#pragma GCC optimize("O3,unroll-loops") #include <bits/stdc++.h> #define fi first #define se second #define pn printf("\n") #define ssize(x) int(x.size()) #define all(x) x.begin(),x.end() #define rall(x) x.rbegin(),x.rend() #define bitcount(x) __builtin_popcount(x) #define bitcountll(x) __builtin_popcountll(x) #define clz(x) __builtin_clz(x) #define ctz(x) __builtin_ctz(x) #define eb emplace_back //~ #define r(x) resize(x) //~ #define rf(x, c) resize(x, c) using namespace std; typedef long long ll; typedef pair<int, int> pii; typedef pair<int, ll> pil; typedef pair<ll, int> pli; typedef pair<ll, ll> pll; typedef double db; typedef long double ldb; #define V vector //~ void read(int &a){ //~ a = 0; char c = _getchar_nolock(); //~ while(c<'0'||'9'<c) c = _getchar_nolock(); //~ while('0'<=c&&c<='9') a = a*10+c-'0', c = _getchar_nolock(); //~ } // random_device rd; // mt19937 rng(rd()); // uniform_int_distribution<int> mrandint(1, (1<<30)-1); // uniform_int_distribution<ll> mrandll(1, 1ll<<60); // int randint(){ return mrandint(rng); } // ll randll(){ return mrandll(rng); } int inf = 2.1e09; ll infll = 2e18; int mod = (1<<23)*119+1; //1e09+7; int add(int a, int b){return a+b >= mod ? a+b - mod : a+b;} int sub(int a, int b){return a-b < 0 ? a-b + mod : a-b;} int mul(int a, int b){return int(a * ll(b) % mod);} int fpow(int a, ll b){ int ret = 1; while(b){ if(b & 1) ret = mul(ret, a); b >>= 1, a = mul(a, a); } return ret; } int inv(int a){ return fpow(a, mod-2); } int coeff(int n, int k, vector<int> &fac, vector<int> &invfac){ if(k < 0 || n < k) return 0; return mul(fac[n], mul(invfac[n-k], invfac[k])); } void calcfac(int n, vector<int> &fac, vector<int> &invfac){ fac[0] = 1, invfac[0] = 1; for(int i = 1; i <= n; ++i) fac[i] = mul(fac[i-1], i); invfac[n] = inv(fac[n]); for(int i = n-1; i; --i) invfac[i] = mul(invfac[i+1], i+1); } struct lz{ double a = 0, b = 0; lz(){} lz(double A, double B) : a(A), b(B) {} lz operator+(const lz &A) const{return lz(a + A.a, b + A.b);} lz operator-(const lz &A) const{return lz(a - A.a, b - A.b);} lz operator*(const lz &A) const{return lz(a * A.a - b * A.b, a * A.b + b * A.a);} lz operator*(const double &A) const{return lz(a * A, b * A);} lz operator/(const double &A) const{return lz(a / A, b / A);} }; void to_lz(V<lz> &t, V<double> &a){ t.resize(ssize(a)); for(int i = 0; i < ssize(a); ++i) t[i].a = a[i]; } struct poly{ const int mxpot = 1<<16; // do zmiany wedle potrzeby vector<lz> roots[2]; void calc_roots(){ double ang; lz wn; roots[0].resize(mxpot), roots[1].resize(mxpot); ang = 2*acos(-1)/mxpot, roots[0][0] = lz(1, 0), roots[1][0] = lz(1, 0); wn = lz(cos(ang), sin(ang)); for(int j = 1; j < mxpot; ++j) roots[0][j] = roots[0][j-1] * wn; wn = lz(cos(-ang), sin(-ang)); for(int j = 1; j < mxpot; ++j) roots[1][j] = roots[1][j-1] * wn; } poly(){ calc_roots(); } void odw(int &x, int p){ int a, b; for(int i = 0; i < p>>1; ++i) a = (x&(1<<i)), b = (x&(1<<(p-i-1))), x -= a + b, x += (a<<(p-2*i-1))+(b>>(p-2*i-1)); } void dft(vector<lz> &a, bool inv){ int n = int(a.size()), p = __builtin_ctz(n), tmp; for(int i = 0; i < n; ++i){ tmp = i, odw(tmp, p); if(i < tmp) swap(a[i], a[tmp]); } int w; for(int dl = 2; dl <= n; dl<<=1){ w = mxpot/dl; for(int i = 0; i < n; i += dl){ for(int k = 0; k < dl>>1; ++k){ lz A = a[i+k], B = a[i+k+(dl>>1)]*roots[inv][w*k]; a[i+k] = A + B; a[i+k+(dl>>1)] = A - B; } } } } void fft(vector<lz> &a, vector<lz> &b){ int n = int(a.size()+b.size()), pot = 1; while(pot < n) pot <<= 1; // n = pot; a.resize(pot), b.resize(pot); dft(a, 0), dft(b, 0); for(int i = 0; i < pot; ++i) a[i] = a[i] * b[i]; dft(a, 1); for(int i = 0; i < pot; ++i) a[i].a = a[i].a/pot, a[i].b = 0; while(n-1 < ssize(a)) a.pop_back(); } } multiplier; void answer(){ int n, t; cin >> n >> t; V<double> p(n); for(int i = 0; i < n; ++i) cin >> p[i]; sort(rall(p)); int k = int(sqrt(n*16))+1; auto get_dp = [&](int l, int r){ // [l, r) int N = r-l; V<double> dp(N<<1|1, 0), tp(N<<1|1, 0); dp[N] = 1; for(int i = l; i < r; ++i){ fill(all(tp), 0); for(int j = N-(i-l); j <= N+(i-l); ++j){ tp[j+1] += dp[j]*p[i]; tp[j-1] += dp[j]*(1-p[i]); } dp = tp; } return dp; }; V<V<double>> dp_list((n+k-1)/k); V<V<lz>> lz_dp((n+k-1)/k); for(int i = 0; i < n; i += k){ dp_list[i/k] = get_dp(i, min(n, i+k)); to_lz(lz_dp[i/k], dp_list[i/k]); } V<lz> dp(1, lz(1, 0)); int start = 0; for(int i = 0; i+1+k <= t; i += k) multiplier.fft(dp, lz_dp[i/k]), start += k; V<double> dpt(ssize(dp)+2*k), dptmp(ssize(dpt)); for(int i = 0; i < ssize(dp); ++i) dpt[i+k] = dp[i].a; double result = 0; int mid = ssize(dpt)/2; for(int i = start; i < min(n, start+k); ++i){ fill(all(dptmp), 0); for(int j = mid-i; j <= mid+i; ++j){ dptmp[j+1] += dpt[j]*p[i]; dptmp[j-1] += dpt[j]*(1-p[i]); } dpt = dptmp; if(t <= i+1){ double sum = 0; for(int j = mid+t; j < ssize(dpt); ++j) sum += dpt[j]; result = max(result, sum); } } multiplier.fft(dp, lz_dp[start/k]); start += k; for(int I = start; I < n; I += k){ dpt.resize(2*k, 0), dptmp.resize(2*k, 0); mid = ssize(dp)/2; double sum = 0; for(int j = mid+t; j < ssize(dp); ++j) sum += dp[j].a; for(int j = mid+t-k; j < min(ssize(dp), mid+t+k-1); ++j) dpt[j-(mid+t-k)] = dp[j].a; for(int i = I; i < min(n, I+k); ++i){ fill(all(dptmp), 0); for(int j = 0; j < 2*k; ++j){ if(j+1 < 2*k){ dptmp[j+1] += dpt[j]*p[i]; if(j == k-1) sum += dpt[j]*p[i]; } if(0 <= j-1){ dptmp[j-1] += dpt[j]*(1-p[i]); if(j == k) sum -= dpt[j]*(1-p[i]); } } dpt = dptmp; result = max(result, sum); } multiplier.fft(dp, lz_dp[I/k]); } cout << setprecision(9) << fixed << result << "\n"; } int main(){ int T = 1; // scanf("%d", &T); ios_base::sync_with_stdio(0); cin.tie(0);// cin >> T; for(++T; --T; ) answer(); 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 | #pragma GCC optimize("O3,unroll-loops") #include <bits/stdc++.h> #define fi first #define se second #define pn printf("\n") #define ssize(x) int(x.size()) #define all(x) x.begin(),x.end() #define rall(x) x.rbegin(),x.rend() #define bitcount(x) __builtin_popcount(x) #define bitcountll(x) __builtin_popcountll(x) #define clz(x) __builtin_clz(x) #define ctz(x) __builtin_ctz(x) #define eb emplace_back //~ #define r(x) resize(x) //~ #define rf(x, c) resize(x, c) using namespace std; typedef long long ll; typedef pair<int, int> pii; typedef pair<int, ll> pil; typedef pair<ll, int> pli; typedef pair<ll, ll> pll; typedef double db; typedef long double ldb; #define V vector //~ void read(int &a){ //~ a = 0; char c = _getchar_nolock(); //~ while(c<'0'||'9'<c) c = _getchar_nolock(); //~ while('0'<=c&&c<='9') a = a*10+c-'0', c = _getchar_nolock(); //~ } // random_device rd; // mt19937 rng(rd()); // uniform_int_distribution<int> mrandint(1, (1<<30)-1); // uniform_int_distribution<ll> mrandll(1, 1ll<<60); // int randint(){ return mrandint(rng); } // ll randll(){ return mrandll(rng); } int inf = 2.1e09; ll infll = 2e18; int mod = (1<<23)*119+1; //1e09+7; int add(int a, int b){return a+b >= mod ? a+b - mod : a+b;} int sub(int a, int b){return a-b < 0 ? a-b + mod : a-b;} int mul(int a, int b){return int(a * ll(b) % mod);} int fpow(int a, ll b){ int ret = 1; while(b){ if(b & 1) ret = mul(ret, a); b >>= 1, a = mul(a, a); } return ret; } int inv(int a){ return fpow(a, mod-2); } int coeff(int n, int k, vector<int> &fac, vector<int> &invfac){ if(k < 0 || n < k) return 0; return mul(fac[n], mul(invfac[n-k], invfac[k])); } void calcfac(int n, vector<int> &fac, vector<int> &invfac){ fac[0] = 1, invfac[0] = 1; for(int i = 1; i <= n; ++i) fac[i] = mul(fac[i-1], i); invfac[n] = inv(fac[n]); for(int i = n-1; i; --i) invfac[i] = mul(invfac[i+1], i+1); } struct lz{ double a = 0, b = 0; lz(){} lz(double A, double B) : a(A), b(B) {} lz operator+(const lz &A) const{return lz(a + A.a, b + A.b);} lz operator-(const lz &A) const{return lz(a - A.a, b - A.b);} lz operator*(const lz &A) const{return lz(a * A.a - b * A.b, a * A.b + b * A.a);} lz operator*(const double &A) const{return lz(a * A, b * A);} lz operator/(const double &A) const{return lz(a / A, b / A);} }; void to_lz(V<lz> &t, V<double> &a){ t.resize(ssize(a)); for(int i = 0; i < ssize(a); ++i) t[i].a = a[i]; } struct poly{ const int mxpot = 1<<16; // do zmiany wedle potrzeby vector<lz> roots[2]; void calc_roots(){ double ang; lz wn; roots[0].resize(mxpot), roots[1].resize(mxpot); ang = 2*acos(-1)/mxpot, roots[0][0] = lz(1, 0), roots[1][0] = lz(1, 0); wn = lz(cos(ang), sin(ang)); for(int j = 1; j < mxpot; ++j) roots[0][j] = roots[0][j-1] * wn; wn = lz(cos(-ang), sin(-ang)); for(int j = 1; j < mxpot; ++j) roots[1][j] = roots[1][j-1] * wn; } poly(){ calc_roots(); } void odw(int &x, int p){ int a, b; for(int i = 0; i < p>>1; ++i) a = (x&(1<<i)), b = (x&(1<<(p-i-1))), x -= a + b, x += (a<<(p-2*i-1))+(b>>(p-2*i-1)); } void dft(vector<lz> &a, bool inv){ int n = int(a.size()), p = __builtin_ctz(n), tmp; for(int i = 0; i < n; ++i){ tmp = i, odw(tmp, p); if(i < tmp) swap(a[i], a[tmp]); } int w; for(int dl = 2; dl <= n; dl<<=1){ w = mxpot/dl; for(int i = 0; i < n; i += dl){ for(int k = 0; k < dl>>1; ++k){ lz A = a[i+k], B = a[i+k+(dl>>1)]*roots[inv][w*k]; a[i+k] = A + B; a[i+k+(dl>>1)] = A - B; } } } } void fft(vector<lz> &a, vector<lz> &b){ int n = int(a.size()+b.size()), pot = 1; while(pot < n) pot <<= 1; // n = pot; a.resize(pot), b.resize(pot); dft(a, 0), dft(b, 0); for(int i = 0; i < pot; ++i) a[i] = a[i] * b[i]; dft(a, 1); for(int i = 0; i < pot; ++i) a[i].a = a[i].a/pot, a[i].b = 0; while(n-1 < ssize(a)) a.pop_back(); } } multiplier; void answer(){ int n, t; cin >> n >> t; V<double> p(n); for(int i = 0; i < n; ++i) cin >> p[i]; sort(rall(p)); int k = int(sqrt(n*16))+1; auto get_dp = [&](int l, int r){ // [l, r) int N = r-l; V<double> dp(N<<1|1, 0), tp(N<<1|1, 0); dp[N] = 1; for(int i = l; i < r; ++i){ fill(all(tp), 0); for(int j = N-(i-l); j <= N+(i-l); ++j){ tp[j+1] += dp[j]*p[i]; tp[j-1] += dp[j]*(1-p[i]); } dp = tp; } return dp; }; V<V<double>> dp_list((n+k-1)/k); V<V<lz>> lz_dp((n+k-1)/k); for(int i = 0; i < n; i += k){ dp_list[i/k] = get_dp(i, min(n, i+k)); to_lz(lz_dp[i/k], dp_list[i/k]); } V<lz> dp(1, lz(1, 0)); int start = 0; for(int i = 0; i+1+k <= t; i += k) multiplier.fft(dp, lz_dp[i/k]), start += k; V<double> dpt(ssize(dp)+2*k), dptmp(ssize(dpt)); for(int i = 0; i < ssize(dp); ++i) dpt[i+k] = dp[i].a; double result = 0; int mid = ssize(dpt)/2; for(int i = start; i < min(n, start+k); ++i){ fill(all(dptmp), 0); for(int j = mid-i; j <= mid+i; ++j){ dptmp[j+1] += dpt[j]*p[i]; dptmp[j-1] += dpt[j]*(1-p[i]); } dpt = dptmp; if(t <= i+1){ double sum = 0; for(int j = mid+t; j < ssize(dpt); ++j) sum += dpt[j]; result = max(result, sum); } } multiplier.fft(dp, lz_dp[start/k]); start += k; for(int I = start; I < n; I += k){ dpt.resize(2*k, 0), dptmp.resize(2*k, 0); mid = ssize(dp)/2; double sum = 0; for(int j = mid+t; j < ssize(dp); ++j) sum += dp[j].a; for(int j = mid+t-k; j < min(ssize(dp), mid+t+k-1); ++j) dpt[j-(mid+t-k)] = dp[j].a; for(int i = I; i < min(n, I+k); ++i){ fill(all(dptmp), 0); for(int j = 0; j < 2*k; ++j){ if(j+1 < 2*k){ dptmp[j+1] += dpt[j]*p[i]; if(j == k-1) sum += dpt[j]*p[i]; } if(0 <= j-1){ dptmp[j-1] += dpt[j]*(1-p[i]); if(j == k) sum -= dpt[j]*(1-p[i]); } } dpt = dptmp; result = max(result, sum); } multiplier.fft(dp, lz_dp[I/k]); } cout << setprecision(9) << fixed << result << "\n"; } int main(){ int T = 1; // scanf("%d", &T); ios_base::sync_with_stdio(0); cin.tie(0);// cin >> T; for(++T; --T; ) answer(); return 0; } |