#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<ll, ll> pii;
template<class T> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; }
#define sz(x) (int)x.size()
const int MAXN = 2e5+5;
const ll INF = 1e18;
void __print(int x) {cerr << x;}
void __print(long x) {cerr << x;}
void __print(long long x) {cerr << x;}
void __print(unsigned x) {cerr << x;}
void __print(unsigned long x) {cerr << x;}
void __print(unsigned long long x) {cerr << x;}
void __print(float x) {cerr << x;}
void __print(double x) {cerr << x;}
void __print(long double x) {cerr << x;}
void __print(char x) {cerr << '\'' << x << '\'';}
void __print(const char *x) {cerr << '\"' << x << '\"';}
void __print(const string &x) {cerr << '\"' << x << '\"';}
void __print(bool x) {cerr << (x ? "true" : "false");}
template<typename T, typename V>
void __print(const pair<T, V> &x);
template<typename T>
void __print(const T &x) {int f = 0; cerr << '{'; for (auto &i: x) cerr << (f++ ? ", " : ""), __print(i); cerr << "}";}
template<typename T, typename V>
void __print(const pair<T, V> &x) {cerr << '{'; __print(x.first); cerr << ", "; __print(x.second); cerr << '}';}
void _print() {cerr << "]\n";}
template <typename T, typename... V>
void _print(T t, V... v) {__print(t); if (sizeof...(v)) cerr << ", "; _print(v...);}
#ifdef DEBUG
#define dbg(x...) cerr << "\e[91m"<<__func__<<":"<<__LINE__<<" [" << #x << "] = ["; _print(x); cerr << "\e[39m" << endl;
#else
#define dbg(x...)
#endif
struct Point{
ll r, c, id, cnt;
bool operator<(const Point& o) const{
if (r != o.r){
return r < o.r;
}
return c < o.c;
}
};
bool byr(const Point& a, const Point& b){
return a.r != b.r ? a.r < b.r : a.c > b.c; // sort by smaller row, then by greater col
}
bool byc(const Point& a, const Point& b){
return a.c != b.c ? a.c < b.c : a.r > b.r; // smaller col, smaller row
}
ll sq(ll x){
return x*x;
}
ll get(pii a, int c){
return a.first * c + a.second;
}
double intersectX(pii a, pii b){
return (double)(b.second - a.second)/(a.first - b.first); // (b2 - b1)/(m1 - m2)
}
/*
ll subtask5(int n, int k, vector<Point>& arr){
vector<vector<ll>> dp(n+1, vector<ll>(k+1, INF)); // dp[i][j] = max cover up to ith point after taking j pictures
for (int i=0; i<=k; i++) dp[0][i] = 0;
vector<deque<pii>> lines(k+1);
vector<int> pj(k+1, 1);
for (int i=1; i<=n; i++){
for (int j=1; j<=k; j++){
ll overlap = i > 1 && arr[i-2].c >= arr[i-1].r ? sq(arr[i-2].c - arr[i-1].r + 1) : 0;
pii cur = {-2*(arr[i-1].r - 1), sq(arr[i-1].r - 1) + dp[i-1][j-1] - overlap};
while (sz(lines[j]) > 1 && intersectX(cur, lines[j][sz(lines[j])-2]) < intersectX(lines[j].back(), lines[j][sz(lines[j])-2])){
lines[j].pop_back();
}
lines[j].push_back(cur);
if (j > 1 && !lines[j].empty()){
while (sz(lines[j]) > 1 && get(lines[j].front(), arr[i-1].c) >= get(lines[j][1], arr[i-1].c)){
lines[j].pop_front();
}
dp[i][j] = sq(arr[i-1].c) + get(lines[j].front(), arr[i-1].c);
}else{
// take all points before in 1 single picture
dp[i][j] = sq(arr[i-1].c - arr[0].r + 1);
}
// for (int ip=1; ip<=i; ip++){
// ll area = sq(arr[i-1].c - arr[ip-1].r + 1);
// overlap = ip > 1 && arr[ip-2].c >= arr[ip-1].r ? sq(arr[ip-2].c - arr[ip-1].r + 1) : 0;
// if (dp[i][j] > dp[ip-1][j-1] + area - overlap){
// dbg(dp[i][j]);
// }
// }
}
}
// for (int i=0; i<=n; i++){
// for (int j=0; j<=k; j++){
// cout << dp[i][j] << " ";
// }
// cout << "\n";
// }
}
*/
pii solve(ll penalty, vector<Point>& arr, int n, int k){
vector<pii> dp(n+1); // dp[i] = min area needed with penalty to cover the first i pictures
deque<pair<pii, ll>> lines;
for (int i=1; i<=n; i++){
ll overlap = i > 1 && arr[i-2].c >= arr[i-1].r ? sq(arr[i-2].c - arr[i-1].r + 1) : 0;
pii cur = {-2*(arr[i-1].r - 1), sq(arr[i-1].r - 1) + dp[i-1].first - overlap + penalty};
while (sz(lines) > 1 && intersectX(cur, lines[sz(lines)-2].first) < intersectX(lines.back().first, lines[sz(lines)-2].first)){
lines.pop_back();
}
lines.push_back({cur, dp[i-1].second + 1});
while (sz(lines) > 1 && get(lines.front().first, arr[i-1].c) >= get(lines[1].first, arr[i-1].c)){
lines.pop_front();
}
dp[i] = {sq(arr[i-1].c) + get(lines.front().first, arr[i-1].c), lines.front().second};
}
return dp[n];
}
long long take_photos(int n, int m, int k, std::vector<int> r, std::vector<int> c) {
// reflect the points under the diag
vector<Point> arr;
int tot = 0;
set<Point> allPoints;
for (int i=0; i<n; i++){
if (r[i] > c[i]){
swap(r[i], c[i]);
}
Point cur = {r[i], c[i], tot, 1};
auto it = allPoints.find(cur);
if (it != allPoints.end()){
arr[(*it).id].cnt++;
}else{
arr.push_back(cur);
allPoints.insert(cur);
tot++;
}
}
n = tot;
sort(arr.begin(), arr.end(), byr);
set<Point, decltype(&byc)> seen(&byc);
for (int i=0; i<n; i++){
auto it = seen.lower_bound(arr[i]);
if (it == seen.end()){
seen.insert(arr[i]);
}else{
// Point upd = *it;
// upd.cnt++;
// seen.erase(it);
// seen.insert(upd);
}
}
arr.clear();
for (Point x : seen){
// cout << x.r << " " << x.c << " " << x.id << " " << x.cnt << "\n";
arr.push_back(x);
}
n = sz(arr);
ll low = 0, high = 1e12;
while (high - low > 1){
ll mid = (low + high)/2;
if (solve(mid, arr, n, k).second < k){
// we can lower the price/penalty per photo
high = mid;
}else{
low = mid;
}
}
// run dp
pii ans = solve(low, arr, n, k);
return ans.first - k * low;
}
