#include "aliens.h"
#include <vector>
#include <algorithm>
#include <iostream>
#include <set>
#include <cassert>
using namespace std;
long long squared(int x) {
return 1ll * x * x;
}
template<typename T>
struct line_elem {
T a;
T b;
T x_start;
bool point_query;
line_elem() : a(0), b(0), x_start(0), point_query(false) {}
line_elem(T x) : a(0), b(0), x_start(x), point_query(true) {}
line_elem(T a, T b, T x_start = 0) : a(a), b(b), x_start(x_start), point_query(false) {}
T get(T x) const {
return a * x + b;
}
bool operator<(const line_elem& rhs) const {
if (point_query || rhs.point_query) {
return x_start < rhs.x_start;
}
return make_pair(a, b) < make_pair(rhs.a, rhs.b);
}
bool operator<(const pair<T, T>& rhs) const {
return make_pair(a, b) < rhs;
}
bool operator<(T x) const {
return x_start < x;
}
};
template<typename T>
T cross_x(const line_elem<T>& X, const line_elem<T>& Y) {
/* Returns the first x >= to when the lines intersect */
T da = X.a - Y.a;
T db = Y.b - X.b;
assert(da != 0);
if (da < 0) {
da = -da;
db = -db;
}
if (db < 0) {
return db / da;
}
return (db + da - 1) / da;
}
template<typename T>
struct line_hull {
/* Implements a max hull, get the max ax+b for all inserted lines. */
bool insert(T a, T b) {
/* Insert line ax+b */
auto ln = line_elem<T>(a, b);
auto it = st.lower_bound(ln);
bool is_last = it == st.end();
if (!is_last && ln.a == it->a) {
return false;
}
auto jt = it;
bool is_first = it == st.begin();
if (!is_first) {
--it;
if (it->a == ln.a) {
if (it == st.begin()) {
is_first = true;
} else {
--it;
}
}
}
T lft_x = is_first ? numeric_limits<T>::min() : cross_x(*it, ln);
T rht_x = is_last ? numeric_limits<T>::max() : cross_x(ln, *jt);
if (rht_x <= lft_x) {
return false;
}
if (!is_first) {
while (it != st.begin()) {
auto kt = it--;
T prev_lft_x = cross_x(*it, *kt);
if (prev_lft_x < lft_x) {
++it;
break;
}
lft_x = cross_x(*it, ln);
}
++it;
}
if (!is_last) {
while (true) {
auto kt = jt++;
if (jt == st.end()) {
break;
}
T prev_rht_x = cross_x(*kt, *jt);
if (rht_x < prev_rht_x) {
break;
}
rht_x = cross_x(ln, *jt);
}
--jt;
}
T mx = numeric_limits<T>::max();
line_elem<T> new_rht;
if (!is_last) {
new_rht = *jt++;
}
st.erase(it, jt);
st.insert(line_elem<T>(a, b, lft_x));
if (!is_last) {
new_rht.x_start = rht_x;
st.insert(new_rht);
}
return true;
}
T get(T x) const {
auto it = st.upper_bound(x);
--it;
return it->get(x);
}
void debug() {
cout << "hull:" << endl;
for (auto i : st) {
cout << i.a << ", " << i.b << ", " << i.x_start << endl;
}
}
set<line_elem<T> > st;
};
long long take_photos(int N, int M, int K, vector<int> R, vector<int> C) {
vector<pair<int, int> > A;
for (int i = 0; i < N; i++) {
int ri = R[i];
int ci = C[i];
if (ri > ci) swap(ri, ci);
A.push_back(make_pair(ci, ci - ri));
}
sort(A.begin(), A.end());
int j = 0;
for (auto p : A) {
while (j > 0 && p.second >= A[j - 1].second + p.first - A[j - 1].first) {
--j;
}
A[j++] = p;
}
A.resize(j);
N = j;
vector<long long> ADJ(N + 1, 0);
for (int i = 1; i < N; i++) {
ADJ[i] = squared(max(0, 1 + A[i].second - A[i].first + A[i - 1].first));
}
long long INF = squared(M + 1);
vector<long long> DP(N + 1, INF);
vector<long long> DP2(N + 1, INF);
DP[0] = 0;
for (int k = 0; k < K; k++) {
line_hull<long long> hl;
//dumb_hull hl;
for (int i = 0; i <= N; i++) {
if (i) {
long long x = A[i - 1].first;
DP[i] = -hl.get(x) + squared(x);
}
/*
for (int j = 0; j < i; j++) {
DP[i] = min(
DP[i],
DP[j] + squared(1 + A[j].second + A[i - 1].first - A[j].first) - ADJ[j]
);
}
*/
// x^2 + bx + c where x = A[i - 1].first
long long sadd = 1 + A[i].second - A[i].first;
long long b = 2 * sadd;
long long c = squared(sadd) + DP[i] - ADJ[i];
hl.insert(-b, -c);
}
DP.swap(DP2);
}
return DP[N];
}
