#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
constexpr int NMAX = 1e6 + 5;
struct Interval {
int st, dr;
bool operator < (const Interval &other) const {
return (st < other.st || (st == other.st && dr > other.dr));
}
};
int N, M, K;
Interval V[NMAX];
LL Cost (int a, int b) {
return 1LL * (b-a+1) * (b-a+1);
}
void Comprim () {
sort(V+1, V+N+1);
vector <Interval> aux;
for (int i = 1; i <= N; ++ i ) {
if (aux.empty()) {
aux.push_back(V[i]);
continue;
}
if (aux.back().dr < V[i].dr)
aux.push_back(V[i]);
}
N = aux.size();
for (int i = 1; i <= N; ++ i )
V[i] = aux[i-1];
}
/*
LL dp[4005][4005];
void Solve_Dp_Brut () {
for (int j = 1; j <= K; ++ j ) {
for (int i = 1; i <= N; ++ i ) {
dp[i][j] = Cost(V[1].st, V[i].dr);
if (j == 1) continue;
dp[i][j] = min(dp[i][j], dp[i-1][j-1] + Cost(V[i].st, V[i].dr));
for (int c = i-1; c >= 1; -- c ) {
int p = V[c+1].st;
int new_capat = V[c].dr;
dp[i][j] = min(dp[i][j], dp[c][j-1] + Cost(p, V[i].dr) - (p <= new_capat ? Cost(p, new_capat) : 0));
}
}
}
}
*/
/*
[dp[c][j-1] - (p <= new_capat ? Cost(p, new_capat) : 0)] + (V[i].dr - p + 1)^2
V[i].dr ^ 2 - 2 * V[i].dr * (p-1) + (p - 1) ^ 2
*/
pair <LL, int> dp[NMAX];
deque <int> D;
pair <LL, LL> Equation (int ind) {
pair <LL, LL> ans;
int p = V[ind+1].st;
int new_capat = V[ind].dr;
ans.second = dp[ind].first - (p <= new_capat ? Cost(p, new_capat) : 0) + 1LL * (p - 1) * (p - 1);
ans.first = - 2 * (p-1);
return ans;
}
long double Intersect (int i, int j) {
pair <LL, LL> pr = Equation(i);
pair <LL, LL> sec = Equation(j);
return 1.0 * (pr.second - sec.second) / (sec.first - pr.first);
}
pair <LL, int> Check (LL penalty) {
dp[0] = {0, 0};
D.clear();
for (int i = 1; i <= N; ++ i ) {
dp[i] = {Cost(V[1].st, V[i].dr) + penalty, 1};
dp[i] = min(dp[i], {dp[i-1].first + Cost(V[i].st, V[i].dr) + penalty, dp[i-1].second + 1});
while (D.size() > 2 && Intersect(D[0], D[1]) < 1.0 * V[i].dr)
D.pop_front();
if (!D.empty()) {
pair <LL, LL> ec = Equation(D.front());
dp[i] = min(dp[i], {ec.first * V[i].dr + ec.second + penalty + 1LL * V[i].dr * V[i].dr, dp[D.front()].second + 1});
}
if (i == N) break;
while (D.size() > 2 && Intersect(i, D.back()) < Intersect(D.back(), D[D.size()-2]))
D.pop_back();
D.push_back(i);
}
return dp[N];
}
LL Solve_Optim () {
LL st = 0, dr = 1e12;
while (st <= dr) {
LL mij = (st + dr) >> 1;
pair <LL, int> pos = Check(mij);
if (pos.second == K)
return (pos.first - 1LL * K * mij);
if (pos.second < K) {
dr = mij - 1;
}
else st = mij + 1;
}
pair <LL, int> pos = Check(st);
return (pos.first - 1LL * K * st);
}
long long take_photos(int n, int m, int k, std::vector<int> r, std::vector<int> c) {
N = n;
M = m;
K = k;
for (int i = 1; i <= N; ++ i ) {
V[i].st = min(r[i-1], c[i-1]);
V[i].dr = max(r[i-1], c[i-1]);
}
Comprim();
///Solve_Dp();
///return dp[N][K];
return Solve_Optim();
}
