#line 1 "test/yc_703_onoff.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/703"
#include <cstdint>
#include <cstdio>
#include <algorithm>
#include <vector>
#line 1 "algorithm/online_to_offline_optimization.cpp"
/**
* @brief オンライン・オフライン変換
* @author えびちゃん
*/
#include <cstddef>
#line 12 "algorithm/online_to_offline_optimization.cpp"
#line 1 "algorithm/monotone_minima.cpp"
/**
* @brief monotone minima
* @author えびちゃん
*/
#line 10 "algorithm/monotone_minima.cpp"
#include <utility>
#line 12 "algorithm/monotone_minima.cpp"
#line 1 "utility/make/fix_point.cpp"
/**
* @brief ラムダ式の再帰
* @author えびちゃん
*/
#ifndef H_make_fix_point
#define H_make_fix_point
#line 10 "utility/make/fix_point.cpp"
template <typename Fn>
class fix_point: Fn {
public:
explicit constexpr fix_point(Fn&& f) noexcept: Fn(std::forward<Fn>(f)) {}
template <typename... Args>
constexpr decltype(auto) operator ()(Args&&... args) const {
return Fn::operator ()(*this, std::forward<Args>(args)...);
}
};
template <typename Fn>
static inline constexpr decltype(auto) make_fix_point(Fn&& f) noexcept {
return fix_point<Fn>{std::forward<Fn>(f)};
}
#endif /* !defined(H_make_fix_point) */
#line 14 "algorithm/monotone_minima.cpp"
template <typename Fn>
auto monotone_minima(Fn&& f, size_t h, size_t w) {
using value_type = decltype(f(h, w));
std::vector<size_t> res(h);
make_fix_point([&](auto dfs, size_t hl, size_t hu, size_t wl, size_t wu) -> void {
if (hl >= hu) return;
size_t hm = (hl+hu) >> 1;
value_type min = f(hm, wl);
res[hm] = wl;
for (size_t j = wl+1; j < wu; ++j) {
value_type cur = f(hm, j);
if (cur < min) {
min = std::move(cur);
res[hm] = j;
}
}
if (hl == hm) return;
dfs(hl, hm, wl, res[hm]+1);
dfs(hm+1, hu, res[hm], wu);
})(0, h, 0, w);
return res;
}
#line 1 "utility/limits.cpp"
/**
* @brief 型依存の定数
* @author えびちゃん
*/
#include <limits>
#line 11 "utility/limits.cpp"
template <typename Tp>
class limits: public std::numeric_limits<Tp> {};
template <typename T1, typename T2>
class limits<std::pair<T1, T2>> {
public:
static constexpr auto min() {
return std::make_pair(limits<T1>::min(), limits<T2>::min());
}
static constexpr auto max() {
return std::make_pair(limits<T1>::max(), limits<T2>::max());
}
};
#line 1 "utility/make/fix_point.cpp"
/**
* @brief ラムダ式の再帰
* @author えびちゃん
*/
#ifndef H_make_fix_point
#define H_make_fix_point
#line 10 "utility/make/fix_point.cpp"
template <typename Fn>
class fix_point: Fn {
public:
explicit constexpr fix_point(Fn&& f) noexcept: Fn(std::forward<Fn>(f)) {}
template <typename... Args>
constexpr decltype(auto) operator ()(Args&&... args) const {
return Fn::operator ()(*this, std::forward<Args>(args)...);
}
};
template <typename Fn>
static inline constexpr decltype(auto) make_fix_point(Fn&& f) noexcept {
return fix_point<Fn>{std::forward<Fn>(f)};
}
#endif /* !defined(H_make_fix_point) */
#line 16 "algorithm/online_to_offline_optimization.cpp"
template <typename Fn>
auto online_to_offline_optimization(Fn&& f, size_t n, decltype(f(n, n)) init = 0) {
// FIXME: コスト関数 f を渡すんじゃなくて induce を渡す設計にしたいかも。
// SMAWK で解きたいとか、別の性質が使えるとかありそう。
// よさげなインタフェースが思いついたら変更する。
using value_type = decltype(f(n, n));
std::vector<value_type> dp(n, limits<value_type>::max());
dp[0] = init;
auto induce = [&](size_t l, size_t m, size_t r) -> void {
auto g = [&](size_t i, size_t j) -> value_type {
return dp[j+l] + f(j+l, i+m);
};
auto argmin = monotone_minima(g, r-m, m-l);
for (size_t i = m; i < r; ++i) {
size_t j = argmin[i-m] + l;
dp[i] = std::min(dp[i], g(i-m, j-l));
}
};
make_fix_point([&](auto& solve, size_t l, size_t r) -> void {
if (l+1 == r) return;
if (l+2 == r) {
if (r <= n) dp[l+1] = std::min(dp[l+1], dp[l] + f(l, l+1));
return;
}
size_t m = (l+r) >> 1;
solve(l, m);
induce(l, m, r);
solve(m, r);
})(0, n);
return dp;
}
#line 9 "test/yc_703_onoff.test.cpp"
int main() {
size_t n;
scanf("%zu", &n);
std::vector<int> a(n), x(n), y(n);
for (auto& ai: a) scanf("%d", &ai);
for (auto& xi: x) scanf("%d", &xi);
for (auto& yi: y) scanf("%d", &yi);
auto f = [&](size_t j, size_t i) {
intmax_t dx = x[j] - a[i-1];
intmax_t dy = y[j];
return dx*dx + dy*dy;
};
auto dp = online_to_offline_optimization(f, n+1);
printf("%jd\n", dp[n]);
}
test/yc_703_onoff.test.cpp