#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
#define int long long
#define _min(x,y) x = min(x,y)
const int INF = 1e18;
const int N = 1e5+4;
int dp[N];
int sm[N];
int us[N];
const int M = 1e6+4;
struct line {
int a=0,b=0, idx = 0;
int f(int x) {
return a*x + b;
}
};
struct chtr{
line ln;
int l,r;
};
int intersect(line &a, line &b){
return (b.b-a.b) / (a.a-b.a);
}
vector<chtr> cht;
int idx = 0;
void add(line ln){
while (cht.size()){
cht.back().r = M-1;
auto [ls,l,r] = cht.back();
int x = intersect(ls, ln);
if (x < l){
cht.pop_back(); continue;
}
if (x>r) return;
cht.back().r = x;
cht.push_back({ln, x+1, r});
break;
}
if (cht.empty()) cht.push_back({ln, 0, M-1});
}
pair<int,int> query(int pt){
int rt = cht.size();
if (idx >= rt) idx = rt - 1;
while (cht[idx].r < pt) idx++;
return {cht[idx].ln.f(pt), cht[idx].ln.idx};
}
int do_dp(vector<array<int,3>> &rgs, int lambda){
cht.clear();
idx = 0;
for (int i=0; i<N; i++) dp[i] = INF; dp[0] = 0;
for (int i=0; i<N; i++) us[i] = 0;
for (int i=1; i<rgs.size(); i++){
// add new ln
line v;
v.b = 0; v.a = 0;
v.b += dp[i-1];
v.b += (rgs[i][0] - 1) * (rgs[i][0] - 1);
v.b -= sm[i];
v.a += 2 * (1 - rgs[i][0]);
v.b += lambda;
v.idx = i;
add(v);
//update ans
int area = rgs[i][1] - rgs[i][0] + 1; area *= area;
auto [mn, idx] = query(rgs[i][1]);
dp[i] = mn + rgs[i][1] * rgs[i][1] + sm[i] - area;
us[i] = us[idx-1] + 1;
}
return us[rgs.size()-1];
}
long long take_photos(signed n, signed m, signed k, std::vector<signed> rw, std::vector<signed> cl) {
vector<array<int,3>> rgs(n);
for (int i=0; i<n; i++){
int l = min(rw[i], cl[i]) + 1; int r = max(rw[i], cl[i]) + 1;
rgs[i] = {l, r, 0};
}
sort(rgs.begin(), rgs.end(), [](array<int,3> &a, array<int,3> &b){
if (a[0] < b[0]) return true;
if (a[0] > b[0]) return false;
return a[1] > b[1];
});
int lastr = -1; vector<array<int,3>> nrgs = {{0,0,0}};
for (int i=0; i<n; i++){
int l = rgs[i][0]; int r = rgs[i][1];
if (r <= lastr) continue;
int area = (r-l+1)*(r-l+1);
if (l <= lastr) area -= (lastr-l+1)*(lastr-l+1);
lastr = r;
nrgs.push_back({l, r, area});
}
rgs = nrgs;
int totarea = 0;
for (int i=rgs.size()-1; i>=0; i--){
int area = rgs[i][1] - rgs[i][0] + 1; area *= area;
sm[i] = totarea + area;
totarea += rgs[i][2];
}
int l=0, r = 2e15;
while (l<r){
int m = (l+r)/2;
if (do_dp(rgs, m) <= k) r = m; else l = m+1;
}
// int ans = 1e18;
// for (int i=0; i<=m*m+1; i++)
// if (do_dp(rgs, i) <= k){
// ans = min(ans, sm[1] + dp[rgs.size()-1] - i * us[rgs.size()-1]);
// }
//for (int i=0; i<m*m+1; i++) cout<<do_dp(rgs,i)<<' '; cout<<'\n';
do_dp(rgs,l);
// cout<<ans<<'\n';
return sm[1] + dp[rgs.size()-1] - l * k;
}
