#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
#define forn(i,n) for(int i=0;i<n;++i)
#define pb push_back
#define all(x) x.begin(), x.end()
#define pi pair<int,int>
#define f first
#define s second
using ll = long long;
#define int ll
const int inf=1e18;
int sq(int x) {
return x*x;
}
struct sgt {
int pointer=0; //Keeps track of the best line from previous query
vector<long long> M; //Holds the slopes of the lines in the envelope
vector<long long> B; //Holds the y-intercepts of the lines in the envelope
//Returns true if either line l1 or line l3 is always better than line l2
bool bad(int l1,int l2,int l3)
{
/*
intersection(l1,l2) has x-coordinate (b1-b2)/(m2-m1)
intersection(l1,l3) has x-coordinate (b1-b3)/(m3-m1)
set the former greater than the latter, and cross-multiply to
eliminate division
*/
return (B[l3]-B[l1])*(M[l1]-M[l2])<(B[l2]-B[l1])*(M[l1]-M[l3]);
}
//Adds a new line (with lowest slope) to the structure
void add(long long m,long long b)
{
//First, let's add it to the end
M.push_back(m);
B.push_back(b);
//If the penultimate is now made irrelevant between the antepenultimate
//and the ultimate, remove it. Repeat as many times as necessary
while (M.size()>=3&&bad(M.size()-3,M.size()-2,M.size()-1))
{
M.erase(M.end()-2);
B.erase(B.end()-2);
}
}
//Returns the minimum y-coordinate of any intersection between a given vertical
//line and the lower envelope
long long query(long long x)
{
//If we removed what was the best line for the previous query, then the
//newly inserted line is now the best for that query
if (pointer>=M.size())
pointer=M.size()-1;
//Any better line must be to the right, since query values are
//non-decreasing
while (pointer<M.size()-1&&
M[pointer+1]*x+B[pointer+1]<M[pointer]*x+B[pointer])
pointer++;
return M[pointer]*x+B[pointer];
}
};
#define int ll
int take_photos(int32_t n, int32_t m, int32_t k, vector<int32_t> r, vector<int32_t> c) {
vector<int> a(m,0);
forn(i,n) {
int x=r[i], y=c[i];
if (x<y) {
a[x]=max(a[x],y-x+1);
} else {
a[y]=max(a[y],x-y+1);
}
}
int x=-inf;
forn(i,m) {
if (!a[i]) continue;
if (a[i]+i<=x) a[i]=0;
else x=a[i]+i;
}
int z=0;
forn(i,m) z+=!!a[i];
if (z<=k) {
int ans=0;
int last=-1;
forn(i,m) {
if (!a[i]) continue;
if (last==-1) {
ans+=a[i]*a[i];
last=i;
continue;
}
if (last + a[last] >= i) {
ans += sq(a[i]) - sq(last+a[last]-i);
last=i;
} else {
ans+=a[i]*a[i];
last=i;
}
}
return ans;
}
if (n*k > 1e7) return -1;
vector<sgt> st(k+1);
forn(i,k+1) st[i].add(0,inf);
vector<pi> v;
forn(i,m) if (a[i]) v.pb({i,i+a[i]});
n=v.size();
st[1].add(-2*v[0].f,sq(v[0].f));
vector<int> dp(k+1,inf);
dp[1]=sq(v[0].s - v[0].f);
for (int i=1; i<n; ++i) {
for (int j=min(1ll*k,i+1); j>0; --j) {
int f=st[j].query(v[i].s) + sq(v[i].s);
int s=dp[j-1] + sq(v[i].s-v[i].f);
if (v[i-1].s > v[i].f) s-=sq(v[i-1].s-v[i].f);
dp[j]=min(f,s);
st[j].add(-2*v[i].f,sq(v[i].f)+s-sq(v[i].s-v[i].f));
}
}
return dp[k];
}
#undef int
