the default Fenwick backend (custom C follows its operation costs).
/// Space: O(r log r) for the dynamic structure and O(R + Q) for the helper.
#include "noya/ds/point_add_range_sum.hpp"
#include "noya/ds/sparse_fenwick_2d.hpp"
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Rectangle-add point-get data structure on sparse coordinates.
/// A half-open rectangle update is its four-corner two-dimensional difference;
/// the value at a point is the inclusive prefix sum of those differences.
/// Register all rectangles before build(), including rectangles added later.
template <class T, class Coordinate = int> struct rectangle_add_point_get_tree {
sparse_fenwick_2d<T, Coordinate> bit;
void reserve_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up) {
if (L == R || dn == up) {
return;
}
bit.reserve(L, dn);
bit.reserve(L, up);
bit.reserve(R, dn);
bit.reserve(R, up);
}
void build() { bit.build(); }
void add_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up, const T &dif) {
if (L == R || dn == up) {
return;
}
bit.add(L, dn, dif);
bit.add(L, up, -dif);
bit.add(R, dn, -dif);
bit.add(R, up, dif);
}
T point_get(Coordinate x, Coordinate y) const {
return bit.prefix_sum_inclusive(x, y);
}
};
/// @brief Offline rectangle-add point-get queries.
/// Add weight w to [l, r) x [d, u), then return the value at every point.
/// Sweep the two vertical edges of each rectangle, maintaining y differences.
/// Coordinates may be negative; weights and their negations are computed in T.
template <class T, class C = noya::fenwick<T>>
std::vector<T>
rectangle_add_point_get(std::vector<std::array<int, 5>> rs,
std::vector<std::array<int, 2>> pt) {
std::vector<T> ans(pt.size());
if (pt.empty()) {
return ans;
}
std::vector<int> ys;
ys.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
ys.push_back(d);
ys.push_back(u);
}
if (ys.empty()) {
return ans;
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
// Right edges swap the y boundaries instead of negating an int weight.
std::vector<std::array<int, 4>> es;
es.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
int a = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
int b = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
es.push_back({l, a, b, w});
es.push_back({r, b, a, w});
}
std::sort(es.begin(), es.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<std::array<int, 3>> qs;
qs.reserve(pt.size());
for (int i = 0; i < int(pt.size()); ++i) {
qs.push_back({pt[i][0], pt[i][1], i});
}
std::sort(qs.begin(), qs.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t ei = 0;
for (auto [x, y, i] : qs) {
while (ei < es.size() && es[ei][0] <= x) {
auto [edge_x, a, b, w] = es[ei++];
T delta(w);
bit.add(a, delta);
bit.add(b, -delta);
}
int end = int(std::upper_bound(ys.begin(), ys.end(), y) - ys.begin());
ans[i] = bit.prod(0, end);
}
return ans;
}
} // namespace noya
#endif // NOYA_RECTANGLE_ADD_POINT_GET_HPP
/// @complexity Time: O(r log(r+1)) preprocessing and O(log^2(r+1)) per rectangle
/// addition or point query; the batch helper is O((R + Q) log(R + Q))
/// with the default Fenwick backend (custom C follows its operation costs).
/// Space: O(r log r) for the dynamic structure and O(R + Q) for the helper.
/// @complexity Time: fenwick O(log n) update/query; block O(1) update and
/// O(sqrt(n)) query. Both initialize in O(n).
/// Space: O(n).
#include <cassert>
#include <vector>
#include <cassert>
#include <numeric>
#include <type_traits>
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
namespace atcoder {
// Reference: https://en.wikipedia.org/wiki/Fenwick_tree
template <class T> struct fenwick_tree {
using U = internal::to_unsigned_t<T>;
public:
fenwick_tree() : _n(0) {}
explicit fenwick_tree(int n) : _n(n), data(n) {}
void add(int p, T x) {
assert(0 <= p && p < _n);
p++;
while (p <= _n) {
data[p - 1] += U(x);
p += p & -p;
}
}
T sum(int l, int r) {
assert(0 <= l && l <= r && r <= _n);
return sum(r) - sum(l);
}
private:
int _n;
std::vector<U> data;
U sum(int r) {
U s = 0;
while (r > 0) {
s += data[r - 1];
r -= r & -r;
}
return s;
}
};
} // namespace atcoder
#include <algorithm>
#include <cassert>
#include <cmath>
#include <vector>
namespace noya {
/// @brief Square-root decomposition for point additions and range sums.
template <class T> struct block {
int V = 0, B = 1;
block() = default;
block(const int &_V) { build(_V); }
std::vector<T> pt, blo;
void build(const int &_V) {
assert(_V >= 0);
V = _V;
B = std::max(1, int(std::sqrt(V)));
pt.assign(V, 0);
blo.assign(V / B + 1, 0);
}
void add(int x, T v) {
assert(0 <= x && x < V);
int bel = x / B;
blo[bel] += v;
pt[x] += v;
}
T query(int x) const {
assert(0 <= x && x <= V);
T res = 0;
int bel = x / B;
for (int i = 0; i < bel; i++)
res += blo[i];
int st = bel * B;
int end = x;
for (int i = st; i < end; i++)
res += pt[i];
return res;
}
/// @brief Sum of [l, r).
T prod(int l, int r) const {
assert(0 <= l && l <= r && r <= V);
if (l == 0) {
return query(r);
}
int lb = l / B, rb = r / B;
T res = 0;
if (lb == rb) {
for (int i = l; i < r; ++i)
res += pt[i];
return res;
}
if (l == lb * B) {
res += blo[lb];
} else {
for (int i = l; i < (lb + 1) * B; ++i)
res += pt[i];
}
for (int i = lb + 1; i < rb; ++i)
res += blo[i];
for (int i = rb * B; i < r; ++i)
res += pt[i];
return res;
}
};
/// @brief Fenwick-tree wrapper with prefix query and half-open prod aliases.
template <class T> struct fenwick : atcoder::fenwick_tree<T> {
using atcoder::fenwick_tree<T>::fenwick_tree;
using atcoder::fenwick_tree<T>::add;
T query(int x) { return this->sum(0, x); }
T prod(int l, int r) { return this->sum(l, r); }
};
} // namespace noya
/// @complexity Time: amortized O(1) registration, O(p log(p+1)) build and
/// O(log^2(p+1)) addition/query. Space: O(1 + p log(p+1)) cells.
#include <algorithm>
#include <cassert>
#include <utility>
#include <vector>
namespace noya {
/// @brief Offline-built sparse two-dimensional Fenwick tree.
/// Register every coordinate that may later be updated, then call build().
/// An x-Fenwick node stores only the registered y-coordinates contributing to
/// it, so the dense coordinate plane is replaced by O(p log p) cells.
template <class T, class Coordinate = int> struct sparse_fenwick_2d {
std::vector<std::pair<Coordinate, Coordinate>> pt;
std::vector<Coordinate> xs;
std::vector<std::vector<Coordinate>> ys;
std::vector<std::vector<T>> dat;
bool ok = false;
/// @brief Remove all registered coordinates and values.
void clear() {
pt.clear();
xs.clear();
ys.clear();
dat.clear();
ok = false;
}
/// @brief Register a coordinate that may be passed to add() after build().
void reserve(Coordinate x, Coordinate y) {
assert(!ok);
pt.emplace_back(x, y);
}
/// @brief Build the compressed Fenwick nodes with every value initially zero.
void build() {
assert(!ok);
xs.reserve(pt.size());
for (auto [x, y] : pt) {
(void)y;
xs.push_back(x);
}
std::sort(xs.begin(), xs.end());
xs.erase(std::unique(xs.begin(), xs.end()), xs.end());
ys.assign(xs.size() + 1, {});
// Global y order makes every node's coordinate list sorted as it is built.
std::sort(pt.begin(), pt.end(), [](const auto &a, const auto &b) {
return a.second < b.second;
});
for (const auto &[x, y] : pt) {
int idx = int(std::lower_bound(xs.begin(), xs.end(), x) - xs.begin());
for (idx++; idx <= int(xs.size()); idx += idx & -idx) {
// A previous visit with this y also populated every higher ancestor.
if (!ys[idx].empty() && ys[idx].back() == y) break;
ys[idx].push_back(y);
}
}
dat.resize(xs.size() + 1);
for (int idx = 1; idx <= int(xs.size()); idx++) {
dat[idx].assign(ys[idx].size() + 1, T{});
}
pt.clear();
pt.shrink_to_fit();
ok = true;
}
/// @brief Add delta at a coordinate registered before build().
void add(Coordinate x, Coordinate y, const T &dif) {
assert(ok);
auto ix = std::lower_bound(xs.begin(), xs.end(), x);
assert(ix != xs.end() && *ix == x);
int xi = int(ix - xs.begin()) + 1;
for (; xi <= int(xs.size()); xi += xi & -xi) {
auto iy = std::lower_bound(ys[xi].begin(), ys[xi].end(), y);
assert(iy != ys[xi].end() && *iy == y);
int yi = int(iy - ys[xi].begin()) + 1;
for (; yi < int(dat[xi].size()); yi += yi & -yi) {
dat[xi][yi] += dif;
}
}
}
/// @brief Sum values at coordinates (px, py) with px < x and py < y.
T prefix_sum(Coordinate x, Coordinate y) const {
assert(ok);
int xi = int(std::lower_bound(xs.begin(), xs.end(), x) - xs.begin());
T res{};
for (; xi > 0; xi -= xi & -xi) {
int yi = int(std::lower_bound(ys[xi].begin(), ys[xi].end(), y) -
ys[xi].begin());
for (; yi > 0; yi -= yi & -yi) {
res += dat[xi][yi];
}
}
return res;
}
/// @brief Sum values at coordinates (px, py) with px <= x and py <= y.
T prefix_sum_inclusive(Coordinate x, Coordinate y) const {
assert(ok);
int xi = int(std::upper_bound(xs.begin(), xs.end(), x) - xs.begin());
T res{};
for (; xi > 0; xi -= xi & -xi) {
int yi = int(std::upper_bound(ys[xi].begin(), ys[xi].end(), y) -
ys[xi].begin());
for (; yi > 0; yi -= yi & -yi) {
res += dat[xi][yi];
}
}
return res;
}
/// @brief Sum values in [l, r) x [dn, up).
T rectangle_sum(Coordinate l, Coordinate r, Coordinate dn,
Coordinate up) const {
assert(ok);
if (l == r || dn == up) return T{};
int xl = int(std::lower_bound(xs.begin(), xs.end(), l) - xs.begin());
int xr = int(std::lower_bound(xs.begin(), xs.end(), r) - xs.begin());
// Cancel shared ancestors of the two prefix paths in each dimension.
auto row_sum = [&](int x) {
const auto &v = ys[x];
int a = int(std::lower_bound(v.begin(), v.end(), dn) - v.begin());
int b = int(std::lower_bound(v.begin(), v.end(), up) - v.begin());
T sum{};
while (a != b) {
if (a < b) {
sum += dat[x][b];
b -= b & -b;
} else {
sum = sum - dat[x][a];
a -= a & -a;
}
}
return sum;
};
T sum{};
while (xl != xr) {
if (xl < xr) {
sum += row_sum(xr);
xr -= xr & -xr;
} else {
sum = sum - row_sum(xl);
xl -= xl & -xl;
}
}
return sum;
}
};
} // namespace noya
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Rectangle-add point-get data structure on sparse coordinates.
/// A half-open rectangle update is its four-corner two-dimensional difference;
/// the value at a point is the inclusive prefix sum of those differences.
/// Register all rectangles before build(), including rectangles added later.
template <class T, class Coordinate = int> struct rectangle_add_point_get_tree {
sparse_fenwick_2d<T, Coordinate> bit;
void reserve_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up) {
if (L == R || dn == up) {
return;
}
bit.reserve(L, dn);
bit.reserve(L, up);
bit.reserve(R, dn);
bit.reserve(R, up);
}
void build() { bit.build(); }
void add_rectangle(Coordinate L, Coordinate dn, Coordinate R,
Coordinate up, const T &dif) {
if (L == R || dn == up) {
return;
}
bit.add(L, dn, dif);
bit.add(L, up, -dif);
bit.add(R, dn, -dif);
bit.add(R, up, dif);
}
T point_get(Coordinate x, Coordinate y) const {
return bit.prefix_sum_inclusive(x, y);
}
};
/// @brief Offline rectangle-add point-get queries.
/// Add weight w to [l, r) x [d, u), then return the value at every point.
/// Sweep the two vertical edges of each rectangle, maintaining y differences.
/// Coordinates may be negative; weights and their negations are computed in T.
template <class T, class C = noya::fenwick<T>>
std::vector<T>
rectangle_add_point_get(std::vector<std::array<int, 5>> rs,
std::vector<std::array<int, 2>> pt) {
std::vector<T> ans(pt.size());
if (pt.empty()) {
return ans;
}
std::vector<int> ys;
ys.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
ys.push_back(d);
ys.push_back(u);
}
if (ys.empty()) {
return ans;
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
// Right edges swap the y boundaries instead of negating an int weight.
std::vector<std::array<int, 4>> es;
es.reserve(rs.size() * 2);
for (auto [l, r, d, u, w] : rs) {
if (l == r || d == u) {
continue;
}
int a = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
int b = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
es.push_back({l, a, b, w});
es.push_back({r, b, a, w});
}
std::sort(es.begin(), es.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<std::array<int, 3>> qs;
qs.reserve(pt.size());
for (int i = 0; i < int(pt.size()); ++i) {
qs.push_back({pt[i][0], pt[i][1], i});
}
std::sort(qs.begin(), qs.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t ei = 0;
for (auto [x, y, i] : qs) {
while (ei < es.size() && es[ei][0] <= x) {
auto [edge_x, a, b, w] = es[ei++];
T delta(w);
bit.add(a, delta);
bit.add(b, -delta);
}
int end = int(std::upper_bound(ys.begin(), ys.end(), y) - ys.begin());
ans[i] = bit.prod(0, end);
}
return ans;
}
} // namespace noya
// NOYA_RECTANGLE_ADD_POINT_GET_HPP