#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
bool cmp(const pair<int, int>& a, const pair<int, int>& b) {
if (a.first != b.first) {
return a.first < b.first;
}
return a.second > b.second;
}
long long divide_up(long long a, long long b) {
if (b < 0) {
a = -a;
b = -b;
}
if (a < 0) {
return a / b;
} else {
return (a + b - 1) / b;
}
}
long long take_photos(int n, int m, int k, vector<int> points_x, vector<int> points_y) {
vector<pair<int, int>> points(n);
for (int i = 0; i < n; i++) {
if (points_x[i] > points_y[i]) {
swap(points_x[i], points_y[i]);
}
points[i] = {points_x[i], points_y[i] + 1};
}
sort(points.begin(), points.end(), cmp);
vector<pair<int, int>> points_fin;
points_fin.push_back({-1, -1});
int max_y = -1;
for (int i = 0; i < n; i++) {
if (max_y < points[i].second) {
max_y = points[i].second;
points_fin.push_back(points[i]);
}
}
points = points_fin;
points_fin.clear();
n = points.size();
vector<long long> l(n), r(n);
for (int i = 0; i < n; i++) {
l[i] = points[i].first;
r[i] = points[i].second;
}
vector<long long> b(n);
for (int i = 0; i < n - 1; i++) {
b[i] = 2 * l[i + 1];
}
k = min(k, n - 1);
vector<long long> dp(n, 2e12);
dp[0] = 0;
for (int p = 0; p < k; p++) {
vector<long long> a(n);
for (int i = 0; i < n - 1; i++) {
a[i] = dp[i] + l[i + 1] * l[i + 1] - max(0ll, r[i] - l[i + 1]) *
max(0ll, r[i] - l[i + 1]);
}
vector<pair<long long, int>> ch; // {x, ind} – начиная с такого r[i] лучше всего переходить из ind
ch.push_back({-1, p});
int cur = 0;
vector<long long> new_dp(n, 2e12);
for (int i = p + 1; i < n; i++) {
while (cur + 1 < ch.size() && ch[cur + 1].first <= r[i]) {
cur++;
}
int j = ch[cur].second;
new_dp[i] = r[i] * r[i] + a[j] - b[j] * r[i];
if (i == n - 1) {
continue;
}
while (ch.size()) {
long long x = ch.back().first;
int j = ch.back().second;
if (a[j] - b[j] * x < a[i] - b[i] * x) {
break;
}
ch.pop_back();
}
if (ch.empty()) {
ch.push_back({-1, i});
} else {
int j = ch.back().second;
ch.push_back({max(divide_up(a[i] - a[j], b[i] - b[j]), -1ll), i});
}
cur = min(cur, int(ch.size()) - 1);
}
dp = new_dp;
}
return dp.back();
}
// dp[p][i] = min_j( dp[p - 1][j] + (r[i] - l[j + 1])^2 - max(0, r[j] - l[j + 1])^2 )
// dp[p][i] = r[i]^2 + min_j( dp[p - 1][j] + l[j + 1]^2 - max(0, r[j] - l[j + 1])^2 - 2 l[j + 1] r[i] )
