#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef vector<ll> vll;
typedef vector<vll> vvll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef pair<ll, ll> pll;
typedef vector<pll> vpll;
typedef vector<vpll> vvpll;
typedef vector<bool> vb;
#define INF(dt) numeric_limits<dt>::max()
#define fi first
#define se second
struct Line {
int m;
ll b;
int b2;
inline pair<ll, int> evaluate(ll x) const {
return {m * x + b, b2};
}
};
/*
Suppose the penalty for using another picture is +y
*/
pll solve_lambda_brute(const vpll& pts, ll m, ll y) {
ll np = pts.size();
// note: stores NEGATIVE area, num pictures
// then, the DP just gets the maximum state
vector<pair<ll, int>> dp(np, {0ll, 0});
deque<Line> ls;
for(ll i = 0ll; i < np; i++) {
ll cl = pts[i].fi, cr = pts[i].se;
Line nl;
if(i == 0ll) {
nl = {- ((int)(pts[i].fi) << 1), y + pts[i].fi * pts[i].fi, -1};
} else {
ll overlap = max(pts[i - 1ll].se - pts[i].fi + 1ll, 0ll);
nl = {- ((int)(pts[i].fi) << 1), dp[i - 1ll].fi - overlap * overlap + y + pts[i].fi * pts[i].fi, dp[i - 1ll].se - 1};
}
while((ll)(ls.size()) >= 2ll) {
Line front = ls.front();
ls.pop_front();
if(front.evaluate(cr + 1ll) < ls.front().evaluate(cr + 1ll)) {
ls.push_front(front);
break;
}
}
while((ll)(ls.size()) >= 2ll) {
Line l1 = ls.back();
ls.pop_back();
Line l2 = ls.back();
/*
If the intersection point is at or on the right of the new line,
delete the back
l1.m x + l1.b = l2.m x + l2.b
x = (l2.b - l1.b) / (l1.m - l2.m)
y >= nl.m * x + nl.b
ineq flips bc l1.m < l2.m
l1.m * (l2.b - l1.b) + l1.b * (l1.m - l2.m) <= nl.m * (l2.b - l1.b) + nl.b * (l1.m - l2.m)
*/
ll l = l1.m * (l2.b - l1.b) + l1.b * (l1.m - l2.m), r = nl.m * (l2.b - l1.b) + nl.b * (l1.m - l2.m);
if(!(l < r || (l == r && l1.b2 >= nl.b2))) {
ls.push_back(l1);
break;
}
}
if((ll)(ls.size()) > 0ll && ls.front().evaluate(cr + 1ll) >= nl.evaluate(cr + 1ll)) ls.pop_back();
ls.push_back(nl);
pair<ll, int> opt = ls.front().evaluate(cr + 1ll);
dp[i] = {opt.fi + (cr + 1ll) * (cr + 1ll), opt.se};
// cerr << y << " " << i << " " << dp[i].fi << " " << -dp[i].se << endl;
}
return dp[np - 1ll];
}
ll take_photos(int i_n, int i_m, int i_k, vi r, vi c) {
ll n = i_n, m = i_m, k = i_k;
// first, we preprocess. remove all redundant points
vpll pts;
for(ll i = 0ll; i < n; i++) {
if(r[i] > c[i]) swap(r[i], c[i]);
pts.push_back({r[i], c[i]});
}
sort(pts.begin(), pts.end(), [](pll p1, pll p2){return p1.fi < p2.fi || (p1.fi == p2.fi && p1.se > p2.se);});
ll max_r = -1ll;
vpll filt_pts;
for(ll i = 0ll; i < n; i++) {
ll cl = pts[i].fi, cr = pts[i].se;
if(cr > max_r) {
filt_pts.push_back(pts[i]);
// cerr << cl << " " << cr << endl;
max_r = cr;
}
}
// Performing lagrangian relaxation
ll lo = 0ll, hi = 1e11;
while(hi - lo > 1ll) {
ll mid = (lo + hi) >> 1ll;
pll cpair = solve_lambda_brute(filt_pts, m, mid);
ll c_k = -cpair.se;
if(c_k >= k) lo = mid;
else hi = mid;
}
return solve_lambda_brute(filt_pts, m, lo).fi - k * lo;
}
