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algebra/test/system_of_linear_equations.test.cpp
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- Last update: 2025-06-28 10:05:47+09:00
- Problem: https://judge.yosupo.jp/problem/system_of_linear_equations
Depends on
- algebra/system_of_linear_equations.hpp
- number_theory/mod_int.hpp
- number_theory/utils.hpp
- template/fastio.hpp
- template/template.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations"
#include "../../template/template.hpp"
#include "../../template/fastio.hpp"
#include "../../algebra/system_of_linear_equations.hpp"
#include "../../number_theory/mod_int.hpp"
int main() {
using M = ModInt<998244353>;
i32 n, m;
rd.read(n, m);
VV<M> a(n, V<M>(m));
REP(i, n) REP(j, m) {
rd.read(a[i][j].val);
}
V<M> b(n);
REP(i, n) {
rd.read(b[i].val);
}
optional<pair<V<M>, VV<M>>> ans = linear_equations(n, m, a, b);
if (ans.has_value()) {
auto [c, d] = *ans;
wr.writeln(LEN(d));
REP(i, m) {
wr.write(c[i].val);
wr.write(" \n"[i + 1 == m]);
}
REP(i, LEN(d)) REP(j, m) {
wr.write(d[i][j].val);
wr.write(" \n"[j + 1 == m]);
}
} else {
wr.writeln(-1);
}
}#line 1 "algebra/test/system_of_linear_equations.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations"
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
#define OVERRIDE(a, b, c, d, ...) d
#define REP2(i, n) for (i32 i = 0; i < (i32)(n); ++i)
#define REP3(i, m, n) for (i32 i = (i32)(m); i < (i32)(n); ++i)
#define REP(...) OVERRIDE(__VA_ARGS__, REP3, REP2)(__VA_ARGS__)
#define PER2(i, n) for (i32 i = (i32)(n)-1; i >= 0; --i)
#define PER3(i, m, n) for (i32 i = (i32)(n)-1; i >= (i32)(m); --i)
#define PER(...) OVERRIDE(__VA_ARGS__, PER3, PER2)(__VA_ARGS__)
#define ALL(x) begin(x), end(x)
#define LEN(x) (i32)(x.size())
using namespace std;
using u32 = unsigned int;
using u64 = unsigned long long;
using i32 = signed int;
using i64 = signed long long;
using f64 = double;
using f80 = long double;
using pi = pair<i32, i32>;
using pl = pair<i64, i64>;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = V<V<T>>;
template <typename T>
using VVV = V<V<V<T>>>;
template <typename T>
using VVVV = V<V<V<V<T>>>>;
template <typename T>
using PQR = priority_queue<T, V<T>, greater<T>>;
template <typename T>
bool chmin(T &x, const T &y) {
if (x > y) {
x = y;
return true;
}
return false;
}
template <typename T>
bool chmax(T &x, const T &y) {
if (x < y) {
x = y;
return true;
}
return false;
}
template <typename T>
i32 lob(const V<T> &arr, const T &v) {
return (i32)(lower_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
i32 upb(const V<T> &arr, const T &v) {
return (i32)(upper_bound(ALL(arr), v) - arr.begin());
}
template <typename T>
V<i32> argsort(const V<T> &arr) {
V<i32> ret(arr.size());
iota(ALL(ret), 0);
sort(ALL(ret), [&](i32 i, i32 j) -> bool {
if (arr[i] == arr[j]) {
return i < j;
} else {
return arr[i] < arr[j];
}
});
return ret;
}
#ifdef INT128
using u128 = __uint128_t;
using i128 = __int128_t;
#endif
[[maybe_unused]] constexpr i32 INF = 1000000100;
[[maybe_unused]] constexpr i64 INF64 = 3000000000000000100;
struct SetUpIO {
SetUpIO() {
#ifdef FAST_IO
ios::sync_with_stdio(false);
cin.tie(nullptr);
#endif
cout << fixed << setprecision(15);
}
} set_up_io;
void scan(char &x) { cin >> x; }
void scan(u32 &x) { cin >> x; }
void scan(u64 &x) { cin >> x; }
void scan(i32 &x) { cin >> x; }
void scan(i64 &x) { cin >> x; }
void scan(f64 &x) { cin >> x; }
void scan(string &x) { cin >> x; }
template <typename T>
void scan(V<T> &x) {
for (T &ele : x) {
scan(ele);
}
}
void read() {}
template <typename Head, typename... Tail>
void read(Head &head, Tail &...tail) {
scan(head);
read(tail...);
}
#define CHAR(...) \
char __VA_ARGS__; \
read(__VA_ARGS__);
#define U32(...) \
u32 __VA_ARGS__; \
read(__VA_ARGS__);
#define U64(...) \
u64 __VA_ARGS__; \
read(__VA_ARGS__);
#define I32(...) \
i32 __VA_ARGS__; \
read(__VA_ARGS__);
#define I64(...) \
i64 __VA_ARGS__; \
read(__VA_ARGS__);
#define F64(...) \
f64 __VA_ARGS__; \
read(__VA_ARGS__);
#define STR(...) \
string __VA_ARGS__; \
read(__VA_ARGS__);
#define VEC(type, name, size) \
V<type> name(size); \
read(name);
#define VVEC(type, name, size1, size2) \
VV<type> name(size1, V<type>(size2)); \
read(name);
#line 3 "template/fastio.hpp"
#include <type_traits>
#line 5 "template/fastio.hpp"
// unable to read INT_MIN (int), LLONG_MIN (long long)
class Reader {
FILE *fp;
static constexpr int BUF = 1 << 18;
char buf[BUF];
char *pl, *pr;
void reread() {
int wd = pr - pl;
std::memcpy(buf, pl, wd);
pl = buf;
pr = buf + wd;
pr += std::fread(pr, 1, BUF - wd, fp);
}
char skip() {
char ch = *pl++;
while (ch <= ' ') {
ch = *pl++;
}
return ch;
}
template <typename T>
void read_unsigned(T &x) {
if (pr - pl < 64) {
reread();
}
x = 0;
char ch = skip();
while ('0' <= ch) {
x = 10 * x + (0xf & ch);
ch = *pl++;
}
}
template <typename T>
void read_signed(T &x) {
if (pr - pl < 64) {
reread();
}
x = 0;
bool neg = false;
char ch = skip();
if (ch == '-') {
ch = *pl++;
neg = true;
}
while ('0' <= ch) {
x = 10 * x + (0xf & ch);
ch = *pl++;
}
if (neg) {
x = -x;
}
}
void read_single(int &x) { read_signed(x); }
void read_single(unsigned &x) { read_unsigned(x); }
void read_single(long &x) { read_signed(x); }
void read_single(unsigned long &x) { read_signed(x); }
void read_single(long long &x) { read_signed(x); }
void read_single(unsigned long long &x) { read_unsigned(x); }
public:
Reader(FILE *fp) : fp(fp), pl(buf), pr(buf) { reread(); }
void read() {}
template <typename Head, typename... Tail>
void read(Head &head, Tail &...tail) {
read_single(head);
read(tail...);
}
};
struct NumberToString {
char buf[10000][4];
constexpr NumberToString() : buf() {
for (int i = 0; i < 10000; ++i) {
int n = i;
for (int j = 3; j >= 0; --j) {
buf[i][j] = '0' + n % 10;
n /= 10;
}
}
}
} constexpr number_to_string_precalc;
class Writer {
FILE *fp;
static constexpr int BUF = 1 << 18;
char buf[BUF];
char *ptr;
void write_u32(unsigned x) {
if ((buf + BUF - ptr) < 32) {
flush();
}
static char sml[12];
int t = 8;
while (x >= 10000) {
unsigned n = x % 10000;
x /= 10000;
std::memcpy(sml + t, number_to_string_precalc.buf[n], 4);
t -= 4;
}
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, sml + (t + 4), 8 - t);
ptr += 8 - t;
}
void write_u64(unsigned long long x) {
if ((buf + BUF - ptr) < 32) {
flush();
}
if (x >= 10000000000000000) {
unsigned long long z = x % 100000000;
x /= 100000000;
unsigned long long y = x % 100000000;
x /= 100000000;
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, number_to_string_precalc.buf[y / 10000], 4);
std::memcpy(ptr + 4, number_to_string_precalc.buf[y % 10000], 4);
std::memcpy(ptr + 8, number_to_string_precalc.buf[z / 10000], 4);
std::memcpy(ptr + 12, number_to_string_precalc.buf[z % 10000], 4);
ptr += 16;
} else {
static char sml[12];
int t = 8;
while (x >= 10000) {
unsigned long long n = x % 10000;
x /= 10000;
std::memcpy(sml + t, number_to_string_precalc.buf[n], 4);
t -= 4;
}
if (x >= 1000) {
std::memcpy(ptr, number_to_string_precalc.buf[x], 4);
ptr += 4;
} else if (x >= 100) {
std::memcpy(ptr, number_to_string_precalc.buf[x] + 1, 3);
ptr += 3;
} else if (x >= 10) {
unsigned q = (x * 103) >> 10;
*ptr++ = q | '0';
*ptr++ = (x - 10 * q) | '0';
} else {
*ptr++ = '0' | x;
}
std::memcpy(ptr, sml + (t + 4), 8 - t);
ptr += 8 - t;
}
}
void write_char(char c) {
if (ptr == buf + BUF) {
flush();
}
*ptr++ = c;
}
template <typename T>
void write_unsigned(T x) {
if constexpr (std::is_same_v<T, unsigned long long> ||
std::is_same_v<T, unsigned long>) {
write_u64(x);
} else {
write_u32(x);
}
}
template <typename T>
void write_signed(T x) {
std::make_unsigned_t<T> y = x;
if (x < 0) {
write_char('-');
y = -y;
}
write_unsigned(y);
}
void write_string(const std::string &s) {
for (char c : s) {
write_char(c);
}
}
void write_single(int x) { write_signed(x); }
void write_single(unsigned x) { write_unsigned(x); }
void write_single(long x) { write_signed(x); }
void write_single(unsigned long x) { write_unsigned(x); }
void write_single(long long x) { write_signed(x); }
void write_single(unsigned long long x) { write_unsigned(x); }
void write_single(char c) { write_char(c); }
void write_single(const std::string &s) { write_string(s); }
public:
Writer(FILE *fp) : fp(fp), ptr(buf) {}
~Writer() { flush(); }
void flush() {
std::fwrite(buf, 1, ptr - buf, fp);
ptr = buf;
}
void write() {}
template <typename Head, typename... Tail>
void write(Head &&head, Tail &&...tail) {
write_single(head);
if (sizeof...(Tail)) {
write_char(' ');
}
write(std::forward<Tail>(tail)...);
}
template <typename... T>
void writeln(T &&...t) {
write(std::forward<T>(t)...);
write_char('\n');
}
};
Reader rd(stdin);
Writer wr(stdout);
#line 2 "algebra/system_of_linear_equations.hpp"
#include <optional>
#line 4 "algebra/system_of_linear_equations.hpp"
template <typename T>
std::optional<std::pair<std::vector<T>, std::vector<std::vector<T>>>>
linear_equations(int n, int m, std::vector<std::vector<T>> a, std::vector<T> b) {
assert((int)a.size() == n);
for (int i = 0; i < n; ++i) {
assert((int)a[i].size() == m);
}
assert((int)b.size() == n);
int li = 0;
std::vector<int> spe(m, -1);
for (int i = 0; li < n && i < m; ++i) {
int row = -1;
for (int j = li; j < n; ++j) {
if (a[j][i] != T()) {
row = j;
break;
}
}
if (row == -1) {
continue;
}
std::swap(a[row], a[li]);
std::swap(b[row], b[li]);
T inv = a[li][i].inv();
for (int j = 0; j < m; ++j) {
a[li][j] *= inv;
}
b[li] *= inv;
for (int j = 0; j < n; ++j) {
if (j == li) {
continue;
}
T cf = a[j][i];
for (int k = 0; k < m; ++k) {
a[j][k] -= a[li][k] * cf;
}
b[j] -= b[li] * cf;
}
spe[i] = li++;
}
for (int i = li; i < n; ++i) {
if (b[i] != T()) {
return std::nullopt;
}
}
std::vector<T> ans(m, T());
std::vector<std::vector<T>> basis(m - li, std::vector<T>(m, T()));
int ph = 0;
for (int i = 0; i < m; ++i) {
if (spe[i] != -1) {
ans[i] = b[spe[i]];
} else {
basis[ph][i] = T(1);
for (int j = 0; j < i; ++j) {
if (int r = spe[j]; r != -1) {
basis[ph][j] = -a[r][i];
}
}
++ph;
}
}
return std::make_pair(ans, basis);
}
#line 2 "number_theory/mod_int.hpp"
#line 2 "number_theory/utils.hpp"
#line 4 "number_theory/utils.hpp"
constexpr bool is_prime(unsigned n) {
if (n == 0 || n == 1) {
return false;
}
for (unsigned i = 2; i * i <= n; ++i) {
if (n % i == 0) {
return false;
}
}
return true;
}
constexpr unsigned mod_pow(unsigned x, unsigned y, unsigned mod) {
unsigned ret = 1, self = x;
while (y != 0) {
if (y & 1) {
ret = (unsigned)((unsigned long long)ret * self % mod);
}
self = (unsigned)((unsigned long long)self * self % mod);
y /= 2;
}
return ret;
}
template <unsigned mod>
constexpr unsigned primitive_root() {
static_assert(is_prime(mod), "`mod` must be a prime number.");
if (mod == 2) {
return 1;
}
unsigned primes[32] = {};
int it = 0;
{
unsigned m = mod - 1;
for (unsigned i = 2; i * i <= m; ++i) {
if (m % i == 0) {
primes[it++] = i;
while (m % i == 0) {
m /= i;
}
}
}
if (m != 1) {
primes[it++] = m;
}
}
for (unsigned i = 2; i < mod; ++i) {
bool ok = true;
for (int j = 0; j < it; ++j) {
if (mod_pow(i, (mod - 1) / primes[j], mod) == 1) {
ok = false;
break;
}
}
if (ok) return i;
}
return 0;
}
// y >= 1
template <typename T>
constexpr T safe_mod(T x, T y) {
x %= y;
if (x < 0) {
x += y;
}
return x;
}
// y != 0
template <typename T>
constexpr T floor_div(T x, T y) {
if (y < 0) {
x *= -1;
y *= -1;
}
if (x >= 0) {
return x / y;
} else {
return -((-x + y - 1) / y);
}
}
// y != 0
template <typename T>
constexpr T ceil_div(T x, T y) {
if (y < 0) {
x *= -1;
y *= -1;
}
if (x >= 0) {
return (x + y - 1) / y;
} else {
return -(-x / y);
}
}
// b >= 1
// returns (g, x) s.t. g = gcd(a, b), a * x = g (mod b), 0 <= x < b / g
// from ACL
template <typename T>
std::pair<T, T> extgcd(T a, T b) {
a = safe_mod(a, b);
T s = b, t = a, m0 = 0, m1 = 1;
while (t) {
T u = s / t;
s -= t * u;
m0 -= m1 * u;
std::swap(s, t);
std::swap(m0, m1);
}
if (m0 < 0) {
m0 += b / s;
}
return std::pair<T, T>(s, m0);
}
// b >= 1
// returns (g, x, y) s.t. g = gcd(a, b), a * x + b * y = g, 0 <= x < b / g, |y| < max(2, |a| / g)
template <typename T>
std::tuple<T, T, T> extgcd2(T a, T b) {
T _a = safe_mod(a, b);
T quot = (a - _a) / b;
T x00 = 0, x01 = 1, y0 = b;
T x10 = 1, x11 = -quot, y1 = _a;
while (y1) {
T u = y0 / y1;
x00 -= u * x10;
x01 -= u * x11;
y0 -= u * y1;
std::swap(x00, x10);
std::swap(x01, x11);
std::swap(y0, y1);
}
if (x00 < 0) {
x00 += b / y0;
x01 -= a / y0;
}
return std::tuple<T, T, T>(y0, x00, x01);
}
// gcd(x, m) == 1
template <typename T>
T inv_mod(T x, T m) {
return extgcd(x, m).second;
}
#line 7 "number_theory/mod_int.hpp"
template <unsigned mod>
struct ModInt {
static_assert(mod != 0, "`mod` must not be equal to 0.");
static_assert(mod < (1u << 31),
"`mod` must be less than (1u << 31) = 2147483648.");
unsigned val;
static constexpr unsigned get_mod() { return mod; }
constexpr ModInt() : val(0) {}
template <typename T, std::enable_if_t<std::is_signed_v<T>> * = nullptr>
constexpr ModInt(T x)
: val((unsigned)((long long)x % (long long)mod + (x < 0 ? mod : 0))) {}
template <typename T, std::enable_if_t<std::is_unsigned_v<T>> * = nullptr>
constexpr ModInt(T x) : val((unsigned)(x % mod)) {}
static constexpr ModInt raw(unsigned x) {
ModInt<mod> ret;
ret.val = x;
return ret;
}
constexpr unsigned get_val() const { return val; }
constexpr ModInt operator+() const { return *this; }
constexpr ModInt operator-() const { return ModInt<mod>(0u) - *this; }
constexpr ModInt &operator+=(const ModInt &rhs) {
val += rhs.val;
if (val >= mod) val -= mod;
return *this;
}
constexpr ModInt &operator-=(const ModInt &rhs) {
val -= rhs.val;
if (val >= mod) val += mod;
return *this;
}
constexpr ModInt &operator*=(const ModInt &rhs) {
val = (unsigned long long)val * rhs.val % mod;
return *this;
}
constexpr ModInt &operator/=(const ModInt &rhs) {
val = (unsigned long long)val * rhs.inv().val % mod;
return *this;
}
friend constexpr ModInt operator+(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) += rhs;
}
friend constexpr ModInt operator-(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) -= rhs;
}
friend constexpr ModInt operator*(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) *= rhs;
}
friend constexpr ModInt operator/(const ModInt &lhs, const ModInt &rhs) {
return ModInt<mod>(lhs) /= rhs;
}
constexpr ModInt pow(unsigned long long x) const {
ModInt<mod> ret = ModInt<mod>::raw(1);
ModInt<mod> self = *this;
while (x != 0) {
if (x & 1) ret *= self;
self *= self;
x >>= 1;
}
return ret;
}
constexpr ModInt inv() const {
static_assert(is_prime(mod), "`mod` must be a prime number.");
assert(val != 0);
return this->pow(mod - 2);
}
friend std::istream &operator>>(std::istream &is, ModInt<mod> &x) {
long long val;
is >> val;
x.val = val % mod + (val < 0 ? mod : 0);
return is;
}
friend std::ostream &operator<<(std::ostream &os, const ModInt<mod> &x) {
os << x.val;
return os;
}
friend bool operator==(const ModInt &lhs, const ModInt &rhs) {
return lhs.val == rhs.val;
}
friend bool operator!=(const ModInt &lhs, const ModInt &rhs) {
return lhs.val != rhs.val;
}
};
template <unsigned mod>
void debug(ModInt<mod> x) {
std::cerr << x.val;
}
#line 6 "algebra/test/system_of_linear_equations.test.cpp"
int main() {
using M = ModInt<998244353>;
i32 n, m;
rd.read(n, m);
VV<M> a(n, V<M>(m));
REP(i, n) REP(j, m) {
rd.read(a[i][j].val);
}
V<M> b(n);
REP(i, n) {
rd.read(b[i].val);
}
optional<pair<V<M>, VV<M>>> ans = linear_equations(n, m, a, b);
if (ans.has_value()) {
auto [c, d] = *ans;
wr.writeln(LEN(d));
REP(i, m) {
wr.write(c[i].val);
wr.write(" \n"[i + 1 == m]);
}
REP(i, LEN(d)) REP(j, m) {
wr.write(d[i][j].val);
wr.write(" \n"[j + 1 == m]);
}
} else {
wr.writeln(-1);
}
}