#pragma once
#include "../graph/graph-template.hpp"
#include "../graph/strongly-connected-components.hpp"
#include "../template.hpp"
/**
* @brief Two Satisfiability(2-SAT)
* ac-libraryと大体同じ
* satisfiable()を呼びtrueが返ってきた場合ansに解が入っている
*/
struct TwoSat {
int n;
Graph<bool> graph;
vector<bool> ans;
TwoSat(int n) : n(n), graph(2 * n), ans(n) {}
void add_clause(int i, bool f, int j, bool g) {
int x = i + (f ? 0 : n), nx = i + (f ? n : 0);
int y = j + (g ? 0 : n), ny = j + (g ? n : 0);
graph.add_directed_edge(nx, y);
graph.add_directed_edge(ny, x);
}
bool satisfiable() {
StronglyConnectedComponents scc(graph);
for(int i = 0; i < n; ++i) {
if(scc.comp[i] == scc.comp[n + i])
return false;
ans[i] = scc.comp[i] > scc.comp[n + i];
}
return true;
}
};
#line 2 "other/fastio.hpp"
// ref: https://maspypy.com/library-checker-many-a-b , Nyaanさん
#line 2 "other/type-utils.hpp"
#include <bits/stdc++.h>
using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using vi = std::vector<int>;
using vii = std::vector<std::vector<int>>;
using pii = std::pair<int, int>;
using vl = std::vector<ll>;
using vll = std::vector<vl>;
using pll = std::pair<ll, ll>;
template <class T>
concept extended_integral =
std::integral<T> || std::same_as<std::remove_cv_t<T>, i128> ||
std::same_as<std::remove_cv_t<T>, u128>;
template <class T>
concept extended_signed_integral =
std::signed_integral<T> || std::same_as<std::remove_cv_t<T>, i128>;
template <class T>
concept extended_unsigned_integral =
std::unsigned_integral<T> || std::same_as<std::remove_cv_t<T>, u128>;
template <class T>
concept Streamable =
requires(std::ostream &os, T &x) { os << x; } || extended_integral<T>;
template <class mint>
concept is_modint = requires(mint &x) {
{ x.val() } -> std::convertible_to<int>;
};
#line 4 "other/fastio.hpp"
namespace fastio {
constexpr int SZ = 1 << 17;
constexpr int offset = 64;
constexpr int mod = 10000;
char in_buf[SZ];
int in_left{}, in_right{};
char out_buf[SZ];
char out_tmp[offset];
int out_right{};
struct Pre {
char num[4 * mod]{};
constexpr Pre() {
for(int i = 0; i < mod; ++i) {
for(int n = i, j = 3; j >= 0; --j, n /= 10)
num[4 * i + j] = '0' + n % 10;
}
}
constexpr const char *operator[](int i) const { return &num[4 * i]; }
} constexpr pre;
void load() {
memmove(in_buf, in_buf + in_left, in_right - in_left);
in_right += -in_left + std::fread(in_buf + in_right - in_left, 1,
SZ - (in_right - in_left), stdin);
in_left = 0;
if(in_right < SZ)
in_buf[in_right++] = '\n';
}
void read(char &c) {
do {
if(in_left == in_right)
load();
c = in_buf[in_left++];
} while(isspace(c));
}
void read(std::string &s) {
s.clear();
char c;
do {
if(in_left == in_right)
load();
c = in_buf[in_left++];
} while(isspace(c));
do {
s += c;
if(in_left == in_right)
load();
c = in_buf[in_left++];
} while(!isspace(c));
}
template <extended_integral T> void read(T &x) {
if(in_right - in_left < offset)
load();
char c;
do
c = in_buf[in_left++];
while(c < '-'); // \n:10 space:32 -:45 '0':48
bool minus{};
if constexpr(extended_signed_integral<T>) {
if(c == '-') {
c = in_buf[in_left++];
minus = true;
}
}
x = 0;
while(c >= '0') {
x = 10 * x + (c & 15);
c = in_buf[in_left++];
}
if constexpr(extended_signed_integral<T>) {
if(minus)
x = -x;
}
}
void flush() { fwrite(out_buf, 1, std::exchange(out_right, 0), stdout); }
void write_range(const char *c, int n) {
int pos{};
while(pos < n) {
if(out_right == SZ)
flush();
int len = std::min(n - pos, SZ - out_right);
memcpy(out_buf + out_right, c + pos, len);
out_right += len;
pos += len;
}
}
void write(char c) {
if(SZ == out_right)
flush();
out_buf[out_right++] = c;
}
void write(const char *c) { write_range(c, strlen(c)); }
void write(const std::string &s) { write_range(s.data(), s.size()); }
template <std::floating_point T> void write(T x) {
int n = std::snprintf(out_tmp, sizeof(out_tmp), "%.16g", x);
write_range(out_tmp, n);
}
void write(bool x) { write(x ? '1' : '0'); }
template <extended_integral T> void write(T x) {
if(x == 0) {
write('0');
}
if constexpr(extended_signed_integral<T>) {
if(x < 0) {
write('-');
x = -x;
}
}
if(SZ - out_right < offset)
flush();
int cur = offset;
for(; x >= 1000; x /= mod) {
cur -= 4;
memcpy(out_tmp + cur, pre[x % mod], 4);
}
if(x >= 100) {
cur -= 3;
memcpy(out_tmp + cur, pre[x % mod] + 1, 3);
} else if(x >= 10) {
cur -= 2;
memcpy(out_tmp + cur, pre[x % mod] + 2, 2);
} else if(x >= 1) {
cur -= 1;
memcpy(out_tmp + cur, pre[x % mod] + 3, 1);
}
write_range(out_tmp + cur, offset - cur);
}
struct Dummy {
// プログラム終了時に出力
~Dummy() { flush(); }
} dummy;
} // namespace fastio
using fastio::write;
#line 4 "template.hpp"
// #pragma GCC target("avx2")
// #pragma GCC optimize("O3")
// #pragma GCC optimize("unroll-loops")
#line 8 "template.hpp"
using namespace std;
#ifdef LOCAL
#include <debug.hpp>
#else
#define debug(...)
#endif
template <Streamable T> void print_one(const T &value) { fastio::write(value); }
template <is_modint T> void print_one(const T &value) {
print_one(value.val());
}
void print() { print_one('\n'); }
template <class T, class... Ts> void print(const T &a, const Ts &...b) {
print_one(a);
((print_one(' '), print_one(b)), ...);
print();
}
template <ranges::range Iterable>
requires(!Streamable<Iterable>)
void print(const Iterable &v) {
for(auto it = v.begin(); it != v.end(); ++it) {
if(it != v.begin())
print_one(' ');
print_one(*it);
}
print();
}
#define all(v) begin(v), end(v)
template <class T> void UNIQUE(T &v) {
ranges::sort(v);
v.erase(unique(all(v)), end(v));
}
template <typename T> inline bool chmax(T &a, T b) {
return ((a < b) ? (a = b, true) : (false));
}
template <typename T> inline bool chmin(T &a, T b) {
return ((a > b) ? (a = b, true) : (false));
}
// https://trap.jp/post/1224/
template <class... T> constexpr auto min(T... a) {
return min(initializer_list<common_type_t<T...>>{a...});
}
template <class... T> constexpr auto max(T... a) {
return max(initializer_list<common_type_t<T...>>{a...});
}
void input() {}
template <class Head, class... Tail> void input(Head &head, Tail &...tail) {
#ifdef LOCAL
cin >> head;
#else
fastio::read(head);
#endif
input(tail...);
}
template <class T> void input(vector<T> &a) {
for(T &x : a)
input(x);
}
#define INT(...) \
int __VA_ARGS__; \
input(__VA_ARGS__)
#define LL(...) \
long long __VA_ARGS__; \
input(__VA_ARGS__)
#define STR(...) \
string __VA_ARGS__; \
input(__VA_ARGS__)
#define REP1_0(n, c) REP1_1(n, c)
#define REP1_1(n, c) \
for(ll REP_COUNTER_##c = 0; REP_COUNTER_##c < (ll)(n); REP_COUNTER_##c++)
#define REP1(n) REP1_0(n, __COUNTER__)
#define REP2(i, a) for(ll i = 0; i < (ll)(a); i++)
#define REP3(i, a, b) for(ll i = (ll)(a); i < (ll)(b); i++)
#define REP4(i, a, b, c) for(ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
ll inf = 3e18;
vl dx = {1, -1, 0, 0};
vl dy = {0, 0, 1, -1};
template <class T> constexpr T floor(T x, T y) noexcept {
return x / y - ((x ^ y) < 0 and x % y);
}
template <class T> constexpr T ceil(T x, T y) noexcept {
return x / y + ((x ^ y) >= 0 and x % y);
}
// yの符号に関わらず非負で定義 \bmod:texコマンド
template <class T> constexpr T bmod(T x, T y) noexcept {
T m = x % y;
return (m < 0) ? m + (y > 0 ? y : -y) : m;
}
template <std::signed_integral T> constexpr int bit_width(T x) noexcept {
return std::bit_width((uint64_t)x);
}
template <std::signed_integral T> constexpr int popcount(T x) noexcept {
return std::popcount((uint64_t)x);
}
constexpr bool kth_bit(auto n, auto k) { return (n >> k) & 1; }
#line 3 "graph/graph-template.hpp"
/**
* @brief Graph Template(グラフテンプレート)
* @see https://ei1333.github.io/library/graph/graph-template.hpp
*/
template <class T = ll> struct Edge {
int from, to;
T cost;
int idx;
Edge() = default;
Edge(int from, int to, T cost = 1, int idx = -1)
: from(from), to(to), cost(cost), idx(idx) {}
Edge &operator=(const int &x) {
to = x;
return *this;
}
operator int() const { return to; }
};
template <class T = ll> struct Graph {
using cost_type = T;
vector<vector<Edge<T>>> g;
int es; // edge_size
Graph() = default;
explicit Graph(int n) : g(n), es(0) {};
int size() const { return ssize(g); }
void add_directed_edge(int from, int to, T cost = 1) {
g[from].emplace_back(from, to, cost, es++);
}
void add_edge(int from, int to, T cost = 1) {
g[from].emplace_back(from, to, cost, es);
g[to].emplace_back(to, from, cost, es++);
}
vector<Edge<T>> &operator[](const int &k) { return g[k]; }
const vector<Edge<T>> &operator[](const int &k) const { return g[k]; }
void read(int m, int padding = -1, bool weighted = false,
bool directed = false) {
for(int i = 0; i < m; ++i) {
int a, b;
T c(1);
input(a, b);
a += padding;
b += padding;
if(weighted)
input(c);
if(directed)
add_directed_edge(a, b, c);
else
add_edge(a, b, c);
}
}
};
#line 3 "graph/strongly-connected-components.hpp"
/**
* @brief 強連結成分分解(Tarjan)
* https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
* comp[i] : 頂点iが属する強連結成分のid
* groups : 各強連結成分の頂点集合
* トポロジカルソートソート済み
* cul_dag() : 強連結成分を1つの頂点に縮約したときのDAGを作る DAGの頂点iが元のグラフのgroups[i]に対応
* verify:https://atcoder.jp/contests/abc357/submissions/66283840
*/
template <class G> struct StronglyConnectedComponents {
const G &g;
vector<int> comp;
vector<vector<int>> groups;
StronglyConnectedComponents(const G &g)
: g(g), comp(g.size()), low(g.size()), ord(g.size(), -1) {
for(int i = 0; i < g.size(); ++i) {
if(ord[i] == -1) {
dfs(i);
}
}
ranges::reverse(groups);
for(int i = 0; i < ssize(groups); ++i) {
for(auto j : groups[i])
comp[j] = i;
}
}
G cul_dag() const {
G dag(groups.size());
for(int i = 0; i < g.size(); ++i) {
int x = comp[i];
for(auto &to : g[i]) {
int y = comp[to];
if(x != y) {
dag.add_directed_edge(x, y, to.cost);
}
}
}
return dag;
}
private:
int id = 0;
vector<int> low, ord; // lowは計算後g.size()に置き換えられる
vector<int> tmp;
void dfs(int cur) {
low[cur] = ord[cur] = id++;
tmp.emplace_back(cur);
for(auto &to : g[cur]) {
if(ord[to] == -1) {
dfs(to);
chmin(low[cur], low[to]);
} else {
chmin(low[cur], ord[to]);
}
}
if(low[cur] == ord[cur]) {
groups.emplace_back();
while(true) {
groups.back().emplace_back(tmp.back());
ord[tmp.back()] =
g.size(); // 今後訪れる頂点はこの強連結成分とは別の強連結成分に属する
tmp.pop_back();
if(groups.back().back() == cur) {
break;
}
}
}
}
};
#line 5 "other/twosat.hpp"
/**
* @brief Two Satisfiability(2-SAT)
* ac-libraryと大体同じ
* satisfiable()を呼びtrueが返ってきた場合ansに解が入っている
*/
struct TwoSat {
int n;
Graph<bool> graph;
vector<bool> ans;
TwoSat(int n) : n(n), graph(2 * n), ans(n) {}
void add_clause(int i, bool f, int j, bool g) {
int x = i + (f ? 0 : n), nx = i + (f ? n : 0);
int y = j + (g ? 0 : n), ny = j + (g ? n : 0);
graph.add_directed_edge(nx, y);
graph.add_directed_edge(ny, x);
}
bool satisfiable() {
StronglyConnectedComponents scc(graph);
for(int i = 0; i < n; ++i) {
if(scc.comp[i] == scc.comp[n + i])
return false;
ans[i] = scc.comp[i] > scc.comp[n + i];
}
return true;
}
};
Two Satisfiability(2-SAT) (other/twosat.hpp)