cplib-cpp
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linear_algebra_matrix/test/linalg_min_plus_semiring.yuki1780.test.cpp
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- Last update: 2023-05-21 18:11:51+09:00
- Problem: https://yukicoder.me/problems/no/1780
Depends on
- linear_algebra_matrix/matrix.hpp
- Min-plus semiring / tropical semiring（(min, +) 半環・トロピカル半環）
(number/min_plus_semiring.hpp)
Code
#define PROBLEM "https://yukicoder.me/problems/no/1780"
#include "../../number/min_plus_semiring.hpp"
#include "../matrix.hpp"
#include <algorithm>
#include <iostream>
#include <string>
#include <tuple>
#include <vector>
using namespace std;
int main() {
constexpr int W = 26;
vector<int> C(W), K(W);
for (auto &x : C) cin >> x, --x;
for (auto &x : K) cin >> x;
int N;
cin >> N;
vector<tuple<string, int, int, int>> SABE;
for (int i = 0; i < N; ++i) {
string s;
int a, b, e;
cin >> s >> a >> b >> e;
--a, --b;
SABE.emplace_back(s, a, b, e);
}
constexpr int dim = 16;
using R = min_plus_semiring<long long>;
auto solve = [&](int init, int c, long long k) -> vector<R> {
auto mat = matrix<R>::Identity(dim);
for (const auto &sabe : SABE) {
string s;
int a, b, e;
std::tie(s, a, b, e) = sabe;
if (count(s.begin(), s.end(), (char)('A' + init)) and R(-e) < mat[a][b]) {
mat[a][b] = -e;
mat[b][a] = -e;
}
}
vector<R> initvec(dim);
initvec[c] = R::id();
return mat.pow_vec(k, initvec);
};
vector<R> rets(dim, R::id());
for (int w = 0; w < W; ++w) {
auto f = solve(w, C[w], K[w]);
for (int i = 0; i < dim; ++i) rets[i] *= f[i];
}
auto ret = *min_element(rets.begin(), rets.end());
cout << (ret != R() ? to_string(-ret.val) : "Impossible") << '\n';
}#line 1 "linear_algebra_matrix/test/linalg_min_plus_semiring.yuki1780.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/1780"
#line 1 "number/min_plus_semiring.hpp"
#include <limits>
// min-plus 半環・トロピカル半環（加法が min, 乗法が plus, 零元が INF, 単位元が 0）
// INF = numeric_limits<T>::max() / 2 を超えたら INF に clamp する
// Verified: https://yukicoder.me/problems/no/1340
template <class T> struct min_plus_semiring {
T val;
static const T _T_inf() {
static_assert(std::numeric_limits<T>::max() > 0,
"std::numeric_limits<>::max() must be properly defined");
return std::numeric_limits<T>::max() / 2;
}
min_plus_semiring() : val(_T_inf()) {}
min_plus_semiring(T x) : val(x) {}
static min_plus_semiring id() { return min_plus_semiring(0); }
min_plus_semiring operator+(const min_plus_semiring &r) const {
return (this->val > r.val ? r.val : this->val);
}
min_plus_semiring &operator+=(const min_plus_semiring &r) { return *this = *this + r; }
min_plus_semiring operator*(const min_plus_semiring &r) const {
if (this->val == _T_inf() or r.val == _T_inf()) return min_plus_semiring();
T tmp = this->val + r.val; // Watch out for overflow
return _T_inf() < tmp ? min_plus_semiring() : min_plus_semiring(tmp);
}
min_plus_semiring &operator*=(const min_plus_semiring &r) { return *this = *this * r; }
bool operator==(const min_plus_semiring &r) const { return this->val == r.val; }
bool operator!=(const min_plus_semiring &r) const { return !(*this == r); }
bool operator<(const min_plus_semiring &x) const { return this->val < x.val; }
bool operator>(const min_plus_semiring &x) const { return this->val > x.val; }
template <class OStream> friend OStream &operator<<(OStream &os, const min_plus_semiring &x) {
return os << x.val;
}
};
#line 2 "linear_algebra_matrix/matrix.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>
namespace matrix_ {
struct has_id_method_impl {
template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type());
template <class T_> static auto check(...) -> std::false_type;
};
template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {};
} // namespace matrix_
template <typename T> struct matrix {
int H, W;
std::vector<T> elem;
typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
inline T &at(int i, int j) { return elem[i * W + j]; }
inline T get(int i, int j) const { return elem[i * W + j]; }
int height() const { return H; }
int width() const { return W; }
std::vector<std::vector<T>> vecvec() const {
std::vector<std::vector<T>> ret(H);
for (int i = 0; i < H; i++) {
std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
}
return ret;
}
operator std::vector<std::vector<T>>() const { return vecvec(); }
matrix() = default;
matrix(int H, int W) : H(H), W(W), elem(H * W) {}
matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
}
template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr>
static T2 _T_id() {
return T2::id();
}
template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr>
static T2 _T_id() {
return T2(1);
}
static matrix Identity(int N) {
matrix ret(N, N);
for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>();
return ret;
}
matrix operator-() const {
matrix ret(H, W);
for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i];
return ret;
}
matrix operator*(const T &v) const {
matrix ret = *this;
for (auto &x : ret.elem) x *= v;
return ret;
}
matrix operator/(const T &v) const {
matrix ret = *this;
const T vinv = _T_id<T>() / v;
for (auto &x : ret.elem) x *= vinv;
return ret;
}
matrix operator+(const matrix &r) const {
matrix ret = *this;
for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i];
return ret;
}
matrix operator-(const matrix &r) const {
matrix ret = *this;
for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i];
return ret;
}
matrix operator*(const matrix &r) const {
matrix ret(H, r.W);
for (int i = 0; i < H; i++) {
for (int k = 0; k < W; k++) {
for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j);
}
}
return ret;
}
matrix &operator*=(const T &v) { return *this = *this * v; }
matrix &operator/=(const T &v) { return *this = *this / v; }
matrix &operator+=(const matrix &r) { return *this = *this + r; }
matrix &operator-=(const matrix &r) { return *this = *this - r; }
matrix &operator*=(const matrix &r) { return *this = *this * r; }
bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
bool operator<(const matrix &r) const { return elem < r.elem; }
matrix pow(int64_t n) const {
matrix ret = Identity(H);
bool ret_is_id = true;
if (n == 0) return ret;
for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
if (!ret_is_id) ret *= ret;
if ((n >> i) & 1) ret *= (*this), ret_is_id = false;
}
return ret;
}
std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const {
matrix x = *this;
while (n) {
if (n & 1) vec = x * vec;
x *= x;
n >>= 1;
}
return vec;
};
matrix transpose() const {
matrix ret(W, H);
for (int i = 0; i < H; i++) {
for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
}
return ret;
}
// Gauss-Jordan elimination
// - Require inverse for every non-zero element
// - Complexity: O(H^2 W)
template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr>
static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
int piv = -1;
for (int j = h; j < mtr.H; j++) {
if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c))))
piv = j;
}
return piv;
}
template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr>
static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
for (int j = h; j < mtr.H; j++) {
if (mtr.get(j, c) != T2()) return j;
}
return -1;
}
matrix gauss_jordan() const {
int c = 0;
matrix mtr(*this);
std::vector<int> ws;
ws.reserve(W);
for (int h = 0; h < H; h++) {
if (c == W) break;
int piv = choose_pivot(mtr, h, c);
if (piv == -1) {
c++;
h--;
continue;
}
if (h != piv) {
for (int w = 0; w < W; w++) {
std::swap(mtr[piv][w], mtr[h][w]);
mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant
}
}
ws.clear();
for (int w = c; w < W; w++) {
if (mtr.at(h, w) != T()) ws.emplace_back(w);
}
const T hcinv = _T_id<T>() / mtr.at(h, c);
for (int hh = 0; hh < H; hh++)
if (hh != h) {
const T coeff = mtr.at(hh, c) * hcinv;
for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff;
mtr.at(hh, c) = T();
}
c++;
}
return mtr;
}
int rank_of_gauss_jordan() const {
for (int i = H * W - 1; i >= 0; i--) {
if (elem[i] != 0) return i / W + 1;
}
return 0;
}
int rank() const { return gauss_jordan().rank_of_gauss_jordan(); }
T determinant_of_upper_triangle() const {
T ret = _T_id<T>();
for (int i = 0; i < H; i++) ret *= get(i, i);
return ret;
}
int inverse() {
assert(H == W);
std::vector<std::vector<T>> ret = Identity(H), tmp = *this;
int rank = 0;
for (int i = 0; i < H; i++) {
int ti = i;
while (ti < H and tmp[ti][i] == T()) ti++;
if (ti == H) {
continue;
} else {
rank++;
}
ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
T inv = _T_id<T>() / tmp[i][i];
for (int j = 0; j < W; j++) ret[i][j] *= inv;
for (int j = i + 1; j < W; j++) tmp[i][j] *= inv;
for (int h = 0; h < H; h++) {
if (i == h) continue;
const T c = -tmp[h][i];
for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c;
for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c;
}
}
*this = ret;
return rank;
}
friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) {
assert(m.W == int(v.size()));
std::vector<T> ret(m.H);
for (int i = 0; i < m.H; i++) {
for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j];
}
return ret;
}
friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) {
assert(int(v.size()) == m.H);
std::vector<T> ret(m.W);
for (int i = 0; i < m.H; i++) {
for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j);
}
return ret;
}
std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; }
std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); }
template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) {
os << "[(" << x.H << " * " << x.W << " matrix)";
os << "\n[column sums: ";
for (int j = 0; j < x.W; j++) {
T s = T();
for (int i = 0; i < x.H; i++) s += x.get(i, j);
os << s << ",";
}
os << "]";
for (int i = 0; i < x.H; i++) {
os << "\n[";
for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
os << "]";
}
os << "]\n";
return os;
}
template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) {
for (auto &v : x.elem) is >> v;
return is;
}
};
#line 5 "linear_algebra_matrix/test/linalg_min_plus_semiring.yuki1780.test.cpp"
#line 7 "linear_algebra_matrix/test/linalg_min_plus_semiring.yuki1780.test.cpp"
#include <iostream>
#include <string>
#include <tuple>
#line 11 "linear_algebra_matrix/test/linalg_min_plus_semiring.yuki1780.test.cpp"
using namespace std;
int main() {
constexpr int W = 26;
vector<int> C(W), K(W);
for (auto &x : C) cin >> x, --x;
for (auto &x : K) cin >> x;
int N;
cin >> N;
vector<tuple<string, int, int, int>> SABE;
for (int i = 0; i < N; ++i) {
string s;
int a, b, e;
cin >> s >> a >> b >> e;
--a, --b;
SABE.emplace_back(s, a, b, e);
}
constexpr int dim = 16;
using R = min_plus_semiring<long long>;
auto solve = [&](int init, int c, long long k) -> vector<R> {
auto mat = matrix<R>::Identity(dim);
for (const auto &sabe : SABE) {
string s;
int a, b, e;
std::tie(s, a, b, e) = sabe;
if (count(s.begin(), s.end(), (char)('A' + init)) and R(-e) < mat[a][b]) {
mat[a][b] = -e;
mat[b][a] = -e;
}
}
vector<R> initvec(dim);
initvec[c] = R::id();
return mat.pow_vec(k, initvec);
};
vector<R> rets(dim, R::id());
for (int w = 0; w < W; ++w) {
auto f = solve(w, C[w], K[w]);
for (int i = 0; i < dim; ++i) rets[i] *= f[i];
}
auto ret = *min_element(rets.begin(), rets.end());
cout << (ret != R() ? to_string(-ret.val) : "Impossible") << '\n';
}