#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<ll, ll> pll;
#define all(x) (x).begin(), (x).end()
#define X first
#define Y second
#define sep ' '
#define debug(x) cerr << #x << ": " << x << endl;
const ll MAXN = 1e6 + 10;
const ll INF = 1e18;
int n, m, k;
pll dp[MAXN];
ll f[MAXN];
ll cov[MAXN];
inline ll pw(ll a) {
return a * a;
}
namespace CHT {
vector<pair<pll, int>> lines;
vector<pll> vec;
inline ll isect(pll a, pll b) {
if (a.X > b.X) swap(a, b);
return (a.Y - b.Y + (a.Y <= b.Y ? 0 : 1) * (b.X - a.X - 1)) / (b.X - a.X);
} // TODO check
inline void clear() {
vec.clear();
lines.clear();
}
inline void add(pair<pll, int> line) {
lines.push_back(line);
while (!vec.empty()) {
ll ind = isect(lines[vec.back().Y].X, line.X);
ll a = lines[vec.back().Y].X.X * ind + lines[vec.back().Y].X.Y;
ll b = line.X.X * ind + line.X.Y;
if (ind < vec.back().X) {
vec.pop_back();
continue;
}
if (a == b && lines[vec.back().Y].Y < line.Y) ind++;
vec.push_back({ind, int(lines.size()) - 1});
break;
}
if (vec.empty())
vec.push_back({-INF, int(lines.size()) - 1});
}
inline int get(ll x) {
auto it = lower_bound(all(vec), pll(x + 1, -INF));
return prev(it) -> Y;
}
}
inline pll calc(ll lambda) {
CHT::clear();
vector<int> opts;
ll mx_f = -1;
for (int i = 0; i < m; i++) {
if (f[i] > mx_f) {
mx_f = f[i];
opts.push_back(i);
}
}
int ind = 0;
for (int e : opts) {
int len = f[e] - e + 1;
dp[ind] = {pw(len + e - opts.front()) + lambda, 1};
if (!(e == opts.front())) {
ll x = (len + e);
int ii = CHT::get(x);
int tlen = len + (e - opts[ii + 1]);
int mn_cov = opts[ii + 1];
pll tmp = {dp[ii].X + pw(tlen) - cov[ii] + lambda, dp[ii].Y + 1};
dp[ind] = min(dp[ind], tmp);
}
if (ind < int(opts.size()) - 1) {
cov[ind] = (f[e] >= opts[ind + 1] ? pw(f[e] - opts[ind + 1] + 1) : 0);
CHT::add({{-2 * opts[ind + 1], 1ll * opts[ind + 1] * opts[ind + 1] + dp[ind].X - cov[ind]}, int(dp[ind].Y)});
}
ind++;
}
return dp[int(opts.size()) - 1];
}
ll take_photos(int n_, int m_, int k_, vector<int> r_, vector<int> c_) {
fill(f, f + MAXN, -1);
n = n_, m = m_, k = k_;
for (int i = 0; i < n; i++) {
int x = r_[i], y = c_[i];
if (y < x) swap(x, y); // TODO: check
f[x] = max(f[x], ll(y));
}
/*
CHT::add({{1, 1}, 0});
CHT::add({{0, 2}, 0});
debug(CHT::get(-1));
*/
ll l = 0, r = INF / n;
while (l < r) {
ll mid = (l + r) / 2;
if (calc(mid).Y > k) l = mid + 1;
else r = mid;
}
return calc(l).X - k * l;
}
