結果
| 問題 | No.1303 Inconvenient Kingdom |
| ユーザー |
 hitonanode
|
| 提出日時 | 2020-11-27 22:37:53 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 5 ms / 3,000 ms |
| コード長 | 18,884 bytes |
| コンパイル時間 | 3,505 ms |
| コンパイル使用メモリ | 242,456 KB |
| 最終ジャッジ日時 | 2025-01-16 08:01:09 |
|
ジャッジサーバーID （参考情報） |
judge3 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 34 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T, typename V>
void ndarray(vector<T>& vec, const V& val, int len) { vec.assign(len, val); }
template <typename T, typename V, typename... Args> void ndarray(vector<T>& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); }
template <typename T> bool chmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; }
template <typename T> bool chmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; }
template <typename T1, typename T2> pair<T1, T2> operator+(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first + r.first, l.second + r.second); }
template <typename T1, typename T2> pair<T1, T2> operator-(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first - r.first, l.second - r.second); }
template <typename T> vector<T> sort_unique(vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <typename T> istream &operator>>(istream &is, vector<T> &vec) { for (auto &v : vec) is >> v; return is; }
template <typename T> ostream &operator<<(ostream &os, const vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
#if __cplusplus >= 201703L
template <typename... T> istream &operator>>(istream &is, tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <typename... T> ostream &operator<<(ostream &os, const tuple<T...> &tpl) { std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os; }
#endif
template <typename T> ostream &operator<<(ostream &os, const deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename T> ostream &operator<<(ostream &os, const set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T, typename TH> ostream &operator<<(ostream &os, const unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; }
template <typename TK, typename TV> ostream &operator<<(ostream &os, const map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <typename TK, typename TV, typename TH> ostream &operator<<(ostream &os, const unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl
#else
#define dbg(x) {}
#endif
template <int mod> struct ModInt {
#if __cplusplus >= 201402L
#define MDCONST constexpr
#else
#define MDCONST
#endif
using lint = long long;
static int get_mod() { return mod; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
std::set<int> fac;
int v = mod - 1;
for (lint i = 2; i * i <= v; i++)
while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < mod; g++) {
bool ok = true;
for (auto i : fac)
if (ModInt(g).power((mod - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val;
MDCONST ModInt() : val(0) {}
MDCONST ModInt &_setval(lint v) {
val = (v >= mod ? v - mod : v);
return *this;
}
MDCONST ModInt(lint v) { _setval(v % mod + mod); }
explicit operator bool() const { return val != 0; }
MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); }
MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); }
MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); }
MDCONST ModInt operator-() const { return ModInt()._setval(mod - val); }
MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); }
friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); }
friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); }
friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); }
MDCONST bool operator==(const ModInt &x) const { return val == x.val; }
MDCONST bool operator!=(const ModInt &x) const { return val != x.val; }
MDCONST bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map<ModInt, T>
friend std::istream &operator>>(std::istream &is, ModInt &x) {
lint t;
return is >> t, x = ModInt(t), is;
}
MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; }
MDCONST lint power(lint n) const {
lint ans = 1, tmp = this->val;
while (n) {
if (n & 1) ans = ans * tmp % mod;
tmp = tmp * tmp % mod, n /= 2;
}
return ans;
}
MDCONST ModInt pow(lint n) const { return power(n); }
MDCONST lint inv() const { return this->power(mod - 2); }
ModInt fac() const {
static std::vector<ModInt> facs;
int l0 = facs.size();
if (l0 > this->val) return facs[this->val];
facs.resize(this->val + 1);
for (int i = l0; i <= this->val; i++) facs[i] = (i == 0 ? ModInt(1) : facs[i - 1] * ModInt(i));
return facs[this->val];
}
ModInt doublefac() const {
lint k = (this->val + 1) / 2;
return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k);
}
ModInt nCr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() / ((*this - r).fac() * r.fac()); }
ModInt sqrt() const {
if (val == 0) return 0;
if (mod == 2) return val;
if (power((mod - 1) / 2) != 1) return 0;
ModInt b = 1;
while (b.power((mod - 1) / 2) == 1) b += 1;
int e = 0, m = mod - 1;
while (m % 2 == 0) m >>= 1, e++;
ModInt x = power((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModInt z = b.power(m);
while (y != 1) {
int j = 0;
ModInt t = y;
while (t != 1) j++, t *= t;
z = z.power(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModInt(std::min(x.val, mod - x.val));
}
};
using mint = ModInt<998244353>;
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]; }
operator std::vector<std::vector<T>>() 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;
}
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));
}
static matrix Identity(int N) {
matrix ret(N, N);
for (int i = 0; i < N; i++) ret.at(i, i) = 1;
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;
for (auto &x : ret.elem) x /= v;
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);
if (n == 0) return ret;
for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
ret *= ret;
if ((n >> i) & 1) ret *= (*this);
}
return ret;
}
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)
matrix gauss_jordan() const {
int c = 0;
matrix mtr(*this);
for (int h = 0; h < H; h++) {
if (c == W) break;
int piv = -1;
for (int j = h; j < H; j++)
if (mtr.get(j, c)) {
piv = j;
break;
}
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) *= -1; // To preserve sign of determinant
}
}
for (int hh = 0; hh < H; hh++)
if (hh != h) {
T coeff = mtr.at(hh, c) * mtr.at(h, c).inv();
for (int w = W - 1; w >= c; w--) { mtr.at(hh, w) -= mtr.at(h, w) * coeff; }
}
c++;
}
return mtr;
}
int rank_of_gauss_jordan() const {
for (int i = H * W - 1; i >= 0; i--)
if (elem[i]) return i / W + 1;
return 0;
}
T determinant_of_upper_triangle() const {
T ret = 1;
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] == 0) ti++;
if (ti == H)
continue;
else
rank++;
ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
T inv = tmp[i][i].inv();
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;
}
friend std::ostream &operator<<(std::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 = 0;
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;
}
friend std::istream &operator>>(std::istream &is, matrix &x) {
for (auto &v : x.elem) is >> v;
return is;
}
};
// Fibonacci numbers f(n) = af(n - 1) + bf(n - 2)
// Example (a = b = 1): 0=>1, 1=>1, 2=>2, 3=>3, 4=>5, ...
template <typename T> T Fibonacci(long long int k, int a = 1, int b = 1) {
matrix<T> mat(2, 2);
mat[0][1] = 1;
mat[1][0] = b;
mat[1][1] = a;
return mat.pow(k + 1)[0][1];
}
// UnionFind Tree (0-indexed), based on size of each disjoint set
struct UnionFind {
std::vector<int> par, cou;
UnionFind(int N = 0) : par(N), cou(N, 1) { iota(par.begin(), par.end(), 0); }
int find(int x) { return (par[x] == x) ? x : (par[x] = find(par[x])); }
bool unite(int x, int y) {
x = find(x), y = find(y);
if (x == y) return false;
if (cou[x] < cou[y]) std::swap(x, y);
par[y] = x, cou[x] += cou[y];
return true;
}
int count(int x) { return cou[find(x)]; }
bool same(int x, int y) { return find(x) == find(y); }
};
mint gyoretsuki(vector<int> vs, vector<pint> edges) {
int D = vs.size();
sort(ALL(vs));
matrix<mint> mat(D - 1, D - 1);
for (auto [u, v] : edges) {
const int i = lower_bound(ALL(vs), u) - vs.begin();
const int j = lower_bound(ALL(vs), v) - vs.begin();
if (i < D - 1) mat[i][i] += 1;
if (j < D - 1) mat[j][j] += 1;
if (i + 1 < D and j + 1 < D) {
mat[i][j] -= 1, mat[j][i] -= 1;
}
}
mat = mat.gauss_jordan();
return mat.determinant_of_upper_triangle();
}
mint solve1(int N, vector<pint> edges) {
if (N <= 1) return 1;
vector d0(N, vector<mint>(N));
vector d1(N, vector<mint>(N));
for (auto [u, v] : edges) {
d0[u][u] += 1;
d0[v][v] += 1;
d0[v][u] -= 1;
d0[u][v] -= 1;
}
REP(i, N) REP(j, i) if (d0[i][j] == 0) {
d1[i][j] -= 1;
d1[j][i] -= 1;
d1[i][i] += 1;
d1[j][j] += 1;
}
d0.resize(N - 1);
d1.resize(N - 1);
mint r0 = 1, r1 = 0;
REP(i, N - 1) d0[i].resize(N - 1), d1[i].resize(N - 1);
dbg(d0);
dbg(d1);
REP(i, N - 1) {
mint p = d0[i][i].inv(), q = -d1[i][i] * p * p;
mint r0new = r0 * d0[i][i];
mint r1new = r1 * d0[i][i] + r0 * d1[i][i];
REP(j, N - 1) {
d1[i][j] = d1[i][j] * p + d0[i][j] * q;
d0[i][j] = d0[i][j] * p;
}
dbg(d0);
dbg(d1);
FOR(l, i + 1, N - 1) {
IFOR(j, i, N - 1) {
d1[l][j] -= d1[l][i] * d0[i][j] + d0[l][i] * d1[i][j];
d0[l][j] -= d0[l][i] * d0[i][j];
}
}
dbg(d0);
dbg(d1);
r0 = r0new, r1 = r1new;
}
dbg(d0);
dbg(d1);
return r0 + r1;
}
int main()
{
int N, M;
cin >> N >> M;
vector<pint> edges;
UnionFind uf1(N);
REP(e, M) {
int u, v;
cin >> u >> v;
u--, v--;
edges.emplace_back(u, v);
uf1.unite(u, v);
}
if (uf1.count(0) == N) {
cout << "0\n" << solve1(N, edges) << '\n';
return 0;
}
vector<pint> sz2r;
REP(i, N) sz2r.emplace_back(uf1.count(i), uf1.find(i));
sz2r = sort_unique(sz2r);
reverse(ALL(sz2r));
map<int, mint> conn_add;
int fuben = 0;
REP(i, N) REP(j, N) fuben += !uf1.same(i, j);
dbg(fuben);
REP(i, N) REP(j, i) if (!uf1.same(i, j)) {
conn_add[uf1.count(i) * uf1.count(j)] += 1;
}
dbg(conn_add);
mint ret = prev(conn_add.end())->second;
map<int, vector<int>> r2is;
map<int, vector<pint>> r2edges;
REP(i, N) r2is[uf1.find(i)].emplace_back(i);
for (auto [u, v] : edges) r2edges[uf1.find(u)].emplace_back(u, v);
dbg(r2is);
dbg(r2edges);
dbg(ret);
for (auto [r, is] : r2is) {
ret *= gyoretsuki(is, r2edges[r]);
dbg(r);
dbg(ret);
}
cout << fuben - prev(conn_add.end())->first * 2 << '\n' << ret << '\n';
}