結果
| 問題 |
No.2405 Minimal Matrix Decomposition
|
| コンテスト | |
| ユーザー |
 tokusakurai
|
| 提出日時 | 2023-08-04 22:17:54 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 427 ms / 2,000 ms |
| コード長 | 12,830 bytes |
| コンパイル時間 | 2,430 ms |
| コンパイル使用メモリ | 210,096 KB |
| 最終ジャッジ日時 | 2025-02-15 22:52:45 |
|
ジャッジサーバーID （参考情報） |
judge4 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 46 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define per(i, n) for (int i = (n)-1; i >= 0; i--)
#define rep2(i, l, r) for (int i = (l); i < (r); i++)
#define per2(i, l, r) for (int i = (r)-1; i >= (l); i--)
#define each(e, v) for (auto &e : v)
#define MM << " " <<
#define pb push_back
#define eb emplace_back
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sz(x) (int)x.size()
using ll = long long;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxheap = priority_queue<T>;
template <typename T>
bool chmax(T &x, const T &y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
bool chmin(T &x, const T &y) {
return (x > y) ? (x = y, true) : false;
}
template <typename T>
int flg(T x, int i) {
return (x >> i) & 1;
}
int pct(int x) { return __builtin_popcount(x); }
int pct(ll x) { return __builtin_popcountll(x); }
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int botbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int botbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
void print(const vector<T> &v, T x = 0) {
int n = v.size();
for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' ');
if (v.empty()) cout << '\n';
}
template <typename T>
void printn(const vector<T> &v, T x = 0) {
int n = v.size();
for (int i = 0; i < n; i++) cout << v[i] + x << '\n';
}
template <typename T>
int lb(const vector<T> &v, T x) {
return lower_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, T x) {
return upper_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
void rearrange(vector<T> &v) {
sort(begin(v), end(v));
v.erase(unique(begin(v), end(v)), end(v));
}
template <typename T>
vector<int> id_sort(const vector<T> &v, bool greater = false) {
int n = v.size();
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; });
return ret;
}
template <typename T>
void reorder(vector<T> &a, const vector<int> &ord) {
int n = a.size();
vector<T> b(n);
for (int i = 0; i < n; i++) b[i] = a[ord[i]];
swap(a, b);
}
template <typename T>
T floor(T x, T y) {
assert(y != 0);
if (y < 0) x = -x, y = -y;
return (x >= 0 ? x / y : (x - y + 1) / y);
}
template <typename T>
T ceil(T x, T y) {
assert(y != 0);
if (y < 0) x = -x, y = -y;
return (x >= 0 ? (x + y - 1) / y : x / y);
}
template <typename S, typename T>
pair<S, T> operator+(const pair<S, T> &p, const pair<S, T> &q) {
return make_pair(p.first + q.first, p.second + q.second);
}
template <typename S, typename T>
pair<S, T> operator-(const pair<S, T> &p, const pair<S, T> &q) {
return make_pair(p.first - q.first, p.second - q.second);
}
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
S a;
T b;
is >> a >> b;
p = make_pair(a, b);
return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, const pair<S, T> &p) {
return os << p.first << ' ' << p.second;
}
struct io_setup {
io_setup() {
ios_base::sync_with_stdio(false);
cin.tie(NULL);
cout << fixed << setprecision(15);
}
} io_setup;
constexpr int inf = (1 << 30) - 1;
constexpr ll INF = (1LL << 60) - 1;
// constexpr int MOD = 1000000007;
constexpr int MOD = 998244353;
struct Runtime_Mod_Int {
int x;
Runtime_Mod_Int() : x(0) {}
Runtime_Mod_Int(long long y) {
x = y % get_mod();
if (x < 0) x += get_mod();
}
static inline int &get_mod() {
static int mod = 0;
return mod;
}
static void set_mod(int md) { get_mod() = md; }
Runtime_Mod_Int &operator+=(const Runtime_Mod_Int &p) {
if ((x += p.x) >= get_mod()) x -= get_mod();
return *this;
}
Runtime_Mod_Int &operator-=(const Runtime_Mod_Int &p) {
if ((x += get_mod() - p.x) >= get_mod()) x -= get_mod();
return *this;
}
Runtime_Mod_Int &operator*=(const Runtime_Mod_Int &p) {
x = (int)(1LL * x * p.x % get_mod());
return *this;
}
Runtime_Mod_Int &operator/=(const Runtime_Mod_Int &p) {
*this *= p.inverse();
return *this;
}
Runtime_Mod_Int &operator++() { return *this += Runtime_Mod_Int(1); }
Runtime_Mod_Int operator++(int) {
Runtime_Mod_Int tmp = *this;
++*this;
return tmp;
}
Runtime_Mod_Int &operator--() { return *this -= Runtime_Mod_Int(1); }
Runtime_Mod_Int operator--(int) {
Runtime_Mod_Int tmp = *this;
--*this;
return tmp;
}
Runtime_Mod_Int operator-() const { return Runtime_Mod_Int(-x); }
Runtime_Mod_Int operator+(const Runtime_Mod_Int &p) const { return Runtime_Mod_Int(*this) += p; }
Runtime_Mod_Int operator-(const Runtime_Mod_Int &p) const { return Runtime_Mod_Int(*this) -= p; }
Runtime_Mod_Int operator*(const Runtime_Mod_Int &p) const { return Runtime_Mod_Int(*this) *= p; }
Runtime_Mod_Int operator/(const Runtime_Mod_Int &p) const { return Runtime_Mod_Int(*this) /= p; }
bool operator==(const Runtime_Mod_Int &p) const { return x == p.x; }
bool operator!=(const Runtime_Mod_Int &p) const { return x != p.x; }
Runtime_Mod_Int inverse() const {
assert(*this != Runtime_Mod_Int(0));
return pow(get_mod() - 2);
}
Runtime_Mod_Int pow(long long k) const {
Runtime_Mod_Int now = *this, ret = 1;
for (; k > 0; k >>= 1, now *= now) {
if (k & 1) ret *= now;
}
return ret;
}
friend ostream &operator<<(ostream &os, const Runtime_Mod_Int &p) { return os << p.x; }
friend istream &operator>>(istream &is, Runtime_Mod_Int &p) {
long long a;
is >> a;
p = Runtime_Mod_Int(a);
return is;
}
};
using mint = Runtime_Mod_Int;
template <typename T>
struct Matrix {
vector<vector<T>> A;
int n, m;
Matrix(int n, int m) : A(n, vector<T>(m, 0)), n(n), m(m) {}
inline const vector<T> &operator[](int k) const { return A[k]; }
inline vector<T> &operator[](int k) { return A[k]; }
static Matrix I(int l) {
Matrix ret(l, l);
for (int i = 0; i < l; i++) ret[i][i] = 1;
return ret;
}
Matrix &operator*=(const Matrix &B) {
assert(m == B.n);
Matrix ret(n, B.m);
for (int i = 0; i < n; i++) {
for (int k = 0; k < m; k++) {
for (int j = 0; j < B.m; j++) ret[i][j] += A[i][k] * B[k][j];
}
}
swap(A, ret.A);
m = B.m;
return *this;
}
Matrix operator*(const Matrix &B) const { return Matrix(*this) *= B; }
Matrix pow(long long k) const {
assert(n == m);
Matrix now = *this, ret = I(n);
for (; k > 0; k >>= 1, now *= now) {
if (k & 1) ret *= now;
}
return ret;
}
bool eq(const T &a, const T &b) const {
return a == b;
// return abs(a-b) <= EPS;
}
// 行基本変形を用いて簡約化を行い、(rank, det) の組を返す
pair<int, T> row_reduction(vector<T> &b) {
assert((int)b.size() == n);
if (n == 0) return make_pair(0, m > 0 ? 0 : 1);
int check = 0, rank = 0;
T det = (n == m ? 1 : 0);
for (int j = 0; j < m; j++) {
int pivot = check;
for (int i = check; i < n; i++) {
if (A[i][j] != 0) pivot = i;
// if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; // T が小数の場合はこちら
}
if (check != pivot) det *= T(-1);
swap(A[check], A[pivot]), swap(b[check], b[pivot]);
if (eq(A[check][j], T(0))) {
det = T(0);
continue;
}
rank++;
det *= A[check][j];
T r = T(1) / A[check][j];
for (int k = j + 1; k < m; k++) A[check][k] *= r;
b[check] *= r;
A[check][j] = T(1);
for (int i = 0; i < n; i++) {
if (i == check) continue;
if (!eq(A[i][j], 0)) {
for (int k = j + 1; k < m; k++) A[i][k] -= A[i][j] * A[check][k];
b[i] -= A[i][j] * b[check];
}
A[i][j] = T(0);
}
if (++check == n) break;
}
return make_pair(rank, det);
}
pair<int, T> row_reduction() {
vector<T> b(n, T(0));
return row_reduction(b);
}
// 行基本変形を行い、逆行列を求める
pair<bool, Matrix> inverse() {
if (n != m) return make_pair(false, Matrix(0, 0));
if (n == 0) return make_pair(true, Matrix(0, 0));
Matrix ret = I(n);
for (int j = 0; j < n; j++) {
int pivot = j;
for (int i = j; i < n; i++) {
if (A[i][j] != 0) pivot = i;
// if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; // T が小数の場合はこちら
}
swap(A[j], A[pivot]), swap(ret[j], ret[pivot]);
if (eq(A[j][j], T(0))) return make_pair(false, Matrix(0, 0));
T r = T(1) / A[j][j];
for (int k = j + 1; k < n; k++) A[j][k] *= r;
for (int k = 0; k < n; k++) ret[j][k] *= r;
A[j][j] = T(1);
for (int i = 0; i < n; i++) {
if (i == j) continue;
if (!eq(A[i][j], T(0))) {
for (int k = j + 1; k < n; k++) A[i][k] -= A[i][j] * A[j][k];
for (int k = 0; k < n; k++) ret[i][k] -= A[i][j] * ret[j][k];
}
A[i][j] = T(0);
}
}
return make_pair(true, ret);
}
// Ax = b の解の 1 つと解空間の基底の組を返す
vector<vector<T>> Gaussian_elimination(vector<T> b) {
row_reduction(b);
vector<vector<T>> ret;
vector<int> p(n, m);
vector<bool> is_zero(m, true);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
if (!eq(A[i][j], T(0))) {
p[i] = j;
break;
}
}
if (p[i] < m) {
is_zero[p[i]] = false;
} else if (!eq(b[i], T(0))) {
return {};
}
}
vector<T> x(m, T(0));
for (int i = 0; i < n; i++) {
if (p[i] < m) x[p[i]] = b[i];
}
ret.push_back(x);
for (int j = 0; j < m; j++) {
if (!is_zero[j]) continue;
x[j] = T(1);
for (int i = 0; i < n; i++) {
if (p[i] < m) x[p[i]] = -A[i][j];
}
ret.push_back(x);
x[j] = T(0);
}
return ret;
}
};
void solve() {
int P;
cin >> P;
mint::set_mod(P);
using mat = Matrix<mint>;
int N, M;
cin >> N >> M;
vector A(N, vector(M, mint(0)));
rep(i, N) rep(j, M) cin >> A[i][j];
auto B = A;
vector<vector<mint>> C(N, vector<mint>(M, 0)), R;
int r = 0;
vector<int> bit;
rep(i, N) {
rep(k, r) {
mint c = A[i][bit[k]] / R[k][bit[k]];
rep(j, M) {
A[i][j] -= R[k][j] * c; //
}
C[i][k] = c;
}
int id = -1;
rep(j, M) {
if (A[i][j] != 0) id = j;
}
if (id != -1) {
R.eb(A[i]);
C[i][r] = 1;
bit.eb(id);
r++;
}
}
rep(i, N) C[i].resize(r);
if (r == 0) {
r++;
if (N * M <= N * r + r * M) {
cout << 1 << '\n';
cout << N MM M << '\n';
rep(i, N) print(B[i]);
return;
} else {
cout << 2 << '\n';
cout << N MM r << '\n';
rep(i, N) print(vector<mint>(r, 0));
cout << r MM M << '\n';
rep(i, r) print(vector<mint>(M, 0));
return;
}
}
if (N * M <= N * r + r * M) {
cout << 1 << '\n';
cout << N MM M << '\n';
rep(i, N) print(B[i]);
return;
} else {
cout << 2 << '\n';
cout << N MM r << '\n';
rep(i, N) print(C[i]);
cout << r MM M << '\n';
rep(i, r) print(R[i]);
return;
}
}
int main() {
int T = 1;
// cin >> T;
while (T--) solve();
}