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‎index.md‎

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path: linear_algebra_matrix/system_of_linear_equations.hpp
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title: linear_algebra_matrix/system_of_linear_equations.hpp
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- icon: ':heavy_check_mark:'
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path: linear_algebra_matrix/upper_triangular_matrix.hpp
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title: "Upper triangular matrix \uFF08\u5B9A\u6570\u6B21\u5143\u4E0A\u4E09\u89D2\
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\u884C\u5217\uFF09"
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pages:
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- icon: ':heavy_check_mark:'
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path: linear_algebra_matrix/test/system_of_linear_equations.test.cpp
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title: linear_algebra_matrix/test/system_of_linear_equations.test.cpp
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- icon: ':heavy_check_mark:'
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path: linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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title: linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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- name: number/test
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pages:
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‎linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp.md‎

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---
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data:
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_extendedDependsOn:
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- icon: ':heavy_check_mark:'
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path: linear_algebra_matrix/upper_triangular_matrix.hpp
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title: "Upper triangular matrix \uFF08\u5B9A\u6570\u6B21\u5143\u4E0A\u4E09\u89D2\
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\u884C\u5217\uFF09"
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- icon: ':heavy_check_mark:'
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path: modint.hpp
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title: modint.hpp
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_extendedRequiredBy: []
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_extendedVerifiedWith: []
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_isVerificationFailed: false
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_pathExtension: cpp
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_verificationStatusIcon: ':heavy_check_mark:'
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attributes:
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'*NOT_SPECIAL_COMMENTS*': ''
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PROBLEM: https://yukicoder.me/problems/no/3530
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links:
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- https://yukicoder.me/problems/no/3530
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bundledCode: "#line 1 \"linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp\"\
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\n#define PROBLEM \"https://yukicoder.me/problems/no/3530\"\n#line 2 \"linear_algebra_matrix/upper_triangular_matrix.hpp\"\
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\n\ntemplate <class T> struct UpperTriangular3d {\n static T explicit_init_required()\
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\ = delete;\n T a00 = this->explicit_init_required(), a01 = this->explicit_init_required(),\n\
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\ a02 = this->explicit_init_required();\n T a11 = this->explicit_init_required(),\
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\ a12 = this->explicit_init_required();\n T a22 = this->explicit_init_required();\n\
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\n UpperTriangular3d operator*(const UpperTriangular3d &r) const {\n \
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\ return UpperTriangular3d{\n .a00 = this->a00 * r.a00,\n \
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\ .a01 = this->a00 * r.a01 + this->a01 * r.a11,\n .a02 = this->a00\
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\ * r.a02 + this->a01 * r.a12 + this->a02 * r.a22,\n .a11 = this->a11\
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\ * r.a11,\n .a12 = this->a11 * r.a12 + this->a12 * r.a22,\n \
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\ .a22 = this->a22 * r.a22,\n };\n }\n\n UpperTriangular3d operator-()\
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\ const {\n return UpperTriangular3d{\n .a00 = -this->a00,\n\
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\ .a01 = -this->a01,\n .a02 = -this->a02,\n .a11\
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\ = -this->a11,\n .a12 = -this->a12,\n .a22 = -this->a22,\n\
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\ };\n }\n\n UpperTriangular3d operator+(const UpperTriangular3d\
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\ &r) const {\n return UpperTriangular3d{\n .a00 = this->a00\
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\ + r.a00,\n .a01 = this->a01 + r.a01,\n .a02 = this->a02\
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\ + r.a02,\n .a11 = this->a11 + r.a11,\n .a12 = this->a12\
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\ + r.a12,\n .a22 = this->a22 + r.a22,\n };\n }\n\n auto\
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\ operator<=>(const UpperTriangular3d &) const = default;\n};\n#line 2 \"modint.hpp\"\
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\n#include <cassert>\n#include <iostream>\n#include <set>\n#include <vector>\n\
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\ntemplate <int md> struct ModInt {\n static_assert(md > 1);\n using lint\
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\ = long long;\n constexpr static int mod() { return md; }\n static int\
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\ get_primitive_root() {\n static int primitive_root = 0;\n if (!primitive_root)\
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\ {\n primitive_root = [&]() {\n std::set<int> fac;\n\
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\ int v = md - 1;\n for (lint i = 2; i * i <= v;\
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\ i++)\n while (v % i == 0) fac.insert(i), v /= i;\n \
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\ if (v > 1) fac.insert(v);\n for (int g = 1; g < md;\
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\ g++) {\n bool ok = true;\n for (auto i\
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\ : fac)\n if (ModInt(g).pow((md - 1) / i) == 1) {\n \
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\ ok = false;\n break;\n \
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\ }\n if (ok) return g;\n \
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\ }\n return -1;\n }();\n }\n return\
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\ primitive_root;\n }\n int val_;\n int val() const noexcept { return\
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\ val_; }\n constexpr ModInt() : val_(0) {}\n constexpr ModInt &_setval(lint\
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\ v) { return val_ = (v >= md ? v - md : v), *this; }\n constexpr ModInt(lint\
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\ v) { _setval(v % md + md); }\n constexpr explicit operator bool() const {\
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\ return val_ != 0; }\n constexpr ModInt operator+(const ModInt &x) const {\n\
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\ return ModInt()._setval((lint)val_ + x.val_);\n }\n constexpr ModInt\
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\ operator-(const ModInt &x) const {\n return ModInt()._setval((lint)val_\
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\ - x.val_ + md);\n }\n constexpr ModInt operator*(const ModInt &x) const\
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\ {\n return ModInt()._setval((lint)val_ * x.val_ % md);\n }\n constexpr\
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\ ModInt operator/(const ModInt &x) const {\n return ModInt()._setval((lint)val_\
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\ * x.inv().val() % md);\n }\n constexpr ModInt operator-() const { return\
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\ ModInt()._setval(md - val_); }\n constexpr ModInt &operator+=(const ModInt\
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\ &x) { return *this = *this + x; }\n constexpr ModInt &operator-=(const ModInt\
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\ &x) { return *this = *this - x; }\n constexpr ModInt &operator*=(const ModInt\
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\ &x) { return *this = *this * x; }\n constexpr ModInt &operator/=(const ModInt\
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\ &x) { return *this = *this / x; }\n friend constexpr ModInt operator+(lint\
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\ a, const ModInt &x) { return ModInt(a) + x; }\n friend constexpr ModInt operator-(lint\
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\ a, const ModInt &x) { return ModInt(a) - x; }\n friend constexpr ModInt operator*(lint\
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\ a, const ModInt &x) { return ModInt(a) * x; }\n friend constexpr ModInt operator/(lint\
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\ a, const ModInt &x) { return ModInt(a) / x; }\n constexpr bool operator==(const\
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\ ModInt &x) const { return val_ == x.val_; }\n constexpr bool operator!=(const\
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\ ModInt &x) const { return val_ != x.val_; }\n constexpr bool operator<(const\
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\ ModInt &x) const {\n return val_ < x.val_;\n } // To use std::map<ModInt,\
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\ T>\n friend std::istream &operator>>(std::istream &is, ModInt &x) {\n \
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\ lint t;\n return is >> t, x = ModInt(t), is;\n }\n constexpr\
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\ friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {\n \
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\ return os << x.val_;\n }\n\n constexpr ModInt pow(lint n) const {\n\
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\ ModInt ans = 1, tmp = *this;\n while (n) {\n if (n\
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\ & 1) ans *= tmp;\n tmp *= tmp, n >>= 1;\n }\n return\
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\ ans;\n }\n\n static constexpr int cache_limit = std::min(md, 1 << 21);\n\
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\ static std::vector<ModInt> facs, facinvs, invs;\n\n constexpr static void\
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\ _precalculation(int N) {\n const int l0 = facs.size();\n if (N\
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\ > md) N = md;\n if (N <= l0) return;\n facs.resize(N), facinvs.resize(N),\
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\ invs.resize(N);\n for (int i = l0; i < N; i++) facs[i] = facs[i - 1]\
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\ * i;\n facinvs[N - 1] = facs.back().pow(md - 2);\n for (int i\
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\ = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);\n for (int\
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\ i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];\n }\n\n constexpr\
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\ ModInt inv() const {\n if (this->val_ < cache_limit) {\n if\
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\ (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};\n while (this->val_\
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\ >= int(facs.size())) _precalculation(facs.size() * 2);\n return invs[this->val_];\n\
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\ } else {\n return this->pow(md - 2);\n }\n }\n\n\
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\ constexpr static ModInt fac(int n) {\n assert(n >= 0);\n if\
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\ (n >= md) return ModInt(0);\n while (n >= int(facs.size())) _precalculation(facs.size()\
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\ * 2);\n return facs[n];\n }\n\n constexpr static ModInt facinv(int\
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\ n) {\n assert(n >= 0);\n if (n >= md) return ModInt(0);\n \
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\ while (n >= int(facs.size())) _precalculation(facs.size() * 2);\n \
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\ return facinvs[n];\n }\n\n constexpr static ModInt doublefac(int n) {\n\
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\ assert(n >= 0);\n if (n >= md) return ModInt(0);\n long\
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\ long k = (n + 1) / 2;\n return (n & 1) ? ModInt::fac(k * 2) / (ModInt(2).pow(k)\
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\ * ModInt::fac(k))\n : ModInt::fac(k) * ModInt(2).pow(k);\n\
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\ }\n\n constexpr static ModInt nCr(int n, int r) {\n assert(n >=\
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\ 0);\n if (r < 0 or n < r) return ModInt(0);\n return ModInt::fac(n)\
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\ * ModInt::facinv(r) * ModInt::facinv(n - r);\n }\n\n constexpr static\
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\ ModInt nPr(int n, int r) {\n assert(n >= 0);\n if (r < 0 or n\
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\ < r) return ModInt(0);\n return ModInt::fac(n) * ModInt::facinv(n - r);\n\
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\ }\n\n static ModInt binom(long long n, long long r) {\n static\
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\ long long bruteforce_times = 0;\n\n if (r < 0 or n < r) return ModInt(0);\n\
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\ if (n <= bruteforce_times or n < (int)facs.size()) return ModInt::nCr(n,\
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\ r);\n\n r = std::min(r, n - r);\n assert((int)r == r);\n\n \
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\ ModInt ret = ModInt::facinv(r);\n for (int i = 0; i < r; ++i) ret\
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\ *= n - i;\n bruteforce_times += r;\n\n return ret;\n }\n\n\
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\ // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)\n\
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\ // Complexity: O(sum(ks))\n // Verify: https://yukicoder.me/problems/no/3178\n\
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\ template <class Vec> static ModInt multinomial(const Vec &ks) {\n \
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\ ModInt ret{1};\n int sum = 0;\n for (int k : ks) {\n \
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\ assert(k >= 0);\n ret *= ModInt::facinv(k), sum += k;\n \
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\ }\n return ret * ModInt::fac(sum);\n }\n template <class... Args>\
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\ static ModInt multinomial(Args... args) {\n int sum = (0 + ... + args);\n\
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\ ModInt result = (1 * ... * ModInt::facinv(args));\n return ModInt::fac(sum)\
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\ * result;\n }\n\n // Catalan number, C_n = binom(2n, n) / (n + 1) = #\
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\ of Dyck words of length 2n\n // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 =\
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\ 14, ...\n // https://oeis.org/A000108\n // Complexity: O(n)\n static\
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\ ModInt catalan(int n) {\n if (n < 0) return ModInt(0);\n return\
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\ ModInt::fac(n * 2) * ModInt::facinv(n + 1) * ModInt::facinv(n);\n }\n\n \
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\ ModInt sqrt() const {\n if (val_ == 0) return 0;\n if (md ==\
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\ 2) return val_;\n if (pow((md - 1) / 2) != 1) return 0;\n ModInt\
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\ b = 1;\n while (b.pow((md - 1) / 2) == 1) b += 1;\n int e = 0,\
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\ m = md - 1;\n while (m % 2 == 0) m >>= 1, e++;\n ModInt x = pow((m\
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\ - 1) / 2), y = (*this) * x * x;\n x *= (*this);\n ModInt z = b.pow(m);\n\
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\ while (y != 1) {\n int j = 0;\n ModInt t = y;\n\
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\ while (t != 1) j++, t *= t;\n z = z.pow(1LL << (e - j\
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\ - 1));\n x *= z, z *= z, y *= z;\n e = j;\n }\n\
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\ return ModInt(std::min(x.val_, md - x.val_));\n }\n};\ntemplate <int\
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\ md> std::vector<ModInt<md>> ModInt<md>::facs = {1};\ntemplate <int md> std::vector<ModInt<md>>\
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\ ModInt<md>::facinvs = {1};\ntemplate <int md> std::vector<ModInt<md>> ModInt<md>::invs\
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\ = {0};\n\nusing ModInt998244353 = ModInt<998244353>;\n// using mint = ModInt<998244353>;\n\
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// using mint = ModInt<1000000007>;\n#line 4 \"linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp\"\
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\n\n#include <algorithm>\n#line 7 \"linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp\"\
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\n#include <map>\n#include <utility>\n#line 10 \"linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp\"\
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\n#include <atcoder/segtree>\n\nusing namespace std;\nusing S = UpperTriangular3d<ModInt998244353>;\n\
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\nS op(const S &l, const S &r) { return l * r; }\nS e() { return S{1, 0, 0, 1,\
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\ 0, 1}; }\n\nS GenR() { return S{ModInt998244353(3) / 4, ModInt998244353(1) /\
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\ 4, 0, 1, 0, 1}; }\nS GenL() { return S{1, 0, 0, ModInt998244353(3) / 4, ModInt998244353(1)\
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\ / 4, 1}; }\n\nModInt998244353 Solve(vector<pair<int, int>> ps) {\n vector<tuple<int,\
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\ int, int>> yxis;\n for (int i = 0; i < (int)ps.size(); ++i) {\n auto\
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\ [x, y] = ps.at(i);\n yxis.emplace_back(y, x, i);\n }\n sort(yxis.begin(),\
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\ yxis.end());\n\n const auto L = GenL(), R = GenR();\n const vector<S>\
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\ init(ps.size(), R);\n atcoder::segtree<S, op, e> seg(init);\n\n int last_x\
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\ = -1e9;\n\n ModInt998244353 ret = 0;\n\n map<int, vector<pair<int, int>>>\
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\ x2yis;\n for (int i = 0; i < (int)ps.size(); ++i) {\n auto [x, y]\
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\ = ps.at(i);\n x2yis[x].emplace_back(y, i);\n }\n for (auto [x,\
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\ yis] : x2yis) {\n\n const ModInt998244353 dx = x - last_x;\n ret\
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\ += dx * seg.all_prod().a02;\n\n for (auto [y, i] : yis) {\n \
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\ const int idx =\n lower_bound(yxis.begin(), yxis.end(), make_tuple(y,\
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\ x, i)) - yxis.begin();\n seg.set(idx, L);\n }\n\n last_x\
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\ = x;\n }\n\n return ret;\n}\n\nint main() {\n int N;\n cin >> N;\n\
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\ vector<pair<int, int>> xy(N);\n\n for (auto &[x, y] : xy) cin >> x >>\
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\ y;\n\n const ModInt998244353 coeff = ModInt998244353(4).pow(N);\n ModInt998244353\
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\ ret1 = Solve(xy) * coeff;\n for (auto &[x, y] : xy) swap(x, y);\n ModInt998244353\
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\ ret2 = Solve(xy) * coeff;\n cout << (ret1 + ret2) * 2 << '\\n';\n}\n"
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code: "#define PROBLEM \"https://yukicoder.me/problems/no/3530\"\n#include \"../upper_triangular_matrix.hpp\"\
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\n#include \"../../modint.hpp\"\n\n#include <algorithm>\n#include <iostream>\n\
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#include <map>\n#include <utility>\n#include <vector>\n#include <atcoder/segtree>\n\
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\nusing namespace std;\nusing S = UpperTriangular3d<ModInt998244353>;\n\nS op(const\
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\ S &l, const S &r) { return l * r; }\nS e() { return S{1, 0, 0, 1, 0, 1}; }\n\
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\nS GenR() { return S{ModInt998244353(3) / 4, ModInt998244353(1) / 4, 0, 1, 0,\
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\ 1}; }\nS GenL() { return S{1, 0, 0, ModInt998244353(3) / 4, ModInt998244353(1)\
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\ / 4, 1}; }\n\nModInt998244353 Solve(vector<pair<int, int>> ps) {\n vector<tuple<int,\
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\ int, int>> yxis;\n for (int i = 0; i < (int)ps.size(); ++i) {\n auto\
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\ [x, y] = ps.at(i);\n yxis.emplace_back(y, x, i);\n }\n sort(yxis.begin(),\
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\ yxis.end());\n\n const auto L = GenL(), R = GenR();\n const vector<S>\
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\ init(ps.size(), R);\n atcoder::segtree<S, op, e> seg(init);\n\n int last_x\
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\ = -1e9;\n\n ModInt998244353 ret = 0;\n\n map<int, vector<pair<int, int>>>\
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\ x2yis;\n for (int i = 0; i < (int)ps.size(); ++i) {\n auto [x, y]\
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\ = ps.at(i);\n x2yis[x].emplace_back(y, i);\n }\n for (auto [x,\
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\ yis] : x2yis) {\n\n const ModInt998244353 dx = x - last_x;\n ret\
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\ += dx * seg.all_prod().a02;\n\n for (auto [y, i] : yis) {\n \
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\ const int idx =\n lower_bound(yxis.begin(), yxis.end(), make_tuple(y,\
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\ x, i)) - yxis.begin();\n seg.set(idx, L);\n }\n\n last_x\
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\ = x;\n }\n\n return ret;\n}\n\nint main() {\n int N;\n cin >> N;\n\
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\ vector<pair<int, int>> xy(N);\n\n for (auto &[x, y] : xy) cin >> x >>\
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\ y;\n\n const ModInt998244353 coeff = ModInt998244353(4).pow(N);\n ModInt998244353\
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\ ret1 = Solve(xy) * coeff;\n for (auto &[x, y] : xy) swap(x, y);\n ModInt998244353\
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\ ret2 = Solve(xy) * coeff;\n cout << (ret1 + ret2) * 2 << '\\n';\n}\n"
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dependsOn:
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- linear_algebra_matrix/upper_triangular_matrix.hpp
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- modint.hpp
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isVerificationFile: true
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path: linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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requiredBy: []
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timestamp: '2026-05-06 21:05:36+09:00'
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verificationStatus: TEST_ACCEPTED
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verifiedWith: []
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documentation_of: linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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layout: document
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redirect_from:
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- /verify/linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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title: linear_algebra_matrix/test/upper_trinaglular_matrix.yuki3530.test.cpp
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---

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