#include "aliens.h"
#include <bits/stdc++.h>
#define x first
#define y second
#define all(v) v.begin(), v.end()
#define chkmin(a, b) a = min(a, b)
#define chkmax(a, b) a = max(a, b)
//#define int ll
using namespace std;
//typedef __int128_t ll;
typedef long long ll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef pair<ll, ll> pii;
typedef vector<pii> vii;
const int MAX_N = 1e5 + 5;
const ll infinity = 1e18;
struct Rmq {
vi loga;
vvi rmq_min, rmq_max;
Rmq() {}
Rmq(vi &v_min, vi &v_max) {
int n = v_min.size();
loga.resize(n + 1);
for (int i = 2; i <= n; i++) loga[i] = loga[i >> 1] + 1;
rmq_min.resize(loga[n] + 1, vi(n));
rmq_max.resize(loga[n] + 1, vi(n));
rmq_min[0] = v_min, rmq_max[0] = v_max;
for (int i = 1; i <= loga[n]; i++) {
for (int j = 0; j < n; j++) {
rmq_min[i][j] = min(rmq_min[i - 1][j], rmq_min[i - 1][min(j + (1 << (i - 1)), n - 1)]);
rmq_max[i][j] = max(rmq_max[i - 1][j], rmq_max[i - 1][min(j + (1 << (i - 1)), n - 1)]);
}
}
}
inline pii query(int l, int r) {
if (r <= l) return {-1, -1};
int x = min(rmq_min[loga[r - l]][l], rmq_min[loga[r - l]][r - (1 << loga[r - l])]);
int y = max(rmq_max[loga[r - l]][l], rmq_max[loga[r - l]][r - (1 << loga[r - l])]);
return {x, y};
}
};
pii dp[MAX_N];
int opt[MAX_N];
int ind[MAX_N];
int r[MAX_N], c[MAX_N];
Rmq rmq;
bitset<MAX_N> inside;
int n, m;
ll Cost(int j, int i) {
auto [left, right] = rmq.query(j, i + 1);
ll x = ll(right - left) * (right - left);
int pre = rmq.query(0, j).y;
if (pre > left) {
x -= ll(pre - left) * (pre - left);
}
return x;
}
pii Solve(ll lambda) {
vector<pair<pii, int>> ranges;
ranges.push_back({{0, n}, 0});
dp[0] = {lambda + Cost(0, 0), 1};
for (int i = 1; i < n; i++) {
while (!ranges.empty()) {
if (ranges.back().x.y <= i) {
ranges.clear();
break;
}
int index = max<int>(ranges.back().x.x, i);
int op = ranges.back().y;
ll x1 = (i ? dp[i - 1].x : 0) + Cost(i, index) + lambda;
ll cnt1 = i ? dp[i - 1].y : 0;
ll x2 = (op ? dp[op - 1].x : 0) + Cost(op, index) + lambda;
ll cnt2 = op ? dp[op - 1].y : 0;
if (pii(x1, cnt1) >= pii(x2, cnt2)) break;
ranges.pop_back();
}
if (ranges.empty()) {
ranges.push_back({{i, n}, i});
} else {
auto [left, right] = ranges.back().x;
chkmax(left, ll(i));
int begin = left, end = right, mid;
int op = ranges.back().y;
while (begin < end) {
mid = (begin + end) >> 1;
ll x1 = (i ? dp[i - 1].x : 0) + Cost(i, mid) + lambda;
ll cnt1 = (i ? dp[i - 1].y : 0) + 1;
ll x2 = (op ? dp[op - 1].x : 0) + Cost(op, mid) + lambda;
ll cnt2 = (op ? dp[op - 1].y : 0) + 1;
if (pii(x1, cnt1) < pii(x2, cnt2)) end = mid;
else begin = mid + 1;
}
ranges.pop_back();
ranges.push_back({{left, end}, op});
if (end != n) ranges.push_back({{end, n}, i});
}
// dp[i] = {infinity, infinity};
// int op = i ? opt[i - 1] : 0;
// for (int l = i; l >= op; l--) {
// ll x = (l ? dp[l - 1].x : 0) + Cost(l, i) + lambda;
// ll cnt = l ? dp[l - 1].y : 0;
// pii p = {x, cnt + 1};
// if (p < dp[i]) {
// dp[i] = p;
// opt[i] = l;
// }
// }
auto it = upper_bound(all(ranges), pair<pii, int>({i, n}, n));
it--;
int l = it->y;
dp[i].x = (l ? dp[l - 1].x : 0) + Cost(l, i) + lambda;
dp[i].y = (l ? dp[l - 1].y : 0) + 1;
// if (opt[i] != l) {
// cout << "";
// }
}
return dp[n - 1];
}
long long take_photos(int32_t N, int32_t M, int32_t k, std::vector<int32_t> R, std::vector<int32_t> C) {
n = N, m = M;
for (int i = 0; i < n; i++) {
r[i] = R[i], c[i] = C[i];
if (r[i] > c[i]) swap(r[i], c[i]);
}
iota(ind, ind + n, 0);
sort(ind, ind + n, [&] (int i, int j) {
return r[i] == r[j] ? c[i] < c[j] : r[i] < r[j];
});
int max_c = 0;
for (int i = 0; i < n; i++) {
chkmax(max_c, c[ind[i]]);
if (max_c > c[ind[i]]) {
inside[ind[i]] = true;
r[ind[i]] = m + 1, c[ind[i]] = m + 1;
}
}
sort(ind, ind + n, [&] (int i, int j) {
return r[i] + c[i] < r[j] + c[j];
});
n -= inside.count();
vi v_min(n), v_max(n);
for (int i = 0; i < n; i++) v_min[i] = r[ind[i]], v_max[i] = c[ind[i]] + 1;
rmq = Rmq(v_min, v_max);
pii p = Solve(0);
if (p.y <= k) return p.x;
ll begin = 0, end = ll(m) * m + 5, mid;
while (begin < end) {
mid = (begin + end) >> 1;
p = Solve(mid);
if (p.y <= k) end = mid;
else begin = mid + 1;
}
return Solve(end).x - k * (end);
}
//6 7 2
//0 3
//4 4
//4 6
//4 5
//4 6
//0 5
//4 10 2
//0 0
//4 4
//5 5
//9 9
