#include "aliens.h"
#include <vector>
#include <algorithm>
#include <iostream>
using namespace std;
/*
Cells (r, c) and (c, r) are equivalent.
(a, b) = (min(r, c), max(r, c))
A square (p, q) covers point (a, b) if and only if p <= a and b <= q
If (a1, b1) and (a2, b2) are points and a1 <= a2 and b2 <= b1, then (a2, b2) is irrelevant.
So, for any (a1, b1) and (a2, b2) a1 < a2 <=> b1 < b2
When checking for overlapping areas, we only have to consider the last square.
Let points be sorted (a[1], b[1]) < (a[2], b[2]) < .... < (a[x], b[x]).
Let dp[i][k] be the minimum area required to cover the first i points using k squares.
dp[0][k] = 0
dp[i][k] = min{dp[j][k-1] + (b[i] - a[j] + 1)^2 - max(0, b[j] - a[j] + 1)^2 | j=1..i-1}
A point (a, b) requires a square of endpoints (a, a) and (b, b) (side = b-a+1)
dp[j][x-1] + sq(b[i]-a[j+1])
dp[j][x-1] + b[i]^2 + a[j+1]^2 - 2*b[i]*a[j+1]
1*[[dp[j][x-1] + a[j+1]^2]] + (-2*b[i])*[[ a[j+1] ]]
*/
vector<int> R, C;
vector<long long> a(1), b(1);
long long sq(long long x)
{
return x*x;
}
const long long INF = 1'000'000'000'001;
long long take_photos(int n, int m, int k, vector<int> r, vector<int> c)
{
// cerr << '\n';
R = r;
C = c;
int I[n];
for(int i = 0; i < n; i++)
I[i] = i;
sort(I, I+n, [] (int x, int y)
{
if(min(R[x], C[x]) == min(R[y], C[y]))
return max(R[x], C[x]) > max(R[y], C[y]);
return min(R[x], C[x]) < min(R[y], C[y]);
});
int maxb = -1;
for(int i:I)
{
// cerr << i << ' ';
if(maxb < max(r[i], c[i]))
{
maxb = max(r[i], c[i]);
a.push_back(min(r[i], c[i]));
b.push_back(max(r[i], c[i]));
}
}
n = a.size() - 1;
// for(int i = 1; i <= n; i++) cerr << a[i] << ' ' << b[i] << '\n';
k = min(k, n);
long long dp[n+1][k+1];
for(int i = 0; i <= n; i++)
for(int j = 0; j <= k; j++)
dp[i][j] = INF;
for(int j = 0; j <= k; j++) dp[0][j] = 0;
for(int i = 1; i <= n; i++) b[i]++;
for(int i = 1; i <= n; i++)
{
dp[i][1] = sq(b[i] - a[1]);
}
for(int x = 2; x <= k; x++)
{
int j1 = 1, j2 = 1;
for(int i = 2; i <= n; i++)
{
while(j1+1 < i && b[j1+1] - a[i] <= 0 && dp[j1][x-1] + sq(b[i]-a[j1+1]) >= dp[j1+1][x-1] + sq(b[i]-a[j1+2]))
j1++;
dp[i][x] = min(dp[i][x], dp[j1][x-1] + sq(b[i]-a[j1+1]));
while(j2+1 < i && dp[j2][x-1] + sq(b[i]-a[j2+1]) >= dp[j2+1][x-1] + sq(b[i] - a[j2+2]))
j2++;
if(b[j2] - a[i] >= 1)
{
dp[i][x] = min(dp[i][x], dp[j2][x-1] + sq(b[i]-a[j2+1]) - sq(b[j2]-a[i]));
}
}
}
/*for(int i = 1; i <= n; i++)
{
// cerr << "i = " << i << ": \n";
dp[i][0] = INF;
dp[i][1] = sq(b[i] - a[1]);
// cerr << dp[i][1] << ' ';
for(int x = 2; x <= k; x++)
{
dp[i][x] = INF;
for(int j = 1; j < i; j++)
{
// cerr << i << ' ' << x << ' ' << j << ' ';
// cerr << dp[j][x-1] << ' ' << sq(b[i] - a[j+1] + 1) << ' ' << sq(max(0LL, b[j] - a[i] + 1)) << '\n';
if(b[j] - a[i] >= 1)
dp[i][x] = min(dp[i][x], dp[j][x-1] + sq(b[i]-a[j+1]) - sq(b[j]-a[i]));
else
dp[i][x] = min(dp[i][x], dp[j][x-1] + sq(b[i]-a[j+1]));
}
// cerr << dp[i][x] << ' ';
}
// cerr << '\n';
}*/
return dp[n][k];
}
