#include using namespace std; #define overload4(_1, _2, _3, _4, name, ...) name #define rep1(n) for(int i = 0; i < (int)(n); ++i) #define rep2(i, n) for(int i = 0; i < (int)(n); ++i) #define rep3(i, a, b) for(int i = (a); i < (int)(b); ++i) #define rep4(i, a, b, c) for(int i = (a); i < (int)(b); i += (c)) #define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__) #define rrep(i,n) for(int i = (int)(n) - 1; i >= 0; --i) #define ALL(a) (a).begin(), (a).end() #define Sort(a) (sort((a).begin(), (a).end())) #define RSort(a) (sort((a).rbegin(), (a).rend())) #define UNIQUE(a) (a.erase(unique((a).begin(), (a).end()), (a).end())) typedef long long int ll; typedef unsigned long long ul; typedef long double ld; typedef vector vi; typedef vector vll; typedef vector vc; typedef vector vst; typedef vector vd; typedef vector vld; typedef pair P; template long long sum(const T& a){ return accumulate(a.begin(), a.end(), 0LL); } template auto min(const T& a){ return *min_element(a.begin(), a.end()); } template auto max(const T& a){ return *max_element(a.begin(), a.end()); } const long long MINF = 0x7fffffffffff; const long long INF = 0x1fffffffffffffff; const long long MOD = 998244353; const long double EPS = 1e-9; const long double PI = acos(-1); template inline bool chmax(T& a, T b) { if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T& a, T b) { if (a > b) { a = b; return 1; } return 0; } template istream &operator>>(istream &is, pair &p){ is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const pair &p){ os << "(" << p.first << ", " << p.second << ")"; return os; } template istream &operator>>(istream &is, vector &v){ for(T &in : v) is >> in; return is; } template ostream &operator<<(ostream &os, const vector &v){ for(int i = 0; i < (int) v.size(); ++i){ os << v[i] << (i + 1 != (int) v.size() ? " " : ""); } return os; } template ostream &operator<<(ostream &os, const map &mp){ for(auto &[key, val] : mp){ os << key << ":" << val << " "; } return os; } template ostream &operator<<(ostream &os, const set &st){ auto itr = st.begin(); for(int i = 0; i < (int)st.size(); ++i){ os << *itr << (i + 1 != (int)st.size() ? " " : ""); itr++; } return os; } template ostream &operator<<(ostream &os, const multiset &st){ auto itr = st.begin(); for(int i = 0; i < (int)st.size(); ++i){ os << *itr << (i + 1 != (int)st.size() ? " " : ""); itr++; } return os; } template ostream &operator<<(ostream &os, queue q){ while(q.size()){ os << q.front() << " "; q.pop(); } return os; } template ostream &operator<<(ostream &os, deque q){ while(q.size()){ os << q.front() << " "; q.pop_front(); } return os; } template ostream &operator<<(ostream &os, stack st){ while(st.size()){ os << st.top() << " "; st.pop(); } return os; } template ostream &operator<<(ostream &os, priority_queue pq){ while(pq.size()){ os << pq.top() << " "; pq.pop(); } return os; } template inline T vin(T& vec, U n) { vec.resize(n); for(int i = 0; i < (int) n; ++i) cin >> vec[i]; return vec; } template inline void vout(T vec, string s = "\n"){ for(auto x : vec) cout << x << s; } template void in(T&... a){ (cin >> ... >> a); } void out(){ cout << '\n'; } template void out(const T& a, const Ts&... b){ cout << a; (cout << ... << (cout << ' ', b)); cout << '\n'; } template void inGraph(vector>& G, U n, U m, bool directed = false){ G.resize(n); for(int i = 0; i < m; ++i){ int a, b; cin >> a >> b; a--, b--; G[a].push_back(b); if(!directed) G[b].push_back(a); } } namespace Geometry{ using T = long double; const T INFT = 9e18; inline constexpr int type(T x, T y){ if(!x && !y) return 0; if(y < 0 || (y == 0 && x > 0)) return -1; return 1; } T absT(T x){ if(x < 0) return -x; return x; } struct Point{ T x, y; Point(T X = 0, T Y = 0) : x(X), y(Y){} inline bool operator==(const Point &other) const { return ((x == other.x) && (y == other.y)); } inline bool operator!=(const Point &other) const { return ((x != other.x) || (y != other.y)); } inline bool operator<(const Point &other) const { int L = type(x, y), R = type(other.x, other.y); if(L != R) return L < R; if(x * other.y == other.x * y) return abs(x + y) < abs(other.x + other.y); return x * other.y > other.x * y; } inline bool operator>(const Point &other) const { int L = type(x, y), R = type(other.x, other.y); if(L != R) return L > R; if(x * other.y == other.x * y) return abs(x + y) > abs(other.x + other.y); return x * other.y < other.x * y; } inline Point operator+() const noexcept { return *this; } inline Point operator-() const noexcept { return Point(-x, -y); } inline Point operator+(const Point &p) const { return Point(x + p.x, y + p.y); } inline Point operator-(const Point &p) const { return Point(x - p.x, y - p.y); } inline Point &operator+=(const Point &p) { return x += p.x, y += p.y, *this; } inline Point &operator-=(const Point &p) { return x -= p.x, y -= p.y, *this; } inline T operator*(const Point &p) const { return x * p.x + y * p.y; } inline Point &operator*=(const T &k) { return x *= k, y *= k, *this; } inline Point operator*(const T &k) { return (*this *= k); } // floor inline Point &operator/=(const T &k) { return x /= k, y /= k, *this; } inline Point operator/(const T &k) { return (*this /= k); } friend inline istream &operator>>(istream &is, Point &p) noexcept { is >> p.x >> p.y; return is; } friend inline ostream &operator<<(ostream &os, const Point &p) noexcept { return os << p.x << " " << p.y; } }; bool angle_equal(const Point &p, const Point &q){ int L = type(p.x, p.y), R = type(q.x, q.y); if(L != R) return false; return p.x * q.y == q.x * p.y; } long double rad2deg(long double rad){ return rad * (long double) 180 / acos(-1); } long double deg2rad(long double deg){ return deg * acosl(-1) / (long double) 180; } Point rotate(Point &p, long double deg){ complex comp(p.x, p.y); comp *= exp(complex(.0, deg2rad(deg))); return Point(comp.real(), comp.imag()); } T cross(const Point &p, const Point &q){ return p.x * q.y - p.y * q.x; } T dot(const Point &p, const Point &q){ return p.x * q.x + p.y * q.y; } T manhattanDist(const Point &p, const Point &q){ return absT(p.x - q.x) + absT(p.y - q.y); } // 2乗 T dist(const Point &p, const Point &q){ return (p.x - q.x) * (p.x - q.x) + (p.y - q.y) * (p.y - q.y); } // 線分 p1-p2 と線分 q1-q2 bool intersection(const Point &p1, const Point &p2, const Point &q1, const Point &q2){ T a = cross(p2 - p1, q1 - p1); T b = cross(p2 - p1, q2 - p1); T c = cross(q2 - q1, p1 - q1); T d = cross(q2 - q1, p2 - q1); if(a == 0 && b == 0){ T e = dot(p2 - p1, q1 - p1); T f = dot(p2 - p1, q2 - p1); if(e > f) swap(e, f); return e <= dist(p1, p2) && 0 <= f; } return a * b <= 0 && c * d <= 0; } // 2倍 T polygonArea(const vector &points){ const int n = points.size(); T res = 0; for(int i = 0; i < n - 1; i++){ res += cross(points[i], points[i + 1]); } res += cross(points[n - 1], points[0]); return absT(res); } vector convexHull(vector points){ vector U, L, res; sort(points.begin(), points.end(), [](Point p, Point q){ return (p.x != q.x ? p.x < q.x : p.y < q.y); }); points.erase(unique(points.begin(), points.end()), points.end()); const int n = points.size(); if((int) points.size() <= 2){ return points; } // lower for(int i = 0; i < n; i++){ int j = L.size(); // 傾きで左回りかをチェック while(j >= 2 && cross(L[j - 1] - L[j - 2], points[i] - L[j - 2]) <= 0){ L.pop_back(); j--; } L.push_back(points[i]); } // upper for(int i = n - 1; i >= 0; i--){ int j = U.size(); while(j >= 2 && cross(U[j - 1] - U[j - 2], points[i] - U[j - 2]) <= 0){ U.pop_back(); j--; } U.push_back(points[i]); } res = L; for(int i = 1; i < (int) U.size() - 1; i++){ res.push_back(U[i]); } return res; } // 点が領域外部: 0, 内部: 1, 境界上: 2 int inCcwConvex(Point p, const vector &points) { const int n = points.size(); T cr1 = cross(points[1] - points[0], p - points[0]); T cr2 = cross(points[n - 1] - points[0], p - points[0]); if(cr1 < 0 || 0 < cr2){ return 0; } int l = 1, r = n - 1; while(abs(r - l) > 1){ int mid = (l + r) / 2; if(cross(p - points[0], points[mid] - points[0]) >= 0){ r = mid; } else{ l = mid; } } T cr = cross(points[l] - p, points[r] - p); if(cr == 0){ return 2; } else if(cr > 0){ if(cr1 == 0 || cr2 == 0){ return 2; } else{ return 1; } } else{ return 0; } } pair> closestPair(vector &points){ const int n = points.size(); assert(n >= 2); vector> sortp(n); for(int i = 0; i < n; i++){ sortp[i] = {points[i], i}; } sort(sortp.begin(), sortp.end(), [](pair p, pair q){ return (p.first.x != q.first.x ? p.first.x < q.first.x : p.first.y < q.first.y); }); int ans1 = -1, ans2 = -1; T min_dist = INFT; auto dfs = [&](auto &self, int l, int r) -> T { if(r - l <= 1){ return INFT; } int mid = (l + r) / 2; T d = min(self(self, l, mid), self(self, mid, r)); vector> tmp; for(int i = l; i < r; i++){ T dx = sortp[mid].first.x - sortp[i].first.x; if(dx * dx < d){ tmp.push_back(sortp[i]); } } sort(tmp.begin(), tmp.end(), [](pair p, pair q){ return p.first.y < q.first.y; }); for(int i = 0; i < (int) tmp.size(); i++){ for(int j = i + 1; j < (int) tmp.size(); j++){ T dy = tmp[j].first.y - tmp[i].first.y; if(dy * dy >= d){ break; } T td = dist(tmp[i].first, tmp[j].first); if(td < d){ d = td; if(d < min_dist){ min_dist = d; ans1 = tmp[i].second; ans2 = tmp[j].second; } } } } return d; }; dfs(dfs, 0, n); return {min_dist, {ans1, ans2}}; } pair> furthestPair(vector &points){ const int n = points.size(); assert(n >= 2); vector convex = convexHull(points); const int m = convex.size(); map, int> mp; for(int i = 0; i < n; i++){ mp[{points[i].x, points[i].y}] = i; } vector idx(m); for(int i = 0; i < m; i++){ idx[i] = mp[{convex[i].x, convex[i].y}]; } if(m == 1){ return {dist(points[0], points[1]), {0, 1}}; }else if(m == 2){ return {dist(convex[0], convex[1]), {idx[0], idx[1]}}; } auto compare = [](Point p, Point q){ return p.x != q.x ? p.x < q.x : p.y < q.y; }; int i = 0, j = 0; for(int k = 0; k < m; k++){ if(compare(convex[k], convex[i])) i = k; if(compare(convex[j], convex[k])) j = k; } int i0 = i, j0 = j; T max_dist = 0; int ans1 = -1, ans2 = -1; while(i != j0 || j != i0){ T d = dist(convex[i], convex[j]); if(d > max_dist){ max_dist = d; ans1 = idx[i]; ans2 = idx[j]; } if(cross(convex[(i + 1) % m] - convex[i], convex[(j + 1) % m] - convex[j]) < 0){ i = (i + 1) % m; }else{ j = (j + 1) % m; } } return {max_dist, {ans1, ans2}}; } } ll T; void input(){ in(T); } ld dp[1 << 14][15][15]; void solve(){ ll n, k; in(n, k); vector p(n); in(p); p.push_back(Geometry::Point(0, 0)); rep(bit, 1 << n) rep(i, n + 1) rep(j, k + 1) dp[bit][i][j] = INF; dp[0][n][k] = 0; rep(bit, 1 << n){ rep(i, n + 1){ rep(j, k + 1){ rep(nxt, n){ if(bit & (1 << nxt)) continue; if(j == 0){ ld d = sqrtl(Geometry::dist(p[i], p[n])) + sqrtl(Geometry::dist(p[n], p[nxt])); chmin(dp[bit | (1 << nxt)][nxt][k - 1], dp[bit][i][j] + d); }else{ ld d = sqrtl(Geometry::dist(p[i], p[nxt])); chmin(dp[bit | (1 << nxt)][nxt][j - 1], dp[bit][i][j] + d); } } } } } ld ans = INF; rep(i, n){ rep(j, k + 1){ chmin(ans, dp[(1 << n) - 1][i][j] + sqrtl(Geometry::dist(p[i], p[n]))); } } out(ans); } int main(){ ios::sync_with_stdio(false); cin.tie(nullptr); cout << fixed << setprecision(20); T = 1; // input(); while(T--) solve(); }