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CF1971_H_±1.cpp
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CF1971_H_±1.cpp
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/**CF1971H
* Standard reduction to 2-SAT decision problem solvable via Kosaraju's algorithm
*/
#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,avx2,fma")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef vector<int> vi;
#define fast_cin() ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
int n;
int num_scc;
const int MAXN = 1002;
vi adjlist[MAXN], adjlist_t[MAXN], scc_index, topo_arr, dfs_num;
vi sccs[MAXN];
void kosaraju(int u, int pass) {
dfs_num[u] = 1;
vi &neighbours = (pass == 1) ? adjlist[u] : adjlist_t[u];
for (auto v : neighbours) {
if (dfs_num[v] == -1) {
kosaraju(v, pass);
}
}
if (pass == 1)
topo_arr.emplace_back(u);
else {
scc_index[u] = num_scc; // colour graph according to num_scc (0 index)
sccs[num_scc].emplace_back(u); // place the node into the scc
}
}
int not_vertex(int a){
if (a < n) return a + n;
else return a - n;
}
int main() {
int tc;
cin >> tc;
while(tc--){
cin >> n;
for (int i=0;i<MAXN;i++){
adjlist[i].clear();
adjlist_t[i].clear();
sccs[i].clear();
}
vector<vector<int>> grid(3, vector<int>(n, 0));
for (int i=0;i<3;i++){
for (int j=0;j<n;j++){
cin >> grid[i][j];
if (grid[i][j] < 0) grid[i][j] = n - grid[i][j] - 1;
else grid[i][j]--;
}
}
for (int j=0; j<n; j++){
for (int i=0; i<3; i++){
for (int k=i+1; k<3; k++){
if (i == k) continue;
// insert (grid[i][j] or grid[k][j])
adjlist[not_vertex(grid[i][j])].emplace_back(grid[k][j]);
adjlist[not_vertex(grid[k][j])].emplace_back(grid[i][j]);
adjlist_t[grid[k][j]].emplace_back(not_vertex(grid[i][j]));
adjlist_t[grid[i][j]].emplace_back(not_vertex(grid[k][j]));
}
}
}
int N = 2*n; // total number of vertices
scc_index.assign(N, -1);
dfs_num.assign(N, -1);
topo_arr.clear();
num_scc = 0;
// Kosaraju First Pass
for (int i = 0; i < N; i++) {
if (dfs_num[i] == -1) {
kosaraju(i, 1);
}
}
// Kosaraju Second Pass, colours the vertices in scc_index
dfs_num.assign(N, -1);
for (int i = N - 1; i >= 0; i--) {
if (dfs_num[topo_arr[i]] == -1) {
kosaraju(topo_arr[i], 2);
num_scc++;
}
}
// Check for literal and its negation being in the same SCC
bool unsat = false;
for (int i=0;i<n; i++){
if (scc_index[i] == scc_index[i+n]){ // Xk and not Xk imply each other
unsat = true;
break;
}
}
cout << (unsat ? "NO" : "YES") << endl;
}
}