lmori's Library

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:heavy_check_mark: verify/LibraryChecker/linear-algebra/RankofMatrix.test.cpp

Depends on

Code

#include "../../../template/template.hpp"
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_rank"
#include "../../../atcoder/modint.hpp"
#include "../../../linear-algebra/Matrix.hpp"
using namespace atcoder;

int main() {
  cin.tie(0)->sync_with_stdio(0);
  int n, m;
  in(n, m);
  Matrix<modint998244353> a(n, m);
  rep(i, n) rep(j, m) {
    int x;
    in(x);
    a[i][j] = x;
  }
  out(a.rank());
}
#line 2 "template/template.hpp"
#pragma region Macros
#include <bits/stdc++.h>
#include <tr2/dynamic_bitset>

using namespace std;
using namespace tr2;
using lint = long long;
using ull = unsigned long long;
using ld = long double;
using int128 = __int128_t;
#define all(x) (x).begin(), (x).end()
#define uniqv(v) v.erase(unique(all(v)), v.end())
#define OVERLOAD_REP(_1, _2, _3, name, ...) name
#define REP1(i, n) for (auto i = std::decay_t<decltype(n)>{}; (i) != (n); ++(i))
#define REP2(i, l, r) for (auto i = (l); (i) != (r); ++(i))
#define rep(...) OVERLOAD_REP(__VA_ARGS__, REP2, REP1)(__VA_ARGS__)
#define logfixed(x) cout << fixed << setprecision(10) << x << endl;

ostream &operator<<(ostream &dest, __int128_t value) {
  ostream::sentry s(dest);
  if (s) {
    __uint128_t tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    int len = end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(ios_base::badbit);
    }
  }
  return dest;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  for (int i = 0; i < (int)v.size(); i++) {
    os << v[i] << (i + 1 != (int)v.size() ? " " : "");
  }
  return os;
}

template <typename T>
ostream &operator<<(ostream &os, const set<T> &set_var) {
  for (auto itr = set_var.begin(); itr != set_var.end(); itr++) {
    os << *itr;
    ++itr;
    if (itr != set_var.end()) os << " ";
    itr--;
  }
  return os;
}

template <typename T>
ostream &operator<<(ostream &os, const unordered_set<T> &set_var) {
  for (auto itr = set_var.begin(); itr != set_var.end(); itr++) {
    os << *itr;
    ++itr;
    if (itr != set_var.end()) os << " ";
    itr--;
  }
  return os;
}

template <typename T, typename U>
ostream &operator<<(ostream &os, const map<T, U> &map_var) {
  for (auto itr = map_var.begin(); itr != map_var.end(); itr++) {
    os << itr->first << " -> " << itr->second << "\n";
  }
  return os;
}

template <typename T, typename U>
ostream &operator<<(ostream &os, const unordered_map<T, U> &map_var) {
  for (auto itr = map_var.begin(); itr != map_var.end(); itr++) {
    os << itr->first << " -> " << itr->second << "\n";
  }
  return os;
}

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &pair_var) {
  os << pair_var.first << " " << pair_var.second;
  return os;
}

void out() { cout << '\n'; }
template <class T, class... Ts>
void out(const T &a, const Ts &...b) {
  cout << a;
  (cout << ... << (cout << ' ', b));
  cout << '\n';
}

void outf() { cout << '\n'; }
template <class T, class... Ts>
void outf(const T &a, const Ts &...b) {
  cout << fixed << setprecision(14) << a;
  (cout << ... << (cout << ' ', b));
  cout << '\n';
}

template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (T &in : v) is >> in;
  return is;
}

inline void in(void) { return; }
template <typename First, typename... Rest>
void in(First &first, Rest &...rest) {
  cin >> first;
  in(rest...);
  return;
}

template <typename T>
bool chmax(T &a, const T &b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <typename T>
bool chmin(T &a, const T &b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}

vector<lint> dx8 = {1, 1, 0, -1, -1, -1, 0, 1};
vector<lint> dy8 = {0, 1, 1, 1, 0, -1, -1, -1};
vector<lint> dx4 = {1, 0, -1, 0};
vector<lint> dy4 = {0, 1, 0, -1};

#pragma endregion
#line 2 "verify/LibraryChecker/linear-algebra/RankofMatrix.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_rank"
#line 1 "atcoder/modint.hpp"



#line 6 "atcoder/modint.hpp"
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "atcoder/internal_math.hpp"



#line 5 "atcoder/internal_math.hpp"

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned long long y = x * _m;
        return (unsigned int)(z - y + (z < y ? _m : 0));
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 1 "atcoder/internal_type_traits.hpp"



#line 7 "atcoder/internal_type_traits.hpp"

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


#line 14 "atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 1 "linear-algebra/Matrix.hpp"
template <class S>
struct Matrix {
 private:
 public:
  vector<vector<S>> A;
  Matrix() {}
  Matrix(int n, int m) : A(n, vector<S>(m)) {}
  Matrix(int n) : A(n, vector<S>(n)) {}

  inline int size() const { return A.size(); }
  inline int height() const { return A.size(); }
  inline int width() const { return A[0].size(); }
  inline const vector<S>& operator[](int h) const { return (A[h]); }
  inline vector<S>& operator[](int h) { return (A[h]); }

  Matrix& operator+=(const Matrix& B) {
    int h = height();
    int w = width();
    for (int i = 0; i < h; i++) {
      for (int j = 0; j < w; j++) {
        (*this)[i][j] += B[i][j];
      }
    }
    return (*this);
  }
  Matrix& operator-=(const Matrix& B) {
    int h = height();
    int w = width();
    for (int i = 0; i < h; i++) {
      for (int j = 0; j < w; j++) {
        (*this)[i][j] -= B[i][j];
      }
    }
    return (*this);
  }

  Matrix& operator*=(const Matrix& B) {
    int h = height();
    int w = B.width();
    int c = width();
    vector<vector<S>> C(h, vector<S>(w));
    for (int i = 0; i < h; i++) {
      for (int j = 0; j < w; j++) {
        for (int k = 0; k < c; k++) {
          C[i][j] = (C[i][j] + (*this)[i][k] * B[k][j]);
        }
      }
    }
    A = move(C);
    return (*this);
  }
  Matrix operator+(const Matrix& B) const { return (Matrix(*this) += B); }
  Matrix operator-(const Matrix& B) const { return (Matrix(*this) -= B); }
  Matrix operator*(const Matrix& B) const { return (Matrix(*this) *= B); }

  int rank() {
    Matrix B(*this);
    if (B.height() == 0 or B.width() == 0) return 0;
    int res = 0;
    int h = height();
    int w = width();
    int ch = 0;
    int cw = 0;
    while (ch < h and cw < w) {
      bool ok = false;
      for (int j = cw; j < w; j++) {
        for (int i = ch; i < h; i++) {
          if (B[i][j] != 0) {
            ok = true;
            swap(B[ch], B[i]);
            S d = B[ch][j];
            for (int j2 = j; j2 < w; j2++) {
              B[ch][j2] /= d;
            }
            for (int i2 = 0; i2 < h; i2++) {
              if (B[i2][j] != 0 and i2 != ch) {
                S m = B[i2][j];
                for (int j2 = j; j2 < w; j2++) {
                  B[i2][j2] -= B[ch][j2] * m;
                }
              }
            }
            res++;
            ch++;
            cw = j + 1;
            break;
          }
        }
        if (ok) break;
      }
      if (!ok) break;
    }
    return res;
  }

  S determinant() {
    Matrix B(*this);
    if (B.height() == 0 or B.width() == 0) return 0;
    assert(B.height() == B.width());
    int h = height();
    int w = width();
    int ch = 0;
    int cw = 0;
    S div = 1;
    bool neg = false;
    while (ch < h and cw < w) {
      bool ok = false;
      for (int j = cw; j < w; j++) {
        for (int i = ch; i < h; i++) {
          if (B[i][j] != 0) {
            ok = true;
            if (ch != i) neg = !neg;
            swap(B[ch], B[i]);
            S d = B[ch][j];
            div /= d;
            for (int j2 = j; j2 < w; j2++) {
              B[ch][j2] /= d;
            }
            for (int i2 = 0; i2 < h; i2++) {
              if (B[i2][j] != 0 and i2 != ch) {
                S m = B[i2][j];
                for (int j2 = j; j2 < w; j2++) {
                  B[i2][j2] -= B[ch][j2] * m;
                }
              }
            }
            ch++;
            cw = j + 1;
            break;
          }
        }
        if (ok) {
          break;
        } else {
          return S(0);
        }
      }
      if (!ok) break;
    }
    S res = (neg ? -B[0][0] : B[0][0]) / div;
    for (int i = 1; i < h; i++) {
      res = res * B[i][i];
    }
    return res;
  }

  pair<bool, Matrix<S>> inverse() {
    int h = height();
    int w = width();
    assert(h == w);
    Matrix<S> B(h, w * 2);
    for (int i = 0; i < h; i++) {
      for (int j = 0; j < w; j++) {
        B[i][j] = (*this)[i][j];
      }
    }
    for (int i = 0; i < h; i++) {
      B[i][i + w] = 1;
    }
    w *= 2;
    int rnk = 0;

    int ch = 0;
    int cw = 0;
    while (ch < h and cw < h) {
      bool ok = false;
      for (int j = cw; j < h; j++) {
        for (int i = ch; i < h; i++) {
          if (B[i][j] != 0) {
            ok = true;
            swap(B[ch], B[i]);
            S d = B[ch][j];
            for (int j2 = j; j2 < w; j2++) {
              B[ch][j2] /= d;
            }
            for (int i2 = 0; i2 < h; i2++) {
              if (B[i2][j] != 0 and i2 != ch) {
                S m = B[i2][j];
                for (int j2 = j; j2 < w; j2++) {
                  B[i2][j2] -= B[ch][j2] * m;
                }
              }
            }
            rnk++;
            ch++;
            cw = j + 1;
            break;
          }
        }
        if (ok) break;
      }
      if (!ok) break;
    }
    Matrix<S> res(h);
    if (rnk == h) {
      for (int i = 0; i < h; i++) {
        for (int j = 0; j < h; j++) {
          res[i][j] = B[i][j + h];
        }
      }
      return {true, res};
    } else {
      return {false, res};
    }
  }

  Matrix<S> linear_equation(vector<S> b) {
    Matrix A(*this);

    int rnk = 0;
    assert(A.height() == b.size());
    int h = height();
    int w = width();
    int ch = 0;
    int cw = 0;
    vector<int> pivot_row(w, -1);
    while (ch < h and cw < w) {
      bool ok = false;
      for (int j = cw; j < w; j++) {
        for (int i = ch; i < h; i++) {
          if (A[i][j] != 0) {
            ok = true;
            swap(A[ch], A[i]);
            swap(b[ch], b[i]);
            S d = A[ch][j];
            for (int j2 = j; j2 < w; j2++) {
              A[ch][j2] /= d;
            }
            b[ch] /= d;
            for (int i2 = 0; i2 < h; i2++) {
              S m = A[i2][j];
              if (A[i2][j] != 0 and i2 != ch) {
                for (int j2 = j; j2 < w; j2++) {
                  A[i2][j2] -= A[ch][j2] * m;
                }
              }
              if (i2 != ch) b[i2] -= b[ch] * m;
            }
            pivot_row[j] = ch;
            rnk++;
            ch++;
            cw = j + 1;
            break;
          }
        }
        if (ok) break;
      }
      if (!ok) break;
    }

    for (int i = rnk; i < h; i++) {
      if (b[i] != 0) return Matrix<S>(0);
    }
    Matrix<S> sol(w - rnk + 1, w);
    int idx = 1;
    for (int j = 0; j < w; j++) {
      if (pivot_row[j] != -1) {
        sol[0][j] = b[pivot_row[j]];
      } else {
        sol[idx][j] = 1;
        for (int i = 0; i < w; i++) {
          if (pivot_row[i] != -1) {
            sol[idx][i] = -A[pivot_row[i]][j];
          }
        }
        idx++;
      }
    }
    return sol;
  }
};
#line 5 "verify/LibraryChecker/linear-algebra/RankofMatrix.test.cpp"
using namespace atcoder;

int main() {
  cin.tie(0)->sync_with_stdio(0);
  int n, m;
  in(n, m);
  Matrix<modint998244353> a(n, m);
  rep(i, n) rep(j, m) {
    int x;
    in(x);
    a[i][j] = x;
  }
  out(a.rank());
}
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