#include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using ll = long long; class range {private: struct I{int x;int operator*(){return x;}bool operator!=(I& lhs){return x ostream& operator<<(ostream& os, const pair& p){ return os << "{" << p.first << ", " << p.second << "}"; } template ostream& operator<<(ostream& os, const vector& obj) { os << "{"; for (const auto& e : obj) os << e << ", "; return os << "}"; } template ostream& operator<<(ostream& os, const set& obj) { os << "set{"; for (const auto& e : obj) os << e << ", "; return os << "}"; } template ostream& operator<<(ostream& os, const map& obj) { os << "map{"; for (const auto& e : obj) os << e << ", "; return os << "}"; } template void take(vector& vec, int n) { vec.resize(n); for (int i = 0; i < n; ++i) cin >> vec[i]; } #ifdef ONLINE_JUDGE #define dump(expr) ; #else #define dump(expr) { cerr << "\033[33m#L" << __LINE__ << ": " << expr << "\033[39m" << endl; } #endif namespace solver { template struct In2 { T1 a; T2 b; friend std::istream& operator>>(std::istream& is, In2& obj) { T1 t1; T2 t2; is >> t1 >> t2; obj = {t1, t2}; return is; } }; template struct In3 { T1 a; T2 b; T3 c; friend std::istream& operator>>(std::istream& is, In3& obj) { T1 t1; T2 t2; T3 t3; is >> t1 >> t2 >> t3; obj = {t1, t2, t3}; return is; } }; int n; vector as; void read() { cin >> n; take(as, 1 << n); } int at(int i, int pos) { return (as[i] >> pos) & 1; } using RetType = void; RetType run() { int N = 1 << n; vector ones(N, 30); for (bool upd = 1; upd; ) { upd = 0; dump(upd) for (int i : range(1, N)) for (int j : range(1, N)) if (i != j) { int k = i ^ j; if (as[i] != -1 && as[j] != -1 && as[k] != -1 && as[k] < (as[i] ^ as[j])) { dump("naive " << i << " " << j << " " << k); cout << "No" << endl; return; } // propagate if (as[k] != -1) continue; int zeroi = 0, zeroj = 0; int pos = ones[k] - 1; if (pos < 0) continue; if (as[i] != -1 && !at(i, pos)) zeroi = 1; if (as[i] == -1 && ones[i] <= pos) zeroi = 1; if (as[j] != -1 && !at(j, pos)) zeroj = 1; if (as[j] == -1 && ones[j] <= pos) zeroj = 1; if (i == 4 && j == 3) { dump(zeroi << " " << zeroj) } if (zeroi && zeroj && as[k] == -1) { ones[k]--; upd = 1; } } } vector res(N); ones[0] = 0; for (int i : range(N)) { if (as[i] == -1) res[i] = (1 << ones[i]) - 1; else res[i] = as[i]; } dump(res); for (int i : range(N)) if (res[i] < (res[0] ^ res[i])) { cout << "No" << endl; return; } for (int i : range(1, N)) for (int j : range(1, N)) if (i != j) { int k = i ^ j; assert(res[k] >= (res[i] ^ res[j])); } cout << "Yes" << endl; for (int i : range(N)) cout << res[i] << " "; cout << endl; } } // namespace template void run(F f) { if constexpr (std::is_same_v) f(); else cout << f() << endl; } int main(int argc, char** argv) { cerr << fixed << setprecision(12); cout << fixed << setprecision(12); int testcase = 1; if (argc > 1) testcase = atoi(argv[1]); while (testcase--) { solver::read(); } run(solver::run); }