#include <bits/stdc++.h> using namespace std; #pragma GCC optimize("O3", "unroll-loops") typedef double ld; void __print(int x) {cerr << x;} void __print(long long x) {cerr << x;} void __print(double x) {cerr << x;} void __print(char x) {cerr << "'" << x << "'";} void __print(const char *x) {cerr << '"' << x << '"';} void __print(const string &x) {cerr << '"' << x << '"';} void __print(bool x) {cerr << (x ? "true" : "false");} template<typename T, typename V> void __print(const pair<T, V> &x) {cerr << '{'; __print(x.first); cerr << ", "; __print(x.second); cerr << '}';} template<typename T> void __print(const T &x) {int f = 0; cerr << '{'; for (auto &i: x) cerr << (f++ ? ", " : ""), __print(i); cerr << "}";} void _print() {cerr << "]\n";} template <typename T, typename... V> void _print(T t, V... v) {__print(t); if (sizeof...(v)) cerr << ", "; _print(v...);} #ifdef LOCAL #define debug(x...) cerr << "[" << #x << "] = ["; _print(x) #else #define debug(x...) #endif using Complex = complex<ld>; void fft(vector<Complex>&a){ int n = (int)a.size(); int L = 31 - __builtin_clz(n); vector<complex<long double>>R(2, 1); vector<Complex>rt(2, 1); for (int k = 2; k < n; k *= 2){ R.resize(n); rt.resize(n); auto x = polar(1.0L, acosl(-1)/k); for (int i = k; i <= 2*k-1; i++){ rt[i] = R[i] = i & 1 ? R[i/2] * x : R[i/2]; } } vector<int>rev(n); for (int i = 0; i<n; i++){ rev[i] = (rev[i/2] | (i&1) << L)/2; } for (int i = 0; i<n; i++){ if (i < rev[i]) swap(a[i], a[rev[i]]); } for (int k = 1; k < n; k *= 2){ for (int i = 0; i<n; i+= 2*k){ for (int j = 0; j < k; j++){ Complex z = rt[j+k] * a[i+j+k]; a[i+j+k] = a[i+j] - z; a[i+j] += z; } } } } vector<ld>mul(vector<ld>&res, vector<ld>&a, vector<ld>&b){ // assert(!a.empty() && !b.empty()); // if (a.empty() || b.empty()) return {}; // int k = 0; // while ((1<<k) < max((int)a.size(), (int)b.size())) k++; // while ((int)a.size() < (1<<k)) a.emplace_back(0); // while ((int)b.size() < (1<<k)) b.emplace_back(0); res.resize((int)a.size()+(int)b.size()-1); int L = 32-__builtin_clz((int)res.size()); int n = (1<<L); vector<Complex>in(n), out(n); copy(a.begin(), a.end(), in.begin()); for (int i = 0; i<(int)b.size(); i++) in[i].imag(b[i]); fft(in); for (auto &x: in) x *= x; for (int i = 0; i<n; i++) { out[i] = in[-i & (n-1)] - conj(in[i]); } fft(out); for (int i = 0; i<(int)res.size(); i++) res[i] = imag(out[i]) / (4*n); return res; } int t; const int sz = 1<<16; vector<ld>tab[2*sz]; int shiftt[2*sz]; void init(int n, vector<ld>&p){ for (int i = sz; i < 2*sz; i++) tab[i].emplace_back(1); for (int i = 0; i<(int)p.size(); i++){ shiftt[i+sz] = 1; tab[i+sz] = {1-p[i], 0, p[i]}; } for (int i = sz-1; i>=1; i--){ mul(tab[i], tab[2*i], tab[2*i+1]); shiftt[i] = shiftt[2*i] + shiftt[2*i+1]; } // debug((int)tab[1].size()); } void get(int x, int lx, int rx, int i, vector<int>&curr){ if (lx > i) return; if (rx <= i) { curr.emplace_back(x); return; } int m = (lx+rx)/2; get(2*x, lx, m, i, curr); get(2*x+1, m+1, rx, i, curr); } vector<ld>c; pair<vector<ld>, int> get(int pos){ vector<int>curr; get(1, 0, sz-1, pos, curr); vector<ld>ans = {1}; int s = 0; for (auto x: curr) s += shiftt[x]; c.clear(); for (int i = (int)curr.size()-1; i>=0; i--){ mul(c, ans, tab[curr[i]]); c.swap(ans); } return {ans, s}; } const ld EPS = 1e-9; void solve(){ int n; cin >> n; cin >> t; cerr << fixed << setprecision(8); vector<ld>p(n); for (auto &x: p) cin >> x; sort(p.rbegin(), p.rend()); init(n+1, p); //parzyste vector<int>even, odd; for (int i = t-1; i < n; i++){ if (i&1) odd.emplace_back(i); else even.emplace_back(i); } ld ans = 0; auto forward = [&](vector<ld>&dp, int shift, int from, int ile, bool best){ vector<ld>new_dp; for (int i = 0; i<ile; i++) new_dp.emplace_back(0); for (auto x: dp) new_dp.emplace_back(x); for (int i = 0; i<ile; i++) new_dp.emplace_back(0); ld res = 0; shift += ile; if (best) { ld curr = 0; for (int i = t + shift; i < (int)new_dp.size(); i++) curr += new_dp[i]; res = max(res, curr); } for (int i = from+1; i <= min(n, from+ile); i++){ // debug(i); vector<ld>dp2((int)new_dp.size()); for (int j = 0; j<(int)new_dp.size(); j++){ dp2[j] = (j > 0 ? new_dp[j-1] : 0) * p[i] + (j + 1 < (int)new_dp.size() ? new_dp[j+1] : 0) * (1-p[i]); } dp2.swap(new_dp); // debug(new_dp); if (best) { ld curr = 0; for (int j = t + shift; j < (int)new_dp.size(); j++) curr += new_dp[j]; res = max(res, curr); } } ld curr = 0; for (int i = t + shift; i < (int)new_dp.size(); i++) curr += new_dp[i]; res = max(res, curr); return res; }; auto check = [&](vector<int>vt){ if (vt.empty()) return; int l = 0, r = (int)vt.size()-1; while (r-l >= 5) { int m = (l+r)/2; auto [dp, shift] = get(vt[m]); ld L = 0; for (int i = t + shift; i < (int)dp.size(); i++) L += dp[i]; ld R = forward(dp, shift, vt[m], 2, false); debug(l, r, L, R); if (L + EPS < R) l = m; else r = m+1; } // debug(l, r, vt[l]); auto [dp, shift] = get(vt[l]); // debug(dp); ans = max(ans, forward(dp, shift, vt[l], 10, true)); // for (int i = l; i <= r; i++) ans = max(ans, get(vt[i])); }; check(even); check(odd); cout << fixed << setprecision(15) << ans << "\n"; } int32_t main(){ ios_base::sync_with_stdio(0); cin.tie(0); solve(); return 0; }
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 | #include <bits/stdc++.h> using namespace std; #pragma GCC optimize("O3", "unroll-loops") typedef double ld; void __print(int x) {cerr << x;} void __print(long long x) {cerr << x;} void __print(double x) {cerr << x;} void __print(char x) {cerr << "'" << x << "'";} void __print(const char *x) {cerr << '"' << x << '"';} void __print(const string &x) {cerr << '"' << x << '"';} void __print(bool x) {cerr << (x ? "true" : "false");} template<typename T, typename V> void __print(const pair<T, V> &x) {cerr << '{'; __print(x.first); cerr << ", "; __print(x.second); cerr << '}';} template<typename T> void __print(const T &x) {int f = 0; cerr << '{'; for (auto &i: x) cerr << (f++ ? ", " : ""), __print(i); cerr << "}";} void _print() {cerr << "]\n";} template <typename T, typename... V> void _print(T t, V... v) {__print(t); if (sizeof...(v)) cerr << ", "; _print(v...);} #ifdef LOCAL #define debug(x...) cerr << "[" << #x << "] = ["; _print(x) #else #define debug(x...) #endif using Complex = complex<ld>; void fft(vector<Complex>&a){ int n = (int)a.size(); int L = 31 - __builtin_clz(n); vector<complex<long double>>R(2, 1); vector<Complex>rt(2, 1); for (int k = 2; k < n; k *= 2){ R.resize(n); rt.resize(n); auto x = polar(1.0L, acosl(-1)/k); for (int i = k; i <= 2*k-1; i++){ rt[i] = R[i] = i & 1 ? R[i/2] * x : R[i/2]; } } vector<int>rev(n); for (int i = 0; i<n; i++){ rev[i] = (rev[i/2] | (i&1) << L)/2; } for (int i = 0; i<n; i++){ if (i < rev[i]) swap(a[i], a[rev[i]]); } for (int k = 1; k < n; k *= 2){ for (int i = 0; i<n; i+= 2*k){ for (int j = 0; j < k; j++){ Complex z = rt[j+k] * a[i+j+k]; a[i+j+k] = a[i+j] - z; a[i+j] += z; } } } } vector<ld>mul(vector<ld>&res, vector<ld>&a, vector<ld>&b){ // assert(!a.empty() && !b.empty()); // if (a.empty() || b.empty()) return {}; // int k = 0; // while ((1<<k) < max((int)a.size(), (int)b.size())) k++; // while ((int)a.size() < (1<<k)) a.emplace_back(0); // while ((int)b.size() < (1<<k)) b.emplace_back(0); res.resize((int)a.size()+(int)b.size()-1); int L = 32-__builtin_clz((int)res.size()); int n = (1<<L); vector<Complex>in(n), out(n); copy(a.begin(), a.end(), in.begin()); for (int i = 0; i<(int)b.size(); i++) in[i].imag(b[i]); fft(in); for (auto &x: in) x *= x; for (int i = 0; i<n; i++) { out[i] = in[-i & (n-1)] - conj(in[i]); } fft(out); for (int i = 0; i<(int)res.size(); i++) res[i] = imag(out[i]) / (4*n); return res; } int t; const int sz = 1<<16; vector<ld>tab[2*sz]; int shiftt[2*sz]; void init(int n, vector<ld>&p){ for (int i = sz; i < 2*sz; i++) tab[i].emplace_back(1); for (int i = 0; i<(int)p.size(); i++){ shiftt[i+sz] = 1; tab[i+sz] = {1-p[i], 0, p[i]}; } for (int i = sz-1; i>=1; i--){ mul(tab[i], tab[2*i], tab[2*i+1]); shiftt[i] = shiftt[2*i] + shiftt[2*i+1]; } // debug((int)tab[1].size()); } void get(int x, int lx, int rx, int i, vector<int>&curr){ if (lx > i) return; if (rx <= i) { curr.emplace_back(x); return; } int m = (lx+rx)/2; get(2*x, lx, m, i, curr); get(2*x+1, m+1, rx, i, curr); } vector<ld>c; pair<vector<ld>, int> get(int pos){ vector<int>curr; get(1, 0, sz-1, pos, curr); vector<ld>ans = {1}; int s = 0; for (auto x: curr) s += shiftt[x]; c.clear(); for (int i = (int)curr.size()-1; i>=0; i--){ mul(c, ans, tab[curr[i]]); c.swap(ans); } return {ans, s}; } const ld EPS = 1e-9; void solve(){ int n; cin >> n; cin >> t; cerr << fixed << setprecision(8); vector<ld>p(n); for (auto &x: p) cin >> x; sort(p.rbegin(), p.rend()); init(n+1, p); //parzyste vector<int>even, odd; for (int i = t-1; i < n; i++){ if (i&1) odd.emplace_back(i); else even.emplace_back(i); } ld ans = 0; auto forward = [&](vector<ld>&dp, int shift, int from, int ile, bool best){ vector<ld>new_dp; for (int i = 0; i<ile; i++) new_dp.emplace_back(0); for (auto x: dp) new_dp.emplace_back(x); for (int i = 0; i<ile; i++) new_dp.emplace_back(0); ld res = 0; shift += ile; if (best) { ld curr = 0; for (int i = t + shift; i < (int)new_dp.size(); i++) curr += new_dp[i]; res = max(res, curr); } for (int i = from+1; i <= min(n, from+ile); i++){ // debug(i); vector<ld>dp2((int)new_dp.size()); for (int j = 0; j<(int)new_dp.size(); j++){ dp2[j] = (j > 0 ? new_dp[j-1] : 0) * p[i] + (j + 1 < (int)new_dp.size() ? new_dp[j+1] : 0) * (1-p[i]); } dp2.swap(new_dp); // debug(new_dp); if (best) { ld curr = 0; for (int j = t + shift; j < (int)new_dp.size(); j++) curr += new_dp[j]; res = max(res, curr); } } ld curr = 0; for (int i = t + shift; i < (int)new_dp.size(); i++) curr += new_dp[i]; res = max(res, curr); return res; }; auto check = [&](vector<int>vt){ if (vt.empty()) return; int l = 0, r = (int)vt.size()-1; while (r-l >= 5) { int m = (l+r)/2; auto [dp, shift] = get(vt[m]); ld L = 0; for (int i = t + shift; i < (int)dp.size(); i++) L += dp[i]; ld R = forward(dp, shift, vt[m], 2, false); debug(l, r, L, R); if (L + EPS < R) l = m; else r = m+1; } // debug(l, r, vt[l]); auto [dp, shift] = get(vt[l]); // debug(dp); ans = max(ans, forward(dp, shift, vt[l], 10, true)); // for (int i = l; i <= r; i++) ans = max(ans, get(vt[i])); }; check(even); check(odd); cout << fixed << setprecision(15) << ans << "\n"; } int32_t main(){ ios_base::sync_with_stdio(0); cin.tie(0); solve(); return 0; } |