This documentation is automatically generated by online-judge-tools/verification-helper
#define PROBLEM "https://judge.yosupo.jp/problem/chromatic_number"
#include "../chromatic_number.hpp"
#include "../../number/factorize.hpp"
#include "../../number/modint_runtime.hpp"
#include "../../random/rand_nondeterministic.hpp"
#include <iostream>
#include <vector>
using namespace std;
int main() {
int N, M;
cin >> N >> M;
vector<int> to(N);
long long md = 4;
do { md = rnd(1LL << 29, 1LL << 30); } while (!is_prime(md));
cerr << md << '\n';
ModIntRuntime::set_mod(md);
while (M--) {
int u, v;
cin >> u >> v;
to[u] |= 1 << v;
to[v] |= 1 << u;
}
cout << ChromaticNumber<ModIntRuntime>(to) << '\n';
}#line 1 "graph/test/chromatic_number.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/chromatic_number"
#line 2 "graph/chromatic_number.hpp"
#include <numeric>
#include <vector>
// CUT begin
// (vertex) chromatic number: (頂点)彩色数
// Verified: https://judge.yosupo.jp/problem/chromatic_number, https://atcoder.jp/contests/abc187/tasks/abc187_f
// Reference:
// [1] A. Bjorklund and T. Husfeldt, "Inclusion--Exclusion Algorithms for Counting Set Partitions,"
// 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).
// - https://www.slideshare.net/wata_orz/ss-12131479
template <typename MODINT, typename Int> int ChromaticNumber(const std::vector<Int> &edge) {
const int V = edge.size(), S = 1 << V;
if (V == 0) return 0;
std::vector<MODINT> f(S), g(S);
for (int s = 0; s < S; s++) g[s] = (__builtin_popcount(s) & 1) ? 1 : -1;
f[0] = 1;
for (int s = 1; s < S; s++) {
int i = __builtin_ctz(s);
f[s] = f[s - (1 << i)] + f[(s - (1 << i)) & ~edge[i]];
}
for (int k = 1; k < V; k++) {
for (int s = 0; s < S; s++) g[s] *= f[s];
if (std::accumulate(g.begin(), g.end(), MODINT()).val()) return k;
}
return V;
};
#line 2 "random/xorshift.hpp"
#include <cstdint>
// CUT begin
uint32_t rand_int() // XorShift random integer generator
{
static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123;
uint32_t t = x ^ (x << 11);
x = y;
y = z;
z = w;
return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8));
}
double rand_double() { return (double)rand_int() / UINT32_MAX; }
#line 3 "number/factorize.hpp"
#include <algorithm>
#include <array>
#include <cassert>
#line 8 "number/factorize.hpp"
namespace SPRP {
// http://miller-rabin.appspot.com/
const std::vector<std::vector<__int128>> bases{
{126401071349994536}, // < 291831
{336781006125, 9639812373923155}, // < 1050535501 (1e9)
{2, 2570940, 211991001, 3749873356}, // < 47636622961201 (4e13)
{2, 325, 9375, 28178, 450775, 9780504, 1795265022} // <= 2^64
};
inline int get_id(long long n) {
if (n < 291831) {
return 0;
} else if (n < 1050535501) {
return 1;
} else if (n < 47636622961201)
return 2;
else { return 3; }
}
} // namespace SPRP
// Miller-Rabin primality test
// https://ja.wikipedia.org/wiki/%E3%83%9F%E3%83%A9%E3%83%BC%E2%80%93%E3%83%A9%E3%83%93%E3%83%B3%E7%B4%A0%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95
// Complexity: O(lg n) per query
struct {
long long modpow(__int128 x, __int128 n, long long mod) noexcept {
__int128 ret = 1;
for (x %= mod; n; x = x * x % mod, n >>= 1) ret = (n & 1) ? ret * x % mod : ret;
return ret;
}
bool operator()(long long n) noexcept {
if (n < 2) return false;
if (n % 2 == 0) return n == 2;
int s = __builtin_ctzll(n - 1);
for (__int128 a : SPRP::bases[SPRP::get_id(n)]) {
if (a % n == 0) continue;
a = modpow(a, (n - 1) >> s, n);
bool may_composite = true;
if (a == 1) continue;
for (int r = s; r--; a = a * a % n) {
if (a == n - 1) may_composite = false;
}
if (may_composite) return false;
}
return true;
}
} is_prime;
struct {
// Pollard's rho algorithm: find factor greater than 1
long long find_factor(long long n) {
assert(n > 1);
if (n % 2 == 0) return 2;
if (is_prime(n)) return n;
long long c = 1;
auto f = [&](__int128 x) -> long long { return (x * x + c) % n; };
for (int t = 1;; t++) {
for (c = 0; c == 0 or c + 2 == n;) c = rand_int() % n;
long long x0 = t, m = std::max(n >> 3, 1LL), x, ys, y = x0, r = 1, g, q = 1;
do {
x = y;
for (int i = r; i--;) y = f(y);
long long k = 0;
do {
ys = y;
for (int i = std::min(m, r - k); i--;)
y = f(y), q = __int128(q) * std::abs(x - y) % n;
g = std::__gcd<long long>(q, n);
k += m;
} while (k < r and g <= 1);
r <<= 1;
} while (g <= 1);
if (g == n) {
do {
ys = f(ys);
g = std::__gcd(std::abs(x - ys), n);
} while (g <= 1);
}
if (g != n) return g;
}
}
std::vector<long long> operator()(long long n) {
std::vector<long long> ret;
while (n > 1) {
long long f = find_factor(n);
if (f < n) {
auto tmp = operator()(f);
ret.insert(ret.end(), tmp.begin(), tmp.end());
} else
ret.push_back(n);
n /= f;
}
std::sort(ret.begin(), ret.end());
return ret;
}
long long euler_phi(long long n) {
long long ret = 1, last = -1;
for (auto p : this->operator()(n)) ret *= p - (last != p), last = p;
return ret;
}
} FactorizeLonglong;
#line 3 "number/modint_runtime.hpp"
#include <iostream>
#include <set>
#line 6 "number/modint_runtime.hpp"
struct ModIntRuntime {
private:
static int md;
public:
using lint = long long;
static int mod() { return md; }
int val_;
static std::vector<ModIntRuntime> &facs() {
static std::vector<ModIntRuntime> facs_;
return facs_;
}
static int &get_primitive_root() {
static int primitive_root_ = 0;
if (!primitive_root_) {
primitive_root_ = [&]() {
std::set<int> fac;
int v = md - 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 < md; g++) {
bool ok = true;
for (auto i : fac)
if (ModIntRuntime(g).power((md - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root_;
}
static void set_mod(const int &m) {
if (md != m) facs().clear();
md = m;
get_primitive_root() = 0;
}
ModIntRuntime &_setval(lint v) {
val_ = (v >= md ? v - md : v);
return *this;
}
int val() const noexcept { return val_; }
ModIntRuntime() : val_(0) {}
ModIntRuntime(lint v) { _setval(v % md + md); }
explicit operator bool() const { return val_ != 0; }
ModIntRuntime operator+(const ModIntRuntime &x) const {
return ModIntRuntime()._setval((lint)val_ + x.val_);
}
ModIntRuntime operator-(const ModIntRuntime &x) const {
return ModIntRuntime()._setval((lint)val_ - x.val_ + md);
}
ModIntRuntime operator*(const ModIntRuntime &x) const {
return ModIntRuntime()._setval((lint)val_ * x.val_ % md);
}
ModIntRuntime operator/(const ModIntRuntime &x) const {
return ModIntRuntime()._setval((lint)val_ * x.inv().val() % md);
}
ModIntRuntime operator-() const { return ModIntRuntime()._setval(md - val_); }
ModIntRuntime &operator+=(const ModIntRuntime &x) { return *this = *this + x; }
ModIntRuntime &operator-=(const ModIntRuntime &x) { return *this = *this - x; }
ModIntRuntime &operator*=(const ModIntRuntime &x) { return *this = *this * x; }
ModIntRuntime &operator/=(const ModIntRuntime &x) { return *this = *this / x; }
friend ModIntRuntime operator+(lint a, const ModIntRuntime &x) {
return ModIntRuntime()._setval(a % md + x.val_);
}
friend ModIntRuntime operator-(lint a, const ModIntRuntime &x) {
return ModIntRuntime()._setval(a % md - x.val_ + md);
}
friend ModIntRuntime operator*(lint a, const ModIntRuntime &x) {
return ModIntRuntime()._setval(a % md * x.val_ % md);
}
friend ModIntRuntime operator/(lint a, const ModIntRuntime &x) {
return ModIntRuntime()._setval(a % md * x.inv().val() % md);
}
bool operator==(const ModIntRuntime &x) const { return val_ == x.val_; }
bool operator!=(const ModIntRuntime &x) const { return val_ != x.val_; }
bool operator<(const ModIntRuntime &x) const {
return val_ < x.val_;
} // To use std::map<ModIntRuntime, T>
friend std::istream &operator>>(std::istream &is, ModIntRuntime &x) {
lint t;
return is >> t, x = ModIntRuntime(t), is;
}
friend std::ostream &operator<<(std::ostream &os, const ModIntRuntime &x) {
return os << x.val_;
}
lint power(lint n) const {
lint ans = 1, tmp = this->val_;
while (n) {
if (n & 1) ans = ans * tmp % md;
tmp = tmp * tmp % md;
n /= 2;
}
return ans;
}
ModIntRuntime pow(lint n) const { return power(n); }
ModIntRuntime inv() const { return this->pow(md - 2); }
static ModIntRuntime fac(int n) {
assert(n >= 0);
if (n >= md) return ModIntRuntime(0);
int l0 = facs().size();
if (l0 > n) return facs()[n];
facs().resize(n + 1);
for (int i = l0; i <= n; i++)
facs()[i] = (i == 0 ? ModIntRuntime(1) : facs()[i - 1] * ModIntRuntime(i));
return facs()[n];
}
static ModIntRuntime facinv(int n) { return ModIntRuntime::fac(n).inv(); }
static ModIntRuntime doublefac(int n) {
assert(n >= 0);
if (n >= md) return ModIntRuntime(0);
long long k = (n + 1) / 2;
return (n & 1)
? ModIntRuntime::fac(k * 2) / (ModIntRuntime(2).pow(k) * ModIntRuntime::fac(k))
: ModIntRuntime::fac(k) * ModIntRuntime(2).pow(k);
}
static ModIntRuntime nCr(int n, int r) {
assert(n >= 0);
if (r < 0 or n < r) return ModIntRuntime(0);
return ModIntRuntime::fac(n) / (ModIntRuntime::fac(r) * ModIntRuntime::fac(n - r));
}
static ModIntRuntime nPr(int n, int r) {
assert(n >= 0);
if (r < 0 or n < r) return ModIntRuntime(0);
return ModIntRuntime::fac(n) / ModIntRuntime::fac(n - r);
}
ModIntRuntime sqrt() const {
if (val_ == 0) return 0;
if (md == 2) return val_;
if (power((md - 1) / 2) != 1) return 0;
ModIntRuntime b = 1;
while (b.power((md - 1) / 2) == 1) b += 1;
int e = 0, m = md - 1;
while (m % 2 == 0) m >>= 1, e++;
ModIntRuntime x = power((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModIntRuntime z = b.power(m);
while (y != 1) {
int j = 0;
ModIntRuntime t = y;
while (t != 1) j++, t *= t;
z = z.power(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModIntRuntime(std::min(x.val_, md - x.val_));
}
};
int ModIntRuntime::md = 1;
#line 2 "random/rand_nondeterministic.hpp"
#include <chrono>
#include <random>
struct rand_int_ {
using lint = long long;
std::mt19937 mt;
// rand_int_() : mt(42) {}
rand_int_() : mt(std::chrono::steady_clock::now().time_since_epoch().count()) {}
lint operator()(lint x) { return this->operator()(0, x); } // [0, x)
lint operator()(lint l, lint r) {
std::uniform_int_distribution<lint> d(l, r - 1);
return d(mt);
}
} rnd;
#line 6 "graph/test/chromatic_number.test.cpp"
#line 9 "graph/test/chromatic_number.test.cpp"
using namespace std;
int main() {
int N, M;
cin >> N >> M;
vector<int> to(N);
long long md = 4;
do { md = rnd(1LL << 29, 1LL << 30); } while (!is_prime(md));
cerr << md << '\n';
ModIntRuntime::set_mod(md);
while (M--) {
int u, v;
cin >> u >> v;
to[u] |= 1 << v;
to[v] |= 1 << u;
}
cout << ChromaticNumber<ModIntRuntime>(to) << '\n';
}