#line 2 "cp-library/src/cp-template.hpp" #include using namespace std; using ll = long long; using ld = long double; using uint = unsigned int; using ull = unsigned long long; using i32 = int; using u32 = unsigned int; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; template < class T > bool chmin(T& a, T b) { if(a > b) { a = b; return true; } return false; } template < class T > bool chmax(T& a, T b) { if(a < b) { a = b; return true; } return false; } template < class T, class U > T ceil (T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template < class T, class U > T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } int popcnt(i32 x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(i64 x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } #line 2 "cp-library/src/utility/rep_itr.hpp" template < class T > struct itr_rep { T i, d; constexpr itr_rep(const T i) noexcept : i(i), d(1) {} constexpr itr_rep(const T i, const T d) noexcept : i(i), d(d) {} void operator++() noexcept { i += d; } constexpr int operator*() const noexcept { return i; } constexpr bool operator!=(const itr_rep x) const noexcept { return d > 0 ? i < x.i : i > x.i; } }; template < class T > struct rep { const itr_rep< T > s, t; constexpr rep(const T t) noexcept : s(0), t(t) {} constexpr rep(const T s, const T t) noexcept : s(s), t(t) {} constexpr rep(const T s, const T t, const T d) noexcept : s(s, d), t(t, d) {} constexpr auto begin() const noexcept { return s; } constexpr auto end () const noexcept { return t; } }; template < class T > struct revrep { const itr_rep < T > s, t; constexpr revrep(const T t) noexcept : s(t - 1, -1), t(-1, -1) {} constexpr revrep(const T s, const T t) noexcept : s(t - 1, -1), t(s - 1, -1) {} constexpr revrep(const T s, const T t, const T d) noexcept : s(t - 1, -d), t(s - 1, -d) {} constexpr auto begin() const noexcept { return s; } constexpr auto end () const noexcept { return t; } }; #line 3 "cp-library/src/utility/io.hpp" /* 128bit integer */ istream& operator>>(istream& is, i128& x) { std::string s; is >> s; int pm = (s[0] == '-'); x = 0; for(int i : rep(pm, int(s.size()))) x = x * 10 + (s[i] - '0'); if(pm) x *= -1; return is; } ostream& operator<<(ostream& os, const i128& x) { if(x == 0) return os << '0'; i128 y = x; if(y < 0) { os << '-'; y *= -1; } std::vector ny; while(y > 0) { ny.push_back(y % 10); y /= 10; } for(int i : revrep(ny.size())) os << ny[i]; return os; } template < class S, class T > istream& operator>>(istream& is, std::pair< S, T >& x) { is >> x.first >> x.second; return is; } template < class S, class T > ostream& operator<<(ostream& os, const std::pair< S, T >& x) { os << x.first << " " << x.second; return os; } namespace scanner { struct sca { template < class T > operator T() { T s; std::cin >> s; return s; } }; struct vec { int n; vec(int n) : n(n) {} template < class T > operator std::vector< T >() { std::vector< T > v(n); for(T& x : v) std::cin >> x; return v; } }; struct mat { int h, w; mat(int h, int w) : h(h), w(w) {} template < class T > operator std::vector< std::vector< T > >() { std::vector m(h, std::vector< T >(w)); for(std::vector< T >& v : m) for(T& x : v) std::cin >> x; return m; } }; struct speedup { speedup() { std::cin.tie(0); std::ios::sync_with_stdio(0); } } speedup_instance; } scanner::sca in() { return scanner::sca(); } scanner::vec in(int n) { return scanner::vec(n); } scanner::mat in(int h, int w) { return scanner::mat(h, w); } namespace printer { void precision(int d) { std::cout << std::fixed << std::setprecision(d); } void flush() { std::cout.flush(); } } template < class T > ostream& operator<<(ostream& os, const std::vector< T > a) { int n = a.size(); for(int i : rep(n)) { os << a[i]; if(i != n - 1) os << ' '; } return os; } int print() { std::cout << '\n'; return 0; } template < class head, class... tail > int print(head&& h, tail&&... t) { std::cout << h; if(sizeof...(tail)) std::cout << ' '; return print(std::forward(t)...); } template < class T > int print_n(const std::vector< T > a) { int n = a.size(); for(int i : rep(n)) std::cout << a[i] << "\n"; return 0; } #line 2 "cp-library/src/utility/key_val.hpp" template < class K, class V > struct key_val { K key; V val; key_val() {} key_val(K key, V val) : key(key), val(val) {} template < std::size_t Index > std::tuple_element_t< Index, key_val >& get() { if constexpr (Index == 0) return key; if constexpr (Index == 1) return val; } }; namespace std { template < class K, class V > struct tuple_size < key_val< K, V > > : integral_constant< size_t, 2 > {}; template < class K, class V > struct tuple_element < 0, key_val< K, V > > { using type = K; }; template < class K, class V > struct tuple_element < 1, key_val< K, V > > { using type = V; }; } #line 2 "cp-library/src/utility/vec_op.hpp" template < class T > key_val< int, T > max_of(const vector< T >& a) { int i = std::max_element(a.begin(), a.end()) - a.begin(); return {i, a[i]}; } template < class T > key_val< int, T > min_of(const vector< T >& a) { int i = std::min_element(a.begin(), a.end()) - a.begin(); return {i, a[i]}; } template < class S, class T > S sum_of(const vector< T >& a) { S sum = 0; for(const T x : a) sum += x; return sum; } template < class S, class T > vector< S > freq_of(const vector< T >& a, T L, T R) { vector< S > res(R - L, S(0)); for(const T x : a) res[x - L] += 1; return res; } template < class S, class T > struct prefix_sum { vector< S > s; prefix_sum(const vector< T >& a) : s(a) { s.insert(s.begin(), S(0)); for(int i : rep(a.size())) s[i + 1] += s[i]; } // [L, R) S sum(int L, int R) { return s[R] - s[L]; } }; #line 3 "cp-library/src/utility/heap.hpp" template < class T > using heap_min = std::priority_queue< T, std::vector< T >, std::greater< T > >; template < class T > using heap_max = std::priority_queue< T, std::vector< T >, std::less< T > >; #line 27 "cp-library/src/cp-template.hpp" #line 1 "cp-library/src/algorithm/bin_search.hpp" template < class T, class F > T bin_search(T ok, T ng, F f) { while(abs(ng - ok) > 1) { T mid = (ok + ng) / 2; (f(mid) ? ok : ng) = mid; } return ok; } template < class T, class F > T bin_search_real(T ok, T ng, F f, int step = 80) { while(step--) { T mid = (ok + ng) / 2; (f(mid) ? ok : ng) = mid; } return ok; } #line 2 "cp-library/src/algorithm/argsort.hpp" template < class T > std::vector< int > argsort(const std::vector< T > &a) { std::vector< int > ids((int)a.size()); std::iota(ids.begin(), ids.end(), 0); std::sort(ids.begin(), ids.end(), [&](int i, int j) { return a[i] < a[j] || (a[i] == a[j] && i < j); }); return ids; } #line 1 "macro.hpp" namespace macro { using size_type = int; template < class container > void sort(container& a) { std::sort(std:: begin(a), std:: end(a)); } template < class container > void rsort(container& a) { std::sort(std::rbegin(a), std::rend(a)); } template < class container > void reverse(container& a) { std::reverse(std::begin(a), std::end(a)); } template < class container > void unique(container& a) { std::sort(std::begin(a), std::end(a)); a.erase(std::unique(std::begin(a), std::end(a)), std::end(a)); } template < class container > container sorted(const container& a) { container b = a; sort(b); return std::move(b); } template < class container > container rsorted(const container& a) { container b = a; rsort(b); return std::move(b); } template < class container, class compare > void sort(container& a, const compare& cmp) { std::sort(std::begin(a), std::end(a), cmp); } template < class container, class compare > container sorted(const container& a, const compare& cmp) { container b = a; sort(b, cmp); return std::move(b); } template < class container, class value > size_type lower_bound(const container& a, const value& x) { return std::lower_bound(std::begin(a), std::end(a), x) - std::begin(a); } template < class container, class value > size_type upper_bound(const container& a, const value& x) { return std::upper_bound(std::begin(a), std::end(a), x) - std::begin(a); } const std::vector> dir4 = { {+1, 0}, {-1, 0}, { 0, +1}, { 0, -1} }; const std::vector> dir8 = { {-1, -1}, {-1, 0}, {-1, +1}, { 0, -1}, { 0, +1}, {+1, -1}, {+1, 0}, {+1, +1} }; #ifdef _DEBUG #define debug(x) std::cout << "[" << __LINE__ << "] " << #x << ": " << x << std::endl #else #define debug(x) #endif template < class container > void concat(container& a, const container& b) { a.insert(std::end(a), std::begin(b), std::end(b)); } std::vector iota(const size_type n) { std::vector I(n); std::iota(std::begin(I), std::end(I), 0); return I; } template < class container > std::vector sort_idx(const container& a) { const size_type n = a.size(); std::vector I = iota(n); std::sort(std::begin(I), std::end(I), [&](size_type i, size_type j) { return a[i] < a[j] or (a[i] == a[j] and i < j); }); return I; } template < class container, class compare > std::vector sort_idx(const container& a, const compare& cmp) { const size_type n = a.size(); std::vector I = iota(n); std::sort(std::begin(I), std::end(I), [&](size_type i, size_type j) { return cmp(a[i], a[j]) or (a[i] == a[j] and i < j); }); return std::move(I); } struct grid { using size_type = int; size_type H, W; grid(const size_type H, const size_type W) : H(H), W(W) {} bool contains(const size_type i, const size_type j) { return 0 <= i and i < H and 0 <= j and j < W; } }; using f64 = long double; template < class T > vector< T >& operator++(vector< T >& a) { for(T& x : a) x++; return a; } template < class T > vector< T >& operator--(vector< T >& a) { for(T& x : a) x--; return a; } template < class T > vector< T > operator++(vector< T >& a, signed) { vector< T > res = a; for(T& x : a) x++; return res; } template < class T > vector< T > operator--(vector< T >& a, signed) { vector< T > res = a; for(T& x : a) x--; return res; } } // namespace macro using namespace macro; #line 2 "cp-library/src/data_structure/fenwick_tree.hpp" template < class comm_monoid > class fenwick_tree { public: using T = typename comm_monoid::set; private: int n, n2; vector< T > data; int ceil_pow2(int n) { int x = 1; while(x < n) x <<= 1; return x; } public: fenwick_tree() : fenwick_tree(0) {} fenwick_tree(int n) : n(n), n2(ceil_pow2(n)), data(n + 1, comm_monoid::id()) { assert(comm_monoid::comm); } fenwick_tree(const vector< T > &a) : n(a.size()), n2(ceil_pow2(n)), data(a) { assert(comm_monoid::comm); data.insert(data.begin(), {comm_monoid::id()}); for(int i = 1; i <= n; i++) { int p = i + (i & -i); if(p <= n) data[p] = comm_monoid::op(data[i], data[p]); } } void add(int i, T x) { for(int p = i + 1; p <= n; p += p & -p) data[p] = comm_monoid::op(data[p], x); } // [0, r) T fold(int r) { T s = comm_monoid::id(); for(int p = r; p > 0; p -= p & -p) s = comm_monoid::op(data[p], s); return s; } // [l, r) T fold(int l, int r) { return comm_monoid::op(comm_monoid::inv(fold(l)), fold(r)); } T get(int i) { return fold(i, i + 1); } void set(int i, T x) { add(i, comm_monoid::op(comm_monoid::inv(get(i)), x)); } template< class func > int search(const func &f) { T s = comm_monoid::id(); if(f(s)) return 0; int i = 0, k = n2; while(k >>= 1) { int p = i | k; if(p <= n && !f(comm_monoid::op(s, data[p]))) s = comm_monoid::op(s, data[i = p]); } return i; } }; #line 1 "cp-library/src/algebra/sum.hpp" template < class T > class sum_monoid { public: using set = T; static constexpr T op(const T &l, const T &r) { return l + r; } static constexpr T id() { return T(0); } static constexpr T inv(const T &x) { return -x; } static constexpr T pow(const T &x, const ll n) { return x * n; } static constexpr bool comm = true; }; #line 5 "A.cpp" int main() { #define int i64 int N = in(); vector> p(N); for(auto &[x, y] : p) x = in(), y = in(); const auto [mx, my] = [&] { vector vx, vy; for(auto [x, y] : p) { vx.push_back(x); vy.push_back(y); } unique(vx); unique(vy); for(auto &[x, y] : p) { x = lower_bound(vx, x); y = lower_bound(vy, y); } return pair{vx.size(), vy.size()}; }(); i64 P = 0, Q = 0; vector> ys(mx); for(auto [x, y] : p) ys[x].push_back(y); fenwick_tree> tree(my); for(int x : rep(mx)) { for(int y : ys[x]) { P += tree.fold(y); Q += tree.fold(y + 1, my); } for(int y : ys[x]) tree.add(y, +1); } auto f = [&](vector v) { sort(v); vector> vec; const int m = v.size(); vec.push_back({v[0], 1}); for(int i : rep(1LL, m)) { if(vec.back().first == v[i]) vec.back().second++; else vec.push_back({v[i], 1}); } i64 sum = 0; for(auto [_, c] : vec) sum += i64(c) * (c - 1) / 2; return i64(m) * (m - 1) / 2 - sum; }; vector X, Y; for(auto [x, y] : p) X.push_back(x), Y.push_back(y); const i64 R = f(X); const i64 S = f(Y); printer::precision(20); print(f64(P - Q) / sqrt(R) / sqrt(S)); }