// competitive-verifier: LOCALCASE test/cpv_local/_data/bzoj/1006
#include "../../../src/fast/rsort32/lib.hpp"
#include "../../../src/graph/chordal/lib.hpp"
#include "../../../src/graph/ds/alist/lib.hpp"
using namespace tifa_libs;
using std::cin, std::cout;
int main() {
u32 n, m;
cin >> n >> m;
alist g(n);
for (u32 i = 0, u, v; i < m; ++i) {
cin >> u >> v;
--u;
--v;
g.add_arc(u, v);
g.add_arc(v, u);
}
flt_ (u32, i, 0, n) ::sort(g[i]);
chordal chd(g);
cout << chd.chromatic_number() << '\n';
}
#line 1 "test/cpv_local/graph/chordal.bzoj1006.cpp"
// competitive-verifier: LOCALCASE test/cpv_local/_data/bzoj/1006
#line 2 "src/fast/rsort32/lib.hpp"
#line 2 "src/util/alias/others/lib.hpp"
#line 2 "src/util/consts/lib.hpp"
#line 2 "src/util/alias/num/lib.hpp"
#line 2 "src/util/util/lib.hpp"
// https://github.com/Tiphereth-A/CP-lib
#include <bits/extc++.h>
// clang-format off
namespace tifa_libs {
#define CEXP constexpr
#define CEXPE constexpr explicit
#define CR const&
#define CP const*
#define PC *const
#define CPC const*const
#define TPN typename
#define NE noexcept
#define CNE const noexcept
#define ND [[nodiscard]]
#define cT_(...) std::conditional_t<sizeof(__VA_ARGS__) <= sizeof(size_t) * 2, __VA_ARGS__, __VA_ARGS__ CR>
// NOLINTNEXTLINE(misc-const-correctness)
#define flt_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i < i##e; ++i)
#define retif_(cond, if_true, ...) if cond return if_true __VA_OPT__(; else return __VA_ARGS__)
#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif
using namespace std::ranges;
using namespace std::literals;
template <class T>
CEXP T abs(T x) NE { retif_((x < 0), -x, x); }
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/num/lib.hpp"
// clang-format off
namespace tifa_libs {
#define mk0_(w, t) using w = t; using c##w = const t
#define mk_(w, t) mk0_(w, t); CEXP w operator""_##w(unsigned long long x) NE { return (w)x; }
mk_(i8, int8_t) mk_(u8, uint8_t) mk_(i16, int16_t) mk_(u16, uint16_t) mk_(i32, int32_t) mk_(u32, uint32_t) mk_(i64, int64_t) mk_(u64, uint64_t) mk_(isz, ssize_t) mk_(usz, size_t) mk_(chr, char) mk_(schr, signed char) mk_(uchr, unsigned char) mk_(sint, signed) mk_(uint, unsigned);
mk0_(i128, __int128_t); mk0_(u128, __uint128_t); mk0_(f32, float); mk0_(f64, double); mk0_(f128, long double);
#undef mk0_
#undef mk_
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/consts/lib.hpp"
// clang-format off
namespace tifa_libs {
using std::numbers::pi_v;
template <std::floating_point FP>
inline FP eps_v = std::sqrt(std::numeric_limits<FP>::epsilon());
template <std::floating_point FP>
CEXP void set_eps(FP v) NE { eps_v<FP> = v; }
CEXP u32 TIME = ((__TIME__[0] & 15) << 20) | ((__TIME__[1] & 15) << 16) | ((__TIME__[3] & 15) << 12) | ((__TIME__[4] & 15) << 8) | ((__TIME__[6] & 15) << 4) | (__TIME__[7] & 15);
CEXP auto STR2U16 = [] { std::array<u32, 65536> table{}; table.fill(-1_u32); flt_ (u32, i, 48, 58) flt_ (u32, j, 48, 58) table[i << 8 | j] = (j & 15) * 10 + (i & 15); return table; }();
inline const auto fn_0 = [](auto&&...) NE {};
inline const auto fn_is0 = [](auto x) NE { return x == 0; };
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/others/lib.hpp"
namespace tifa_libs {
template <class T>
struct chash {
CEXP static u64 C = u64(pi_v<f128> * 2e18) | 71;
CEXP u64 operator()(T x) CNE { return __builtin_bswap64(((u64)x ^ TIME) * C); }
};
// clang-format off
#define mk_(w, t) using w = t; using c##w = const t;
mk_(strn, std::string) mk_(strnv, std::string_view)
#undef mk_
template <class T> struct edge_t { T w; u32 u, v; CEXP auto operator<=>(edge_t CR) const = default; }; template <class T> using cedge_t = const edge_t<T>;
template <class T> struct pt3 { T _0, _1, _2; CEXP auto operator<=>(pt3 CR) const = default; }; template <class T> using cpt3 = const pt3<T>;
template <class T> struct pt4 { T _0, _1, _2, _3; CEXP auto operator<=>(pt4 CR) const = default; }; template <class T> using cpt4 = const pt4<T>;
#define mkT_(w, t, ...) template <class T> using w = t __VA_OPT__(, ) __VA_ARGS__; template <class T> using c##w = const t __VA_OPT__(, ) __VA_ARGS__;
mkT_(ptt, std::pair<T, T>) mkT_(alc, std::pmr::polymorphic_allocator<T>) mkT_(vec, std::vector<T>) mkT_(vvec, vec<vec<T>>) mkT_(v3ec, vvec<vec<T>>) mkT_(vecpt, vec<ptt<T>>) mkT_(vvecpt, vvec<ptt<T>>) mkT_(ptvec, ptt<vec<T>>) mkT_(ptvvec, ptt<vvec<T>>)
#undef mkT_
template <class T> using itl = std ::initializer_list<T>;
template <class T, usz ext = std::dynamic_extent> using spn = std::span<T const, ext>;
template <class T, usz N> using arr = std::array<T, N>; template <class T, usz N> using carr = std::array<const T, N>;
template <class U, class T> using vecp = vec<std::pair<U, T>>; template <class U, class T> using vvecp = vvec<std::pair<U, T>>;
template <class U, class T> using vvecp = vvec<std::pair<U, T>>; template <class U, class T> using vvvecp = vvec<vvec<std::pair<U, T>>>;
#ifdef PB_DS_ASSOC_CNTNR_HPP
template <class T, class C = std::less<T>> using set = __gnu_pbds::tree<T, __gnu_pbds::null_type, C>;
template <class K, class V, class C = std::less<K>> using map = __gnu_pbds::tree<K, V, C>;
// hset<u64> s({}, {}, {}, {}, {1<<16});
template <class T, class HF = chash<T>> using hset = __gnu_pbds::gp_hash_table<T, __gnu_pbds::null_type, HF>;
// hmap<u64, int> s({}, {}, {}, {}, {1<<16});
template <class K, class V, class HF = chash<K>> using hmap = __gnu_pbds::gp_hash_table<K, V, HF>;
#else
using std::set, std::map;
template <class T, class HF = chash<T>> using hset = std::unordered_set<T, HF>;
template <class K, class V, class HF = chash<K>> using hmap = std::unordered_map<K, V, HF>;
#endif
#ifdef PB_DS_PRIORITY_QUEUE_HPP
template <class T, class C = std::less<T>> using pq = __gnu_pbds::priority_queue<T, C>;
#else
template <class T, class C = std::less<T>> using pq = std::priority_queue<T, vec<T>, C>;
#endif
template <class T> using pqg = pq<T, std::greater<T>>;
// clang-format on
#define mk1_(V, A, T) using V##A = V<T>;
#define mk_(V, A, T) mk1_(V, A, T) mk1_(c##V, A, T)
#define mk(A, T) mk_(edge_t, A, T) mk_(ptt, A, T) mk_(pt3, A, T) mk_(pt4, A, T) mk_(vec, A, T) mk_(vvec, A, T) mk_(v3ec, A, T) mk_(vecpt, A, T) mk_(vvecpt, A, T) mk_(ptvec, A, T) mk_(ptvvec, A, T) mk1_(spn, A, T) mk1_(itl, A, T)
mk(b, bool) mk(c, chr) mk(i, i32) mk(u, u32) mk(ii, i64) mk(uu, u64) mk(t, isz) mk(z, usz) mk(f, f32) mk(d, f64) mk(s, strn);
#undef mk
#undef mk_
#undef mk1_
} // namespace tifa_libs
#line 4 "src/fast/rsort32/lib.hpp"
namespace tifa_libs {
template <class C>
requires(std::is_array_v<C> && std::integral<decltype(std::declval<C>()[0])> && sizeof(std::declval<C>()[0]) == 4) || (std::contiguous_iterator<TPN C::iterator> && std::integral<TPN C::value_type> && sizeof(TPN C::value_type) == 4)
void rsort32(C& a) NE {
if (a.size() <= 1) return;
if (a.size() <= 200'000) {
std::ranges::sort(a);
return;
}
arr<u32, 256> _0{}, _1{}, _2{}, _3{};
cu32 n = (u32)a.size();
vecu b(n);
u32 *a_ = (u32*)a.data(), *b_ = (u32*)b.data();
for (cu32 *_ = a_ + n, *i = a_; i < _; ++i) ++_0[*i & 255], ++_1[*i >> 8 & 255], ++_2[*i >> 16 & 255], ++_3[*i >> 24 & 255];
flt_ (u32, i, 1, 256) _0[i] += _0[i - 1], _1[i] += _1[i - 1], _2[i] += _2[i - 1], _3[i] += _3[i - 1];
for (u32 CP i = a_ + n; --i >= a_;) b_[--_0[*i & 255]] = *i;
for (u32 CP i = b_ + n; --i >= b_;) a_[--_1[*i >> 8 & 255]] = *i;
for (u32 CP i = a_ + n; --i >= a_;) b_[--_2[*i >> 16 & 255]] = *i;
for (u32 CP i = b_ + n; --i >= b_;) a_[--_3[*i >> 24 & 255]] = *i;
if CEXP (std::is_signed_v<TPN C::value_type>) {
u32 i = n;
while (i && a[i - 1] < 0) --i;
rotate(a_, a_ + n, a_ + i);
}
}
template <class C>
requires(std::is_array_v<C> && std::integral<decltype(std::declval<C>()[0])> && sizeof(std::declval<C>()[0]) == 4) || range<C>
void sort(C& a) NE {
if CEXP (std::is_array_v<C> || (std::contiguous_iterator<TPN C::iterator> && std::integral<TPN C::value_type> && sizeof(TPN C::value_type) == 4)) rsort32(a);
else std::ranges::sort(a);
}
} // namespace tifa_libs
#line 2 "src/graph/chordal/lib.hpp"
#line 2 "src/graph/ds/graph_c/lib.hpp"
#line 2 "src/util/traits/others/lib.hpp"
// clang-format off
#line 4 "src/util/traits/others/lib.hpp"
namespace tifa_libs {
//! only for template without non-type argument
template <class, template <class...> class> CEXP bool specialized_from_v = false;
template <template <class...> class T, class... Args> CEXP bool specialized_from_v<T<Args...>, T> = true;
static_assert(specialized_from_v<vecu, std::vector>);
template <class T> concept container_c = common_range<T> && !std::is_array_v<std::remove_cvref_t<T>> && !std::same_as<std::remove_cvref_t<T>, strn> && !std::same_as<std::remove_cvref_t<T>, strnv>;
template <class T> concept istream_c = std::derived_from<T, std::istream> || std::derived_from<T, std::wistream> || requires(T is) { is.peek(); };
template <class T> concept ostream_c = std::derived_from<T, std::ostream> || std::derived_from<T, std::wostream> || requires(T os) { os.flush(); };
} // namespace tifa_libs
// clang-format on
#line 4 "src/graph/ds/graph_c/lib.hpp"
namespace tifa_libs {
namespace graph_info_impl_ {
struct graph_info_tag_base {};
template <class... Info>
requires(std::derived_from<Info, graph_info_tag_base> && ...)
struct graph_tag_base : Info... {
CEXP graph_tag_base(auto&&... args) NE : Info(std::forward<decltype(args)>(args)...)... {}
protected:
CEXP void add_arc(auto&&... args) NE { (Info::add_arc(std::forward<decltype(args)>(args)...), ...); }
CEXP void del_arc(auto&&... args) NE { (Info::del_arc(std::forward<decltype(args)>(args)...), ...); }
};
template <class T, class = std::is_void<T>::type>
struct E;
template <class T>
struct E<T, std::false_type> {
u32 to;
T cost;
CEXP E() = default;
CEXP E(u32 v, cT_(T) c) NE : to(v), cost(c) {}
CEXP operator u32() CNE { return to; }
};
template <class T>
struct E<T, std::true_type> {
u32 to;
CEXP E() = default;
CEXP E(u32 v) NE : to(v) {}
CEXP operator u32() CNE { return to; }
};
} // namespace graph_info_impl_
namespace graph_impl_ {
template <class etag_t>
struct graph : etag_t {
using Et = etag_t::val_t;
CEXP graph() = default;
CEXPE graph(u32 n, auto&&... e_args) NE : etag_t(n, std::forward<decltype(e_args)>(e_args)...) {}
CEXP void add_edge(u32 u, u32 v, auto&&... args) NE { etag_t::add_arc(u, v, std::forward<decltype(args)>(args)...), etag_t::add_arc(v, u, std::forward<decltype(args)>(args)...); }
template <class F>
CEXP void foreach(u32 u, F&& f) CNE {
if CEXP (std::is_void_v<Et>)
for (auto v : (*this)[u]) f(v);
else
for (auto [v, w] : (*this)[u]) f(v, w);
}
};
} // namespace graph_impl_
// clang-format off
#define CONCEPT_GRAPH(name, base) \
template <class T> concept name##_c = specialized_from_v<T, base>; \
template <class T> concept u##name##_c = name##_c<T> && std::same_as<TPN T::Et, void>; \
template <class T> concept w##name##_c = name##_c<T> && !std::same_as<TPN T::Et, void>
// clang-format on
CONCEPT_GRAPH(graph, graph_impl_::graph);
} // namespace tifa_libs
#line 5 "src/graph/chordal/lib.hpp"
namespace tifa_libs {
template <graph_c G> // prefer alist
class chordal {
G CR g;
vecu deg;
public:
vecu peo, rnk;
// @param g simple UNDIRECTED graph
//! g[i] MUST be sorted
CEXPE chordal(G CR g) NE : g(g), deg(g.vsize()), peo(g.vsize()), rnk(g.vsize()) {
cu32 n = g.vsize();
vecu l(n * 2 + 1), r, idx(n);
std::iota(begin(l), end(l), 0), r = l;
auto ins = [&](u32 i, u32 j) NE { r[l[i] = l[j]] = i, r[l[j] = i] = j; };
auto del = [&](u32 i) NE { r[l[i]] = r[i], l[r[i]] = l[i]; };
flt_ (u32, i, 0, n) ins(i, n);
u32 li = n;
flt_ (u32, i, 0, n) {
++li;
while (l[li] == li) --li;
u32 v = l[li];
for (idx[v] = -1_u32, del(v); u32 to : g[v])
if (~idx[to]) ++deg[to], del(to), ins(to, n + (++idx[to]));
peo[i] = v;
}
reverse(peo);
flt_ (u32, i, 0, n) rnk[peo[i]] = i;
}
template <bool find_indcycle = false>
std::conditional_t<find_indcycle, std::optional<vecu>, bool> is_chordal_graph() CNE {
std::optional<vecu> ret;
auto has_edge = [&](u32 u, u32 v) { return binary_search(g[u], v, {}, [](auto CR e) { return (u32)e; }); };
for (u32 u : peo) {
vecu s;
for (s.reserve(g.deg_out(u)); auto v : g[u])
if (rnk[u] < rnk[v])
if (s.push_back(v); rnk[s.back()] < rnk[s[0]]) swap(s[0], s.back());
flt_ (u32, j, 1, (u32)s.size())
if (!has_edge(s[0], s[j])) {
if CEXP (!find_indcycle) return false;
else {
cu32 x = s[j], y = s[0], z = u;
vecu pre(peo.size(), -1_u32);
std::queue<u32> q({x});
while (!q.empty()) {
cu32 t = q.front();
if (q.pop(); has_edge(t, y)) {
pre[y] = t;
ret.emplace({y});
while (ret->back() != x) ret->push_back(pre[ret->back()]);
ret->push_back(z);
return ret;
}
for (cu32 u : g[t])
if (u != z && !has_edge(u, z) && !~pre[u]) pre[u] = t, q.push(u);
}
}
}
}
if CEXP (find_indcycle) return ret;
else return true;
}
// @return {x}, which $\{x\}+N(x)$ be a maximal clique
ND CEXP vecu maximal_cliques() CNE {
vecu fst(peo.size(), -1_u32), n(peo.size()), res;
for (cu32 u : peo)
for (auto v : g[u])
if (rnk[u] < rnk[(u32)v])
if (++n[u]; !~fst[u] || rnk[(u32)v] < rnk[fst[u]]) fst[u] = (u32)v;
for (vecb vis(peo.size()); cu32 u : peo) {
if (vis[u]) continue;
res.push_back(u), vis[fst[u]] = n[u] > n[fst[u]];
}
return res;
}
ND CEXP u32 chromatic_number() CNE { return max(deg) + 1; }
ND CEXP vecu max_independent_set() CNE {
vecu res;
for (vecb vis(peo.size()); cu32 u : peo) {
if (vis[u]) continue;
for (res.push_back(u); u32 v : g[u]) vis[v] = true;
}
return res;
}
};
} // namespace tifa_libs
#line 2 "src/graph/ds/alist/lib.hpp"
#line 4 "src/graph/ds/alist/lib.hpp"
namespace tifa_libs {
namespace alist_impl_ {
template <class T, class... Info>
class alist_tag : public graph_info_impl_::graph_tag_base<Info...> {
using base_t = graph_info_impl_::graph_tag_base<Info...>;
vvec<graph_info_impl_::E<T>> g;
protected:
using val_t = T;
CEXPE alist_tag(u32 n) NE : base_t(n), g(n) {}
public:
CEXP void build() CNE {}
CEXP void add_arc(u32 u, auto&&... args) NE { base_t::add_arc(u, std::forward<decltype(args)>(args)...), g[u].emplace_back(std::forward<decltype(args)>(args)...); }
ND CEXP u32 vsize() CNE { return (u32)g.size(); }
ND CEXP u32 deg_out(u32 u) CNE { return (u32)g[u].size(); }
CEXP auto CR operator[](u32 u) CNE { return g[u]; }
CEXP auto& operator[](u32 u) NE { return g[u]; }
};
} // namespace alist_impl_
template <class Et = void, class... Info>
using alist = graph_impl_::graph<alist_impl_::alist_tag<Et, Info...>>;
} // namespace tifa_libs
#line 6 "test/cpv_local/graph/chordal.bzoj1006.cpp"
using namespace tifa_libs;
using std::cin, std::cout;
int main() {
u32 n, m;
cin >> n >> m;
alist g(n);
for (u32 i = 0, u, v; i < m; ++i) {
cin >> u >> v;
--u;
--v;
g.add_arc(u, v);
g.add_arc(v, u);
}
flt_ (u32, i, 0, n) ::sort(g[i]);
chordal chd(g);
cout << chd.chromatic_number() << '\n';
}
test/cpv_local/graph/chordal.bzoj1006.cpp