/** * date : 2026-09-19 16:41:46 * author : Nyaan */ #define NDEBUG using namespace std; // intrinstic #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // utility namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template using V = vector; template using VV = vector>; using vi = vector; using vl = vector; using vd = V; using vs = V; using vvi = vector>; using vvl = vector>; template using minpq = priority_queue, greater>; template struct P : pair { template constexpr P(Args... args) : pair(args...) {} using pair::first; using pair::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P; using pi = P; using vp = V; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template int sz(const T &t) { return t.size(); } template inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline T Max(const vector &v) { return *max_element(begin(v), end(v)); } template inline T Min(const vector &v) { return *min_element(begin(v), end(v)); } template inline long long Sum(const vector &v) { return accumulate(begin(v), end(v), 0LL); } template int lb(const vector &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template int ub(const vector &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { unsigned long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template pair mkp(const T &t, const U &u) { return make_pair(t, u); } template vector mkrui(const vector &v, bool rev = false) { vector ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template vector mkuni(const vector &v) { vector ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template vector mkord(int N, F f) { vector ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template vector mkinv(vector &v) { int max_val = *max_element(begin(v), end(v)); vector inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector mkiota(int n) { vector ret(n); iota(begin(ret), end(ret), 0); return ret; } template T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template bool nxp(T &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template vector> product(const vector &a) { vector> ret; vector v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I, const function &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I = T{1}) { return Power(a, n, I, function{[](T &) -> void {}}); } template T Rev(const T &v) { T res = v; reverse(begin(res), end(res)); return res; } template vector Transpose(const vector &v) { using U = typename T::value_type; if(v.empty()) return {}; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { res[j][i] = v[i][j]; } } return res; } template vector Rotate(const vector &v, int clockwise = true) { using U = typename T::value_type; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { if (clockwise) { res[W - 1 - j][i] = v[i][j]; } else { res[j][H - 1 - i] = v[i][j]; } } } return res; } } // namespace Nyaan // bit operation namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return __builtin_popcountll(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template inline int gbit(const T &a, int i) { return (a >> i) & 1; } template inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan // inout namespace Nyaan { template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan // debug #ifdef NyaanDebug #define trc(...) (void(0)) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) (void(0)) #else #define trc2(...) (void(0)) #endif // macro #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } using namespace std; namespace internal { unsigned long long non_deterministic_seed() { unsigned long long m = chrono::duration_cast( chrono::high_resolution_clock::now().time_since_epoch()) .count(); m ^= 9845834732710364265uLL; m ^= m << 24, m ^= m >> 31, m ^= m << 35; return m; } unsigned long long deterministic_seed() { return 88172645463325252UL; } // 64 bit の seed 値を生成 (手元では seed 固定) // 連続で呼び出すと同じ値が何度も返ってくるので注意 // #define RANDOMIZED_SEED するとシードがランダムになる unsigned long long seed() { #if defined(DETERMINISTIC_SEED) return deterministic_seed(); #elif defined(NyaanLocal) && !defined(RANDOMIZED_SEED) return deterministic_seed(); #else return non_deterministic_seed(); #endif } } // namespace internal namespace my_rand { using i64 = long long; using u64 = unsigned long long; // [0, 2^64 - 1) u64 rng() { static u64 _x = internal::seed(); return _x ^= _x << 7, _x ^= _x >> 9; } // [l, r] i64 rng(i64 l, i64 r) { assert(l <= r); return l + rng() % u64(r - l + 1); } // [l, r) i64 randint(i64 l, i64 r) { assert(l < r); return l + rng() % u64(r - l); } // choose n numbers from [l, r) without overlapping vector randset(i64 l, i64 r, i64 n) { assert(l <= r && n <= r - l); unordered_set s; for (i64 i = n; i; --i) { i64 m = randint(l, r + 1 - i); if (s.find(m) != s.end()) m = r - i; s.insert(m); } vector ret; for (auto& x : s) ret.push_back(x); sort(begin(ret), end(ret)); return ret; } // [0.0, 1.0) double rnd() { return rng() * 5.42101086242752217004e-20; } // [l, r) double rnd(double l, double r) { assert(l < r); return l + rnd() * (r - l); } template void randshf(vector& v) { int n = v.size(); for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]); } } // namespace my_rand using my_rand::randint; using my_rand::randset; using my_rand::randshf; using my_rand::rnd; using my_rand::rng; using namespace std; struct Timer { chrono::high_resolution_clock::time_point st; Timer() { reset(); } void reset() { st = chrono::high_resolution_clock::now(); } long long elapsed() { auto ed = chrono::high_resolution_clock::now(); return chrono::duration_cast(ed - st).count(); } long long operator()() { return elapsed(); } }; // using namespace std; namespace TwoSatImpl { namespace internal { template struct csr { vector start; vector elist; csr(int n, const vector>& edges) : start(n + 1), elist(edges.size()) { for (auto e : edges) { start[e.first + 1]++; } for (int i = 1; i <= n; i++) { start[i] += start[i - 1]; } auto counter = start; for (auto e : edges) { elist[counter[e.first]++] = e.second; } } }; struct scc_graph { public: scc_graph(int n) : _n(n) {} int num_vertices() { return _n; } void add_edge(int from, int to) { edges.push_back({from, {to}}); } pair> scc_ids() { auto g = csr(_n, edges); int now_ord = 0, group_num = 0; vector visited, low(_n), ord(_n, -1), ids(_n); visited.reserve(_n); auto dfs = [&](auto self, int v) -> void { low[v] = ord[v] = now_ord++; visited.push_back(v); for (int i = g.start[v]; i < g.start[v + 1]; i++) { auto to = g.elist[i].to; if (ord[to] == -1) { self(self, to); low[v] = min(low[v], low[to]); } else { low[v] = min(low[v], ord[to]); } } if (low[v] == ord[v]) { while (true) { int u = visited.back(); visited.pop_back(); ord[u] = _n; ids[u] = group_num; if (u == v) break; } group_num++; } }; for (int i = 0; i < _n; i++) { if (ord[i] == -1) dfs(dfs, i); } for (auto& x : ids) { x = group_num - 1 - x; } return {group_num, ids}; } vector> scc() { auto ids = scc_ids(); int group_num = ids.first; vector counts(group_num); for (auto x : ids.second) counts[x]++; vector> groups(ids.first); for (int i = 0; i < group_num; i++) { groups[i].reserve(counts[i]); } for (int i = 0; i < _n; i++) { groups[ids.second[i]].push_back(i); } return groups; } void add_node_size(int m) { _n += m; } int size() { return _n; } private: int _n; struct edge { int to; }; vector> edges; }; } // namespace internal struct two_sat { public: two_sat() : _n(0), built(false), scc(0) {} two_sat(int n) : _n(n), built(false), scc(2 * n) {} int add_var() { scc.add_node_size(2); return _n++; } // (not i) は ~i で渡す void add_clause(int i, int j) { i = max(2 * i, -1 - 2 * i); j = max(2 * j, -1 - 2 * j); assert(0 <= i && i < 2 * _n); assert(0 <= j && j < 2 * _n); scc.add_edge(i, j ^ 1); scc.add_edge(j, i ^ 1); } void if_then(int i, int j) { add_clause(~i, j); } void set_val(int i) { add_clause(i, i); } // (not i) は ~i で渡す void at_most_one(const vector& nodes) { if ((int)nodes.size() <= 1) return; int cur = ~nodes[0]; for (int i = 2; i < (int)nodes.size(); i++) { int nxt = add_var(), n_i = ~nodes[i]; add_clause(cur, n_i); add_clause(cur, nxt); add_clause(n_i, nxt); cur = ~nxt; } add_clause(cur, ~nodes[1]); } bool satisfiable() { _answer.resize(_n); built = true; auto id = scc.scc_ids().second; for (int i = 0; i < _n; i++) { if (id[2 * i] == id[2 * i + 1]) { _answer.clear(); return false; } _answer[i] = id[2 * i] < id[2 * i + 1]; } return true; } vector answer() { if (!built) satisfiable(); return _answer; } private: int _n; vector _answer; bool built; internal::scc_graph scc; }; } // namespace TwoSatImpl using TwoSatImpl::two_sat; /** * @brief 2-SAT * @docs docs/math/two-sat.md */ using namespace Nyaan; vvi sub(int N, vi X, vi Y, vi Z) { assert(sz(X) == N); assert(sz(Y) == N); assert(sz(Z) == 2 * N - 1); int maxX = Max(X), minY = Min(Y); trc(X, Y, Z); vvi A(N, vi(N, -1)); rep(i, N) if (X[i] == 1) { if (minY == 0) return {}; rep(j, N) A[i][j] = 1; } rep(j, N) if (Y[j] == 0) { if (maxX == 1) return {}; rep(i, N) A[i][j] = 0; } VV freeX(N), freeY(N), freeZ(2 * N - 1); vi baseZ(2 * N - 1); rep(z, 2 * N - 1) { int s = 0; rep(i, N) { int j = z - i; if (!(0 <= j and j < N)) continue; if (A[i][j] == -1) { freeX[i].emplace_back(i, j); freeY[j].emplace_back(i, j); freeZ[z].emplace_back(i, j); } else { s ^= A[i][j]; } } if (freeZ[z].empty() and s != Z[z]) return {}; baseZ[z] = s; } auto rand_choice = [](V& v) { assert(!v.empty()); return v[rng(0, sz(v) - 1)]; }; rep(t, 10) { // 斜めをいい感じに決める rep(z, 2 * N - 1) { int val = baseZ[z]; each2(i, j, freeZ[z]) A[i][j] = rng() % 2, val ^= A[i][j]; if (val != Z[z]) { auto [i, j] = rand_choice(freeZ[z]); A[i][j] ^= 1; } } vi valx(N), valy(N); rep(i, N) rep(j, N) valx[i] += A[i][j], valy[j] += A[i][j]; auto flip = [&](int i, int j) { valx[i] -= A[i][j], valy[j] -= A[i][j]; A[i][j] ^= 1; valx[i] += A[i][j], valy[j] += A[i][j]; }; vi ord = mkiota(2 * N); rep(_, 2 * N + 1) { int upd = false; randshf(ord); each(k, ord) { if (k < N) { int i = k; if (X[i] == 0 and valx[i] == N) { upd = true; auto [_, j] = rand_choice(freeX[i]); int z = i + j; if (sz(freeZ[z]) == 1) continue; int ii, jj; do { tie(ii, jj) = rand_choice(freeZ[z]); } while (i == ii); flip(i, j), flip(ii, jj); } } else { int j = k - N; if (Y[j] == 1 and valy[j] == 0) { upd = true; auto [i, _] = rand_choice(freeY[j]); int z = i + j; if (sz(freeZ[z]) == 1) continue; int ii, jj; do { tie(ii, jj) = rand_choice(freeZ[z]); } while (i == ii); flip(i, j), flip(ii, jj); } } } if (!upd) return A; } } return {}; } void q() { rng(); ini(N); vi X(N), Y(N), Z(2 * N - 1); in(X, Y, Z); vvi ans(N, vi(N)); rep(b, 30) { vi x(N), y(N), z(2 * N - 1); rep(i, N) x[i] = gbit(X[i], b); rep(i, N) y[i] = gbit(Y[i], b); rep(i, 2 * N - 1) z[i] = gbit(Z[i], b); auto a = sub(N, x, y, z); trc(a); if (a.empty()) die(-1); rep(i, N) { int s = 1; rep(j, N) s &= a[i][j]; assert(s == x[i]); } rep(j, N) { int s = 0; rep(i, N) s |= a[i][j]; assert(s == y[j]); } rep(k, 2 * N - 1) { int s = 0; rep(i, N) { int j = k - i; if (0 <= j and j < N) s ^= a[i][j]; } assert(s == z[k]); } rep(i, N) rep(j, N) ans[i][j] += a[i][j] << b; } each(v, ans) out(v); } void test() {} void Nyaan::solve() { #ifdef NyaanLocal // test(); trc2("test OK"); #endif Timer timer; int t = 1; in(t); while (t--) q(); trc2(timer()); }