#pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; //#define int long long typedef long long ll; typedef unsigned long long ul; typedef unsigned int ui; //constexpr ll mod = 998244353; constexpr ll mod = 1000000007; const ll INF = mod * mod; typedef pairP; #define rep(i,n) for(int i=0;i=0;i--) #define Rep(i,sta,n) for(int i=sta;i=1;i--) #define Rep1(i,sta,n) for(int i=sta;i<=n;i++) #define all(v) (v).begin(),(v).end() typedef pair LP; template void chmin(T& a, T b) { a = min(a, b); } template void chmax(T& a, T b) { a = max(a, b); } template void cinarray(vector& v) { rep(i, v.size())cin >> v[i]; } template void coutarray(vector& v) { rep(i, v.size()) { if (i > 0)cout << " "; cout << v[i]; } cout << "\n"; } ll mod_pow(ll x, ll n, ll m = mod) { if (n < 0) { ll res = mod_pow(x, -n, m); return mod_pow(res, m - 2, m); } if (abs(x) >= m)x %= m; if (x < 0)x += m; //if (x == 0)return 0; ll res = 1; while (n) { if (n & 1)res = res * x % m; x = x * x % m; n >>= 1; } return res; } //mod should be <2^31 struct modint { int n; modint() :n(0) { ; } modint(ll m) { if (m < 0 || mod <= m) { m %= mod; if (m < 0)m += mod; } n = m; } operator int() { return n; } }; bool operator==(modint a, modint b) { return a.n == b.n; } bool operator<(modint a, modint b) { return a.n < b.n; } modint operator+=(modint& a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= (int)mod; return a; } modint operator-=(modint& a, modint b) { a.n -= b.n; if (a.n < 0)a.n += (int)mod; return a; } modint operator*=(modint& a, modint b) { a.n = ((ll)a.n * b.n) % mod; return a; } modint operator+(modint a, modint b) { return a += b; } modint operator-(modint a, modint b) { return a -= b; } modint operator*(modint a, modint b) { return a *= b; } modint operator^(modint a, ll n) { if (n == 0)return modint(1); modint res = (a * a) ^ (n / 2); if (n % 2)res = res * a; return res; } ll inv(ll a, ll p) { return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p); } modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); } modint operator/=(modint& a, modint b) { a = a / b; return a; } const int max_n = 1 << 21; modint fact[max_n], factinv[max_n]; void init_f() { fact[0] = modint(1); for (int i = 0; i < max_n - 1; i++) { fact[i + 1] = fact[i] * modint(i + 1); } factinv[max_n - 1] = modint(1) / fact[max_n - 1]; for (int i = max_n - 2; i >= 0; i--) { factinv[i] = factinv[i + 1] * modint(i + 1); } } modint comb(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[b] * factinv[a - b]; } modint combP(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[a - b]; } ll gcd(ll a, ll b) { a = abs(a); b = abs(b); if (a < b)swap(a, b); while (b) { ll r = a % b; a = b; b = r; } return a; } using ld = long double; //typedef long double ld; typedef pair LDP; const ld eps = 1e-10; const ld pi = acosl(-1.0); template void addv(vector& v, int loc, T val) { if (loc >= v.size())v.resize(loc + 1, 0); v[loc] += val; } /*const int mn = 2000005; bool isp[mn]; vector ps; void init() { fill(isp + 2, isp + mn, true); for (int i = 2; i < mn; i++) { if (!isp[i])continue; ps.push_back(i); for (int j = 2 * i; j < mn; j += i) { isp[j] = false; } } }*/ //[,val) template auto prev_itr(set& st, T val) { auto res = st.lower_bound(val); if (res == st.begin())return st.end(); res--; return res; } //[val,) template auto next_itr(set& st, T val) { auto res = st.lower_bound(val); return res; } using mP = pair; mP operator+(mP a, mP b) { return { a.first + b.first,a.second + b.second }; } mP operator+=(mP& a, mP b) { a = a + b; return a; } mP operator-(mP a, mP b) { return { a.first - b.first,a.second - b.second }; } mP operator-=(mP& a, mP b) { a = a - b; return a; } LP operator+(LP a, LP b) { return { a.first + b.first,a.second + b.second }; } LP operator+=(LP& a, LP b) { a = a + b; return a; } LP operator-(LP a, LP b) { return { a.first - b.first,a.second - b.second }; } LP operator-=(LP& a, LP b) { a = a - b; return a; } mt19937 mt(time(0)); const string drul = "DRUL"; string senw = "SENW"; //DRUL,or SENW int dx[4] = { 1,0,-1,0 }; int dy[4] = { 0,1,0,-1 }; //----------------------------------------- typedef long double Data; typedef vector Array; typedef vector mat; bool is_zero(Data dat) { return abs(dat) < eps; } mat operator-(mat a) { rep(i, a.size())rep(j, a[0].size())a[i][j] = -a[i][j]; return a; } mat operator+(mat lhs, mat& rhs) { rep(i, lhs.size())rep(j, lhs[0].size())lhs[i][j] += rhs[i][j]; return lhs; } mat operator-(mat lhs, mat& rhs) { rep(i, lhs.size())rep(j, lhs[0].size())lhs[i][j] -= rhs[i][j]; return lhs; } mat operator*(const mat& lhs, const mat& rhs) { mat ret(lhs.size(), Array(rhs[0].size(), 0)); rep(i, lhs.size())rep(j, rhs[0].size())rep(k, rhs.size()) { ret[i][j] += lhs[i][k] * rhs[k][j]; } return ret; } //no verify int rankMat(mat a) { const int n = a.size(), m = a[0].size(); int r = 0; for (int i = 0; r < n && i < m; ++i) { int pivot = r; for (int j = r + 1; j < n; ++j) if (abs(a[j][i]) > abs(a[pivot][i]))pivot = j; swap(a[pivot], a[r]); if (is_zero(a[r][i]))continue; for (int k = m - 1; k >= i; --k) a[r][k] = a[r][k] / a[r][i]; for (int j = r + 1; j < n; ++j) { for (int k = m - 1; k >= i; --k) { a[j][k] = fma(-a[r][k], a[j][i], a[j][k]); } } ++r; } return r; } mat scalar(int sz, Data k) { mat ret(sz, Array(sz, 0)); rep(i, sz)ret[i][i] = k; return ret; } //行列累乗 mat operator^(const mat& lhs, const ll n) { if (n == 0)return scalar(lhs.size(), 1); mat ret = (lhs * lhs) ^ (n / 2); if (n % 2) { ret = ret * lhs; } return ret; } Data det(mat a) { const int n = a.size(); Data D = Data(1); for (int i = 0; i < n; ++i) { int pivot = i; for (int j = i + 1; j < n; ++j) { if (abs(a[j][i]) > abs(a[pivot][i]))pivot = j; } swap(a[pivot], a[i]); D = D * a[i][i] * Data(i != pivot ? -1 : 1); if (is_zero(a[i][i]))break; Data coef = (Data)1 / a[i][i]; for (int j = i + 1; j < n; ++j) { if (a[j][i] == (Data)0)continue; for (int k = n - 1; k >= i; --k) { a[j][k] = a[j][k] - a[i][k] * a[j][i] * coef; } } } return D; } pair> LUPDecomposition(mat a) { int n = a.size(); vector perm(n); iota(begin(perm), end(perm), 0); rep(i, n) { int pivot = i; for (int j = i + 1; j < n; ++j) if (abs(a[j][i]) > abs(a[pivot][i]))pivot = j; swap(a[pivot], a[i]); swap(perm[pivot], perm[i]); for (int j = i + 1; j < n; ++j) { a[j][i] /= a[i][i]; for (int k = i + 1; k < n; ++k) { a[j][k] -= a[i][k] * a[j][i]; } } } return make_pair(a, perm); } Array LUPBackSubstitution(mat& LU, vector& perm, Array a) { int n = LU.size(); Array tmp(n); rep(i, n)tmp[i] = a[perm[i]]; swap(tmp, a); rep(i, n) { rep(j, i) { a[i] -= a[j] * LU[i][j]; } } for (int i = n - 1; i >= 0; --i) { for (int j = i + 1; j < n; ++j) { a[i] -= a[j] * LU[i][j]; } a[i] /= LU[i][i]; } return a; } //Ax=bのxを返す Array calc(mat A, Array b) { pair> p = LUPDecomposition(A); return LUPBackSubstitution(p.first, p.second, b); } void solve() { int n, m; cin >> n >> m; vector c(n); rep1(i, n-2)cin >> c[i]; mat A(n - 1, Array(n - 1)); Array b(n - 1); rep(i, n - 1) { A[i][i] += 1; rep1(j, m) { int to = i + j; if (to >= n)to = 2 * (n - 1) - to; if (to < n - 1) { A[i][to] -= 1 / (ld)m; b[i] += c[to]/(ld)m; } } } auto a = calc(A, b); //coutarray(a); vector dp(n); per(i, n - 1) { if (i + m >= n - 1) { dp[i] = 0; } else { //decide ld val1 = INF; rep1(j, m) { chmin(val1, a[i + j] + c[i + j]); } //random ld val2 = 0; rep1(j, m) { val2 += (dp[i + j] + c[i + j]) / (ld)m; } dp[i] = min(val1, val2); } } cout << dp[0] << "\n"; } signed main() { ios::sync_with_stdio(false); cin.tie(0); cout << fixed << setprecision(10); //init_f(); //init(); //expr(); //while(true) //int t; cin >> t; rep(i, t) solve(); return 0; }