#line 1 "/home/maspy/compro/library/my_template.hpp" #include using namespace std; using ll = long long; using pi = pair; using vi = vector; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vec(type, name, ...) vector name(__VA_ARGS__) #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) #define FOR_(n) for (ll _ = 0; (_) < (ll)(n); ++(_)) #define FOR(i, n) for (ll i = 0; (i) < (ll)(n); ++(i)) #define FOR3(i, m, n) for (ll i = (m); (i) < (ll)(n); ++(i)) #define FOR_R(i, n) for (ll i = (ll)(n)-1; (i) >= 0; --(i)) #define FOR3_R(i, m, n) for (ll i = (ll)(n)-1; (i) >= (ll)(m); --(i)) #define FOR_subset(t, s) for (ll t = s; t >= 0; t = (t == 0 ? -1 : (t - 1) & s)) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll template T SUM(vector &A) { T sum = T(0); for (auto &&a: A) sum += a; return sum; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()) int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } template T ceil(T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } template pair divmod(T x, U y) { T q = floor(x, y); return {q, x - q * y}; } ll binary_search(function check, ll ok, ll ng) { assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; if (check(x)) ok = x; else ng = x; } return ok; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } vi s_to_vi(const string& S, char first_char) { vi A(S.size()); FOR(i, S.size()) { A[i] = S[i] - first_char; } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } template vc bincount(const vc &A, int size) { vc C(size); for (auto &&x: A) { ++C[x]; } return C; } template vector argsort(const vector &A) { // stable vector ids(A.size()); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return A[i] < A[j] || (A[i] == A[j] && i < j); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { int n = len(A); assert(len(I) == n); vc B(n); FOR(i, n) B[i] = A[I[i]]; return B; } #line 1 "/home/maspy/compro/library/other/io.hpp" // based on yosupo's fastio #include namespace detail { template std::true_type check_value(int); template std::false_type check_value(long); } // namespace detail template struct is_modint : decltype(detail::check_value(0)) {}; template using is_modint_t = enable_if_t::value>; template using is_not_modint_t = enable_if_t::value>; struct Scanner { FILE *fp; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) { bool sep = false; for (size_t i = st; i < ed; i++) { if (isspace(line[i])) { sep = true; break; } } if (!sep) reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); } if (neg) ref = -ref; return true; } template * = nullptr> bool read_single(T &ref) { long long val = 0; bool f = read_single(val); ref = T(val); return f; } bool read_single(double &ref) { string s; if (!read_single(s)) return false; ref = std::stod(s); return true; } bool read_single(char &ref) { string s; if (!read_single(s) || s.size() != 1) return false; ref = s[0]; return true; } template bool read_single(vector &ref) { for (auto &d: ref) { if (!read_single(d)) return false; } return true; } template bool read_single(pair &p) { return (read_single(p.first) && read_single(p.second)); } template bool read_single(tuple &p) { return (read_single(get<0>(p)) && read_single(get<1>(p)) && read_single(get<2>(p))); } template bool read_single(tuple &p) { return (read_single(get<0>(p)) && read_single(get<1>(p)) && read_single(get<2>(p)) && read_single(get<3>(p))); } void read() {} template void read(H &h, T &... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE *fp) : fp(fp) {} }; struct Printer { Printer(FILE *_fp) : fp(_fp) {} ~Printer() { flush(); } static constexpr size_t SIZE = 1 << 15; FILE *fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write(const char &val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write('0'); return; } if (val < 0) { write('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char(0x30 | (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write(const string &s) { for (char c: s) write(c); } void write(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write(s[i]); } void write(const double &x) { ostringstream oss; oss << setprecision(15) << x; string s = oss.str(); write(s); } void write(const long double &x) { ostringstream oss; oss << setprecision(15) << x; string s = oss.str(); write(s); } template * = nullptr> void write(T &ref) { write(ref.val); } template void write(const vector &val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } template void write(const pair &val) { write(val.first); write(' '); write(val.second); } template void write(const tuple &val) { auto &[a, b, c] = val; write(a); write(' '); write(b); write(' '); write(c); } template void write(const tuple &val) { auto &[a, b, c, d] = val; write(a); write(' '); write(b); write(' '); write(c); write(' '); write(d); } template void write(const array &val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } void write(i128 val) { string s; bool negative = 0; if(val < 0){ negative = 1; val = -val; } while (val) { s += '0' + int(val % 10); val /= 10; } if(negative) s += "-"; reverse(all(s)); if (len(s) == 0) s = "0"; write(s); } }; Scanner scanner = Scanner(stdin); Printer printer = Printer(stdout); void flush() { printer.flush(); } void print() { printer.write('\n'); } template void print(Head &&head, Tail &&... tail) { printer.write(head); if (sizeof...(Tail)) printer.write(' '); print(forward(tail)...); } void read() {} template void read(Head &head, Tail &... tail) { scanner.read(head); read(tail...); } #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 1 "/home/maspy/compro/library/dp/monge.hpp" template class LARSCH { struct reduce_row; struct reduce_col; struct reduce_row { int n; std::function f; int cur_row; int state; std::unique_ptr rec; reduce_row(int n_) : n(n_), f(), cur_row(0), state(0), rec() { const int m = n / 2; if (m != 0) { rec = std::make_unique(m); } } void set_f(std::function f_) { f = f_; if (rec) { rec->set_f([&](int i, int j) -> T { return f(2 * i + 1, j); }); } } int get_argmin() { const int cur_row_ = cur_row; cur_row += 1; if (cur_row_ % 2 == 0) { const int prev_argmin = state; const int next_argmin = [&]() { if (cur_row_ + 1 == n) { return n - 1; } else { return rec->get_argmin(); } }(); state = next_argmin; int ret = prev_argmin; for (int j = prev_argmin + 1; j <= next_argmin; j += 1) { if (f(cur_row_, ret) > f(cur_row_, j)) { ret = j; } } return ret; } else { if (f(cur_row_, state) <= f(cur_row_, cur_row_)) { return state; } else { return cur_row_; } } } }; struct reduce_col { int n; std::function f; int cur_row; std::vector cols; reduce_row rec; reduce_col(int n_) : n(n_), f(), cur_row(0), cols(), rec(n) {} void set_f(std::function f_) { f = f_; rec.set_f([&](int i, int j) -> T { return f(i, cols[j]); }); } int get_argmin() { const int cur_row_ = cur_row; cur_row += 1; const auto cs = [&]() -> std::vector { if (cur_row_ == 0) { return {{0}}; } else { return {{2 * cur_row_ - 1, 2 * cur_row_}}; } }(); for (const int j: cs) { while ([&]() { const int size = cols.size(); return size != cur_row_ && f(size - 1, cols.back()) > f(size - 1, j); }()) { cols.pop_back(); } if (cols.size() != n) { cols.push_back(j); } } return cols[rec.get_argmin()]; } }; std::unique_ptr base; public: LARSCH(int n, std::function f) : base(std::make_unique(n)) { base->set_f(f); } int get_argmin() { return base->get_argmin(); } }; /* https://dic.kimiyuki.net/d-edge-shortest-path-monge 上凸関数 calc_L(lambda) の最大値を求める問題に帰着 |f| の上限 f_lim も渡す */ template ll d_edge_shortest_path_monge(ll N, F f, ll d, ll f_lim) { const ll INF = 1LL << 60; auto calc_L = [&](ll lambda) -> ll { auto cost = [&](int frm, int to) -> ll { return f(frm, to) + lambda; }; vector DP(N, INF); DP[0] = 0; LARSCH larsch(N - 1, [&](int i, int j) { // dp[j] から dp[i+1] への遷移 if (i + 1 <= j) return INF; return DP[j] + cost(j, i + 1); }); FOR3(v, 1, N) { ll frm = larsch.get_argmin(); DP[v] = DP[frm] + cost(frm, v); } ll ANS = DP.back() - lambda * d; return ANS; }; ll ANS = -INF; ll L = -3 * f_lim - 10; ll R = 3 * f_lim + 10; ll x = binary_search([&](ll x) { return calc_L(x - 1) <= calc_L(x); }, L, R); return calc_L(x); } #line 4 "main.cpp" template bool check_monge(int N, F f) { FOR(l, N + 1) FOR(k, l) FOR(j, k) FOR(i, j) { ll lhs = f(i, l) + f(j, k); ll rhs = f(i, k) + f(j, l); if (lhs < rhs) return false; } return true; } template vi shortest_path_monge(ll N, F f) { vector DP(N + 1, INF); DP[0] = 0; LARSCH larsch(N, [&](int i, int j) { // dp[j] から dp[i+1] への遷移 if (i + 1 <= j) return INF; return DP[j] + f(j, i + 1); }); FOR3(r, 1, N + 1) { ll l = larsch.get_argmin(); DP[r] = DP[l] + f(l, r); } return DP; } void solve() { LL(N); VEC(ll, A, N); VEC(ll, X, N); VEC(ll, Y, N); auto f = [&](ll i, ll j) -> ll { ll a = A[j - 1]; ll x = X[i], y = Y[i]; ll dx = abs(a - x); ll dy = abs(y); return dx * dx * dx + dy * dy * dy; }; const ll INF = 1LL << 60; print(shortest_path_monge(N, f).back()); } signed main() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << setprecision(15); ll T = 1; // LL(T); FOR(_, T) solve(); return 0; }