// #pragma comment(linker, "/stack:200000000") #include #include #include namespace suisen { // ! utility template using constraints_t = std::enable_if_t, std::nullptr_t>; template constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) { if constexpr (cond_v) { return std::forward(then); } else { return std::forward(or_else); } } // ! function template using is_same_as_invoke_result = std::is_same, ReturnType>; template using is_uni_op = is_same_as_invoke_result; template using is_bin_op = is_same_as_invoke_result; template using is_comparator = std::is_same, bool>; // ! integral template >> constexpr int bit_num = std::numeric_limits>::digits; template struct is_nbit { static constexpr bool value = bit_num == n; }; template static constexpr bool is_nbit_v = is_nbit::value; // ? template struct safely_multipliable {}; template <> struct safely_multipliable { using type = long long; }; template <> struct safely_multipliable { using type = __int128_t; }; template <> struct safely_multipliable { using type = unsigned long long; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = float; }; template <> struct safely_multipliable { using type = double; }; template <> struct safely_multipliable { using type = long double; }; template using safely_multipliable_t = typename safely_multipliable::type; } // namespace suisen // ! type aliases using i128 = __int128_t; using u128 = __uint128_t; using ll = long long; using uint = unsigned int; using ull = unsigned long long; template using vec = std::vector; template using vec2 = vec>; template using vec3 = vec>; template using vec4 = vec>; template using pq_greater = std::priority_queue, std::greater>; template using umap = std::unordered_map; // ! macros (capital: internal macro) #define OVERLOAD2(_1,_2,name,...) name #define OVERLOAD3(_1,_2,_3,name,...) name #define OVERLOAD4(_1,_2,_3,_4,name,...) name #define REP4(i,l,r,s) for(std::remove_reference_t>i=(l);i<(r);i+=(s)) #define REP3(i,l,r) REP4(i,l,r,1) #define REP2(i,n) REP3(i,0,n) #define REPINF3(i,l,s) for(std::remove_reference_t>i=(l);;i+=(s)) #define REPINF2(i,l) REPINF3(i,l,1) #define REPINF1(i) REPINF2(i,0) #define RREP4(i,l,r,s) for(std::remove_reference_t>i=(l)+fld((r)-(l)-1,s)*(s);i>=(l);i-=(s)) #define RREP3(i,l,r) RREP4(i,l,r,1) #define RREP2(i,n) RREP3(i,0,n) #define rep(...) OVERLOAD4(__VA_ARGS__, REP4 , REP3 , REP2 )(__VA_ARGS__) #define rrep(...) OVERLOAD4(__VA_ARGS__, RREP4 , RREP3 , RREP2 )(__VA_ARGS__) #define repinf(...) OVERLOAD3(__VA_ARGS__, REPINF3, REPINF2, REPINF1)(__VA_ARGS__) #define CAT_I(a, b) a##b #define CAT(a, b) CAT_I(a, b) #define UNIQVAR(tag) CAT(tag, __LINE__) #define loop(n) for (std::remove_reference_t> UNIQVAR(loop_variable) = n; UNIQVAR(loop_variable) --> 0;) #define all(iterable) (iterable).begin(), (iterable).end() #define input(type, ...) type __VA_ARGS__; read(__VA_ARGS__) // ! I/O utilities // pair template std::ostream& operator<<(std::ostream& out, const std::pair &a) { return out << a.first << ' ' << a.second; } // tuple template std::ostream& operator<<(std::ostream& out, const std::tuple &a) { if constexpr (N >= std::tuple_size_v>) { return out; } else { out << std::get(a); if constexpr (N + 1 < std::tuple_size_v>) { out << ' '; } return operator<<(out, a); } } // vector template std::ostream& operator<<(std::ostream& out, const std::vector &a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } // array template std::ostream& operator<<(std::ostream& out, const std::array &a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } inline void print() { std::cout << '\n'; } template inline void print(const Head &head, const Tail &...tails) { std::cout << head; if (sizeof...(tails)) std::cout << ' '; print(tails...); } template auto print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") -> decltype(std::cout << *v.begin(), void()) { for (auto it = v.begin(); it != v.end();) { std::cout << *it; if (++it != v.end()) std::cout << sep; } std::cout << end; } // pair template std::istream& operator>>(std::istream& in, std::pair &a) { return in >> a.first >> a.second; } // tuple template std::istream& operator>>(std::istream& in, std::tuple &a) { if constexpr (N >= std::tuple_size_v>) { return in; } else { return operator>>(in >> std::get(a), a); } } // vector template std::istream& operator>>(std::istream& in, std::vector &a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } // array template std::istream& operator>>(std::istream& in, std::array &a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } template void read(Args &...args) { ( std::cin >> ... >> args ); } // ! integral utilities // Returns pow(-1, n) template constexpr inline int pow_m1(T n) { return -(n & 1) | 1; } // Returns pow(-1, n) template <> constexpr inline int pow_m1(bool n) { return -int(n) | 1; } // Returns floor(x / y) template constexpr inline T fld(const T x, const T y) { return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y; } template constexpr inline T cld(const T x, const T y) { return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y; } template > = nullptr> constexpr inline int popcount(const T x) { return __builtin_popcount(x); } template > = nullptr> constexpr inline int popcount(const T x) { return __builtin_popcount(x); } template > = nullptr> constexpr inline int popcount(const T x) { return __builtin_popcountll(x); } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clzll(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctzll(x) : suisen::bit_num; } template constexpr inline int floor_log2(const T x) { return suisen::bit_num - 1 - count_lz(x); } template constexpr inline int ceil_log2(const T x) { return floor_log2(x) + ((x & -x) != x); } template constexpr inline int kth_bit(const T x, const unsigned int k) { return (x >> k) & 1; } template constexpr inline int parity(const T x) { return popcount(x) & 1; } struct all_subset { struct all_subset_iter { const int s; int t; constexpr all_subset_iter(int s) : s(s), t(s + 1) {} constexpr auto operator*() const { return t; } constexpr auto operator++() {} constexpr auto operator!=(std::nullptr_t) { return t ? (--t &= s, true) : false; } }; int s; constexpr all_subset(int s) : s(s) {} constexpr auto begin() { return all_subset_iter(s); } constexpr auto end() { return nullptr; } }; // ! container template > = nullptr> auto priqueue_comp(const Comparator comparator) { return std::priority_queue, Comparator>(comparator); } template auto isize(const Iterable &iterable) -> decltype(int(iterable.size())) { return iterable.size(); } template > = nullptr> auto generate_vector(int n, Gen generator) { std::vector v(n); for (int i = 0; i < n; ++i) v[i] = generator(i); return v; } template auto generate_range_vector(T l, T r) { return generate_vector(r - l, [l](int i) { return l + i; }); } template auto generate_range_vector(T n) { return generate_range_vector(0, n); } template void sort_unique_erase(std::vector &a) { std::sort(a.begin(), a.end()); a.erase(std::unique(a.begin(), a.end()), a.end()); } template auto foreach_adjacent_values(InputIterator first, InputIterator last, BiConsumer f) -> decltype(f(*first++, *last), void()) { if (first != last) for (auto itr = first, itl = itr++; itr != last; itl = itr++) f(*itl, *itr); } template auto foreach_adjacent_values(Container c, BiConsumer f) -> decltype(c.begin(), c.end(), void()){ foreach_adjacent_values(c.begin(), c.end(), f); } // ! other utilities // x <- min(x, y). returns true iff `x` has chenged. template inline bool chmin(T &x, const T &y) { if (y >= x) return false; x = y; return true; } // x <- max(x, y). returns true iff `x` has chenged. template inline bool chmax(T &x, const T &y) { if (y <= x) return false; x = y; return true; } namespace suisen {} using namespace suisen; using namespace std; struct io_setup { io_setup(int precision = 20) { std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout << std::fixed << std::setprecision(precision); } } io_setup_ {}; // ! code from here #include #include #include #include #include namespace suisen { struct BipartiteMatching { static constexpr int ABSENT = -1; BipartiteMatching() {} BipartiteMatching(int n, int m) : _n(n), _m(m), _to_r(_n, ABSENT), _to_l(_m, ABSENT), _g(n + m) {} void add_edge(int fr, int to) { _g[fr].push_back(to), _f = -1; } template int solve() { if (_f >= 0) return _f; static std::mt19937 rng(std::random_device{}()); if constexpr (shuffle) for (auto &adj : _g) std::shuffle(adj.begin(), adj.end(), rng); std::vector vis(_n, false); auto dfs = [&, this](auto dfs, int u) -> bool { if (std::exchange(vis[u], true)) return false; for (int v : _g[u]) if (_to_l[v] == ABSENT) return _to_r[u] = v, _to_l[v] = u, true; for (int v : _g[u]) if (dfs(dfs, _to_l[v])) return _to_r[u] = v, _to_l[v] = u, true; return false; }; for (bool upd = true; std::exchange(upd, false);) { vis.assign(_n, false); for (int i = 0; i < _n; ++i) if (_to_r[i] == ABSENT) upd |= dfs(dfs, i); } return _f = _n - std::count(_to_r.begin(), _to_r.end(), ABSENT); } std::vector> max_matching() { if (_f < 0) _f = solve(); std::vector> res; res.reserve(_f); for (int i = 0; i < _n; ++i) if (_to_r[i] != ABSENT) res.emplace_back(i, _to_r[i]); return res; } std::vector> min_edge_cover() { auto res = max_matching(); std::vector vl(_n, false), vr(_n, false); for (const auto &[u, v] : res) vl[u] = vr[v] = true; for (int u = 0; u < _n; ++u) for (int v : _g[u]) if (not (vl[u] and vr[v])) { vl[u] = vr[v] = true; res.emplace_back(u, v); } return res; } std::vector min_vertex_cover() { if (_f < 0) _f = solve(); std::vector> g(_n + _m); std::vector cl(_n, true), cr(_m, false); for (int u = 0; u < _n; ++u) for (int v : _g[u]) { if (_to_r[u] == v) { g[v + _n].push_back(u); cl[u] = false; } else { g[u].push_back(v + _n); } } std::vector vis(_n + _m, false); std::deque dq; for (int i = 0; i < _n; ++i) if (cl[i]) { dq.push_back(i); vis[i] = true; } while (dq.size()) { int u = dq.front(); dq.pop_front(); for (int v : g[u]) { if (vis[v]) continue; vis[v] = true; (v < _n ? cl[v] : cr[v - _n]) = true; dq.push_back(v); } } std::vector res; for (int i = 0; i < _n; ++i) if (not cl[i]) res.push_back(i); for (int i = 0; i < _m; ++i) if (cr[i]) res.push_back(_n + i); return res; } std::vector max_independent_set() { std::vector use(_n + _m, true); for (int v : min_vertex_cover()) use[v] = false; std::vector res; for (int i = 0; i < _n + _m; ++i) if (use[i]) res.push_back(i); return res; } int left_size() const { return _n; } int right_size() const { return _m; } std::pair size() const { return { _n, _m }; } int right(int l) const { return _to_r[l]; } int left(int r) const { return _to_l[r]; } const auto graph() const { return _g; } auto reversed_graph() const { std::vector> h(_m); for (int i = 0; i < _n; ++i) for (int j : _g[i]) h[j].push_back(i); return h; } private: int _n, _m; std::vector _to_r, _to_l; std::vector> _g; int _f = 0; }; } // namespace suisen #include namespace suisen { static std::optional> bipartite_coloring(const std::vector>& g, int col0 = 0, int col1 = 1) { const int n = g.size(); int uncolored = 2; while (uncolored == col0 or uncolored == col1) ++uncolored; std::vector color(n, uncolored); for (int i = 0; i < n; ++i) { if (color[i] != uncolored) continue; color[i] = col0; std::deque dq { i }; while (dq.size()) { int u = dq.front(); dq.pop_front(); for (int v : g[u]) { if (color[v] == uncolored) { dq.push_back(v); color[v] = color[u] ^ col0 ^ col1; } else if (color[v] == color[u]) { return std::nullopt; } } } } return color; } } // namespace suisen int main() { input(int, n); vector a(n); read(a); map> mp; rep(i, n) { mp[a[i]].push_back(i); } vector> g(n); rep(i, n) { rep(bit, 30) { for (int j : mp[a[i] ^ (1 << bit)]) { g[i].push_back(j); } } } auto bip = *bipartite_coloring(g); vector id(n); int l = 0, r = 0; rep(i, n) { if (bip[i] == 0) { id[i] = l++; } else { id[i] = r++; } } BipartiteMatching m(l, r); rep(i, n) { if (bip[i] == 1) continue; for (int j : g[i]) { m.add_edge(id[i], id[j]); } } print(m.max_independent_set().size()); return 0; }