// // 一般化 Convex Hull Trick // Monge 性を満たす関数群 f_1, f_2, ..., f_N // // verified // COLOCON 2018 Final C - スペースエクスプローラー高橋君 // https://beta.atcoder.jp/contests/colopl2018-final-open/tasks/colopl2018_final_c // // AtCoder EDPC Z - Frog 3 // https://atcoder.jp/contests/dp/tasks/dp_z // // yukicoder No.705 ゴミ拾い Hard // https://yukicoder.me/problems/no/705 // #include using namespace std; // Convex Hull Trick /* Func:Monge 性を仮定 - MIN: クエリ x の取りうる最小値 - MAX: クエリ x の取りうる最大値 (f_i(MAX) がオーバーフローしないように注意) - INF: 最大値 - insert (Func f_i): add f_i, O(log N) - query (x): min_i{ f_i(x) }, O(log N) */ template struct CHT { using Func = function; struct Node { Func func; Node *left, *right; Node(const Func& f) : left(nullptr), right(nullptr) { func = f; } }; const T MIN, MAX, INF; Node* root; CHT(T MIN, T MAX, T INF) : MIN(MIN), MAX(MAX), INF(INF), root(nullptr) { } Node* insert(Node* p, T low, T high, Func& f) { if (!p) return new Node(f); if (p->func(low) <= f(low) && p->func(high) <= f(high)) return p; if (p->func(low) >= f(low) && p->func(high) >= f(high)) { p->func = f; return p; } T mid = (low + high) / 2; if (p->func(mid) > f(mid)) swap(p->func, f); if (p->func(low) >= f(low)) p->left = insert(p->left, low, mid, f); else p->right = insert(p->right, mid, high, f); return p; } void insert(Func f) { root = insert(root, MIN, MAX, f); } T query(Node* p, T low, T high, T x) { if (!p) return INF; if (low == high) return p->func(x); T mid = (low + high) / 2; if (x <= mid) return min(p->func(x), query(p->left, low, mid, x)); else return min(p->func(x), query(p->right, mid, high, x)); } T query(T x) { return query(root, MIN, MAX, x); } }; //------------------------------// // Examples //------------------------------// // COLOCON 2018 Final C - スペースエクスプローラー高橋君 void COLOCON_2018_final_C() { long long N; cin >> N; vector a(N), res(N); for (int i = 0; i < N; ++i) cin >> a[i]; CHT cht(0, 210000, 1LL<<60); for (long long i = 0; i < N; ++i) { auto func = [i, &a](long long x) -> long long { return (x - i) * (x - i) + a[i]; }; cht.insert(func); } for (long long i = 0; i < N; ++i) res[i] = cht.query(i); for (int i = 0; i < N; ++i) cout << res[i] << endl; } // AtCoder EDPC Z - Frog 3 void EDPC_Z() { long long N, C; cin >> N >> C; vector H(N); for (int i = 0; i < N; i++) cin >> H[i]; const long long INF = 1LL<<60; const long long MAX = 1LL<<40; vector dp(N, INF); /*   dp[i] = min_j(dp[j] + (H[j] - H[i])² + C) */ dp[0] = 0; CHT cht(0, MAX, INF); auto func = [&H, C](long long x) -> long long { return (x - H[0]) * (x - H[0]) + C; }; cht.insert(func); for (int i = 1; i < N; i++) { long long val = cht.query(H[i]); dp[i] = min(dp[i], val); auto func = [&dp, &H, i, C](long long x) -> long long { return (x - H[i]) * (x - H[i]) + dp[i] + C; }; cht.insert(func); } long long res = dp[N-1]; cout << res << endl; } // yukicoder No.705 ゴミ拾い Hard void yukicoder_705() { int N; cin >> N; vector A(N), X(N), Y(N); for (int i = 0; i < N; i++) cin >> A[i]; for (int i = 0; i < N; i++) cin >> X[i]; for (int i = 0; i < N; i++) cin >> Y[i]; const long long INF = 1LL<<60; const long long MAX = 110000; vector dp(N+1, INF); CHT cht(0, MAX, INF); dp[0] = 0; for (int i = 1; i <= N; i++) { auto func = [&dp, &X, &Y, i](long long x) -> long long { long long dx = abs(x - X[i-1]), dy = abs(Y[i-1]); return dp[i-1] + dx*dx*dx + dy*dy*dy; }; cht.insert(func); dp[i] = cht.query(A[i-1]); } cout << dp[N] << endl; } int main() { //COLOCON_2018_final_C(); //EDPC_Z(); yukicoder_705(); }