#include "aliens.h"
/*
* Author: Nonoze
* Created: Sunday 29/03/2026
*/
#include <bits/stdc++.h>
using namespace std;
#ifndef DEBUG
#define dbg(...)
#endif
// #define cout cerr << "OUT: "
#define endl '\n'
#define endlfl '\n' << flush
#define quit(x) return (void)(cout << x << endl)
template<typename T> void read(T& x) { cin >> x; }
template<typename T1, typename T2> void read(pair<T1, T2>& p) { read(p.first), read(p.second); }
template<typename T> void read(vector<T>& v) { for (auto& x : v) read(x); }
template<typename T1, typename T2> void read(T1& x, T2& y) { read(x), read(y); }
template<typename T1, typename T2, typename T3> void read(T1& x, T2& y, T3& z) { read(x), read(y), read(z); }
template<typename T1, typename T2, typename T3, typename T4> void read(T1& x, T2& y, T3& z, T4& zz) { read(x), read(y), read(z), read(zz); }
template<typename T> void print(vector<T>& v) { for (auto& x : v) cout << x << ' '; cout << endl; }
#define sz(x) (int)(x.size())
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define make_unique(v) sort(all(v)), v.erase(unique(all(v)), (v).end())
#define pb push_back
#define mp(a, b) make_pair(a, b)
#define fi first
#define se second
#define cmin(a, b) a = min(a, b)
#define cmax(a, b) a = max(a, b)
#define YES cout << "YES" << endl
#define NO cout << "NO" << endl
#define QYES quit("YES")
#define QNO quit("NO")
#define int long long
#define double long double
const int inf = numeric_limits<int>::max() / 4;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
const int MOD = 1e9+7, LOG=20;
int intersect(pair<int, int> a, pair<int, int> b) {
if (a.fi>b.fi) swap(a, b);
if (a.se<b.fi) return 0;
return (a.se-b.fi+1)*(a.se-b.fi+1);
}
int get_area(pair<int, int> a) {
return (a.se-a.fi+1)*(a.se-a.fi+1);
}
int get_area(pair<int, int> a, pair<int, int> b) {
return get_area(a)+get_area(b)-intersect(a, b);
}
struct Line {
int m, b;
Line(int _m, int _b) {
m=_m, b=_b;
}
int eval(int x) {
return m*x+b;
}
};
bool bad(Line a, Line b, Line c) {
return (__int128_t)(c.b-a.b)*(a.m-b.m)<(__int128_t)(b.b-a.b)*(a.m-c.m);
}
int take_photos(signed n, signed m, signed k, vector<signed> r, vector<signed> c) {
vector<pair<int, int>> a;
for (int i=0; i<n; i++) a.pb({min(r[i], c[i]), max(r[i], c[i])});
sort(all(a), [](pair<int, int> x, pair<int, int> y) {
if (x.fi==y.fi) return x.se>y.se;
return x.fi<y.fi;
});
{
vector<pair<int, int>> b;
for (auto& p: a) {
if (b.empty() || b.back().se<p.se) b.pb(p);
}
swap(a, b);
}
n=sz(a);
int tot=0;
for (auto& p: a) tot+=get_area(p);
for (int i=0; i<n-1; i++) tot-=intersect(a[i], a[i+1]);
if (n<=k) return tot;
set<int> neww;
for (int i=1; i<n; i++) {
if (a[i].fi>a[i-1].se+1) a.pb({a[i-1].se+1, a[i].fi-1}), neww.insert({a[i-1].se+1});
}
sort(all(a)); n=sz(a);
vector<vector<int>> dp(n+1, vector<int>(k, 0));
vector<bool> none(n);
vector<int> pref(n); for (int i=1; i<n; i++) {
if (neww.count(a[i].fi)) none[i]=1;
pref[i]=pref[i-1]+a[i].fi*(a[i].se-a[i-1].se)*2 + (none[i]?(a[i].se-a[i].fi+1)*(a[i].se-a[i].fi+1):0);
}
for (int i=n-1; i>=0; i--) {
if (none[i]) {
dp[i][k-1]=dp[i+1][k-1];
continue;
}
dp[i][k-1]=pref[n-1]-pref[i]+a[i].fi*(a[i].se-a[n-1].se)*2;
}
for (int j=k-2; j>=0; j--) {
deque<Line> hull;
for (int i=n-1; i>=0; i--) {
if (none[i]) {
dp[i][j]=dp[i+1][j];
continue;
}
// add line: y=dp[end+1][j+1]+pref[end], x=-a[end].se*2
Line cur(-a[i].se*2, dp[i+1][j+1]+pref[i]);
while (sz(hull)>=2 && bad(hull[sz(hull)-2], hull.back(), cur)) hull.pop_back();
hull.pb(cur);
// best end: dp[end+1][j+1]+pref[end]-a[i].fi*a[end].se*2 - pref[i] + a[i].fi*a[i].se*2
while (sz(hull)>1 && hull.front().eval(a[i].fi)>=hull[1].eval(a[i].fi)) hull.pop_front();
if (!hull.empty()) dp[i][j]=hull.front().eval(a[i].fi) - pref[i] + a[i].fi*a[i].se*2;
else dp[i][j]=0;
}
}
return dp[0][0]+tot;
}
