rectangle_sum.hpp¶
对静态带权点集回答轴对齐矩形内权值和；适合所有点预先给出、随后只有矩形查询的题目。
Complexity: Time: \(O((P + Q) \log (P + Q))\) with the default Fenwick backend. Custom \(C\) follows its construction, add, and prod costs. Space: \(O(P + Q)\).
AC 记录：rectangle_sum。
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O((P + Q) log(P + Q)) with the default Fenwick backend.
/// Custom C follows its construction, add, and prod costs.
/// Space: O(P + Q).
#include "noya/ds/point_add_range_sum.hpp"
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Offline rectangle sum. Points (x, y, weight), queries [l, r) x [d, u).
template <class T, class C = noya::fenwick<T>>
std::vector<T> rectangle_sum(std::vector<std::array<int, 3>> pt,
std::vector<std::array<int, 4>> qs) {
std::vector<T> ans(qs.size());
if (pt.empty() || qs.empty()) {
return ans;
}
std::sort(pt.begin(), pt.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<int> ys;
ys.reserve(pt.size());
for (auto [x, y, w] : pt) {
ys.push_back(y);
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
for (auto &[x, y, w] : pt) {
y = int(std::lower_bound(ys.begin(), ys.end(), y) - ys.begin());
}
std::vector<std::array<int, 4>> events;
events.reserve(qs.size() * 2);
for (int i = 0; i < int(qs.size()); ++i) {
auto [l, r, d, u] = qs[i];
d = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
u = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
if (l == r || d == u) continue;
events.push_back({l, d, u, ~i});
events.push_back({r, d, u, i});
}
if (events.empty()) {
return ans;
}
std::sort(events.begin(), events.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t p = 0;
for (auto [x, d, u, idx] : events) {
// Strictly smaller x preserves both half-open rectangle boundaries.
while (p < pt.size() && pt[p][0] < x) {
bit.add(pt[p][1], T(pt[p][2]));
++p;
}
if (idx >= 0)
ans[idx] += bit.prod(d, u);
else
ans[~idx] -= bit.prod(d, u);
}
return ans;
}
} // namespace noya
// NOYA_RECTANGLE_SUM_HPP
#ifndef NOYA_RECTANGLE_SUM_HPP
#define NOYA_RECTANGLE_SUM_HPP 1
/// @complexity Time: O((P + Q) log(P + Q)) with the default Fenwick backend.
/// Custom C follows its construction, add, and prod costs.
/// Space: O(P + Q).
#include "noya/ds/point_add_range_sum.hpp"
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Offline rectangle sum. Points (x, y, weight), queries [l, r) x [d, u).
template <class T, class C = noya::fenwick<T>>
std::vector<T> rectangle_sum(std::vector<std::array<int, 3>> pt,
std::vector<std::array<int, 4>> qs) {
std::vector<T> ans(qs.size());
if (pt.empty() || qs.empty()) {
return ans;
}
std::sort(pt.begin(), pt.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<int> ys;
ys.reserve(pt.size());
for (auto [x, y, w] : pt) {
ys.push_back(y);
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
for (auto &[x, y, w] : pt) {
y = int(std::lower_bound(ys.begin(), ys.end(), y) - ys.begin());
}
std::vector<std::array<int, 4>> events;
events.reserve(qs.size() * 2);
for (int i = 0; i < int(qs.size()); ++i) {
auto [l, r, d, u] = qs[i];
d = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
u = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
if (l == r || d == u) continue;
events.push_back({l, d, u, ~i});
events.push_back({r, d, u, i});
}
if (events.empty()) {
return ans;
}
std::sort(events.begin(), events.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t p = 0;
for (auto [x, d, u, idx] : events) {
// Strictly smaller x preserves both half-open rectangle boundaries.
while (p < pt.size() && pt[p][0] < x) {
bit.add(pt[p][1], T(pt[p][2]));
++p;
}
if (idx >= 0)
ans[idx] += bit.prod(d, u);
else
ans[~idx] -= bit.prod(d, u);
}
return ans;
}
} // namespace noya
#endif // NOYA_RECTANGLE_SUM_HPP
/// @complexity Time: O((P + Q) log(P + Q)) with the default Fenwick backend.
/// Custom C follows its construction, add, and prod costs.
/// Space: O(P + Q).
/// @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
#include <algorithm>
#include <array>
#include <cstddef>
#include <vector>
namespace noya {
/// @brief Offline rectangle sum. Points (x, y, weight), queries [l, r) x [d, u).
template <class T, class C = noya::fenwick<T>>
std::vector<T> rectangle_sum(std::vector<std::array<int, 3>> pt,
std::vector<std::array<int, 4>> qs) {
std::vector<T> ans(qs.size());
if (pt.empty() || qs.empty()) {
return ans;
}
std::sort(pt.begin(), pt.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
std::vector<int> ys;
ys.reserve(pt.size());
for (auto [x, y, w] : pt) {
ys.push_back(y);
}
std::sort(ys.begin(), ys.end());
ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
for (auto &[x, y, w] : pt) {
y = int(std::lower_bound(ys.begin(), ys.end(), y) - ys.begin());
}
std::vector<std::array<int, 4>> events;
events.reserve(qs.size() * 2);
for (int i = 0; i < int(qs.size()); ++i) {
auto [l, r, d, u] = qs[i];
d = int(std::lower_bound(ys.begin(), ys.end(), d) - ys.begin());
u = int(std::lower_bound(ys.begin(), ys.end(), u) - ys.begin());
if (l == r || d == u) continue;
events.push_back({l, d, u, ~i});
events.push_back({r, d, u, i});
}
if (events.empty()) {
return ans;
}
std::sort(events.begin(), events.end(),
[](const auto &a, const auto &b) { return a[0] < b[0]; });
C bit(int(ys.size()));
std::size_t p = 0;
for (auto [x, d, u, idx] : events) {
// Strictly smaller x preserves both half-open rectangle boundaries.
while (p < pt.size() && pt[p][0] < x) {
bit.add(pt[p][1], T(pt[p][2]));
++p;
}
if (idx >= 0)
ans[idx] += bit.prod(d, u);
else
ans[~idx] -= bit.prod(d, u);
}
return ans;
}
} // namespace noya
// NOYA_RECTANGLE_SUM_HPP