#include "aliens.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef double db;
typedef string str;
typedef pair<int,int> pi;
typedef pair<ll,ll> pl;
typedef pair<db,db> pd;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<db> vd;
typedef vector<str> vs;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<pd> vpd;
#define mp make_pair
#define f first
#define s second
#define sz(x) (int)x.size()
#define all(x) begin(x), end(x)
#define rall(x) (x).rbegin(), (x).rend()
#define rsz resize
#define ins insert
#define ft front()
#define bk back()
#define pf push_front
#define pb push_back
#define eb emplace_back
#define lb lower_bound
#define ub upper_bound
#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define trav(a,x) for (auto& a: x)
const int MOD = 1e9+7; // 998244353;
const int MX = 2e5+5;
const ll INF = 1e18;
const ld PI = acos((ld)-1);
const int xd[4] = {1,0,-1,0}, yd[4] = {0,1,0,-1};
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());
template<class T> bool ckmin(T& a, const T& b) {
return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) {
return a < b ? a = b, 1 : 0; }
int pct(int x) { return __builtin_popcount(x); }
int bits(int x) { return 31-__builtin_clz(x); } // floor(log2(x))
int cdiv(int a, int b) { return a/b+!(a<0||a%b == 0); } // division of a by b rounded up, assumes b > 0
int fstTrue(function<bool(int)> f, int lo, int hi) {
hi ++; assert(lo <= hi); // assuming f is increasing
while (lo < hi) { // find first index such that f is true
int mid = (lo+hi)/2;
f(mid) ? hi = mid : lo = mid+1;
}
return lo;
}
// INPUT
template<class A> void re(complex<A>& c);
template<class A, class B> void re(pair<A,B>& p);
template<class A> void re(vector<A>& v);
template<class A, size_t SZ> void re(array<A,SZ>& a);
template<class T> void re(T& x) { cin >> x; }
void re(db& d) { str t; re(t); d = stod(t); }
void re(ld& d) { str t; re(t); d = stold(t); }
template<class H, class... T> void re(H& h, T&... t) { re(h); re(t...); }
template<class A> void re(complex<A>& c) { A a,b; re(a,b); c = {a,b}; }
template<class A, class B> void re(pair<A,B>& p) { re(p.f,p.s); }
template<class A> void re(vector<A>& x) { trav(a,x) re(a); }
template<class A, size_t SZ> void re(array<A,SZ>& x) { trav(a,x) re(a); }
// TO_STRING
#define ts to_string
str ts(char c) { return str(1,c); }
str ts(bool b) { return b ? "true" : "false"; }
str ts(const char* s) { return (str)s; }
str ts(str s) { return s; }
template<class A> str ts(complex<A> c) {
stringstream ss; ss << c; return ss.str(); }
str ts(vector<bool> v) {
str res = "{"; F0R(i,sz(v)) res += char('0'+v[i]);
res += "}"; return res; }
template<size_t SZ> str ts(bitset<SZ> b) {
str res = ""; F0R(i,SZ) res += char('0'+b[i]);
return res; }
template<class A, class B> str ts(pair<A,B> p);
template<class T> str ts(T v) { // containers with begin(), end()
bool fst = 1; str res = "{";
for (const auto& x: v) {
if (!fst) res += ", ";
fst = 0; res += ts(x);
}
res += "}"; return res;
}
template<class A, class B> str ts(pair<A,B> p) {
return "("+ts(p.f)+", "+ts(p.s)+")"; }
// OUTPUT
template<class A> void pr(A x) { cout << ts(x); }
template<class H, class... T> void pr(const H& h, const T&... t) {
pr(h); pr(t...); }
void ps() { pr("\n"); } // print w/ spaces
template<class H, class... T> void ps(const H& h, const T&... t) {
pr(h); if (sizeof...(t)) pr(" "); ps(t...); }
// DEBUG
void DBG() { cerr << "]" << endl; }
template<class H, class... T> void DBG(H h, T... t) {
cerr << ts(h); if (sizeof...(t)) cerr << ", ";
DBG(t...); }
#ifdef LOCAL // compile with -DLOCAL
#define dbg(...) cerr << "LINE(" << __LINE__ << ") -> [" << #__VA_ARGS__ << "]: [", DBG(__VA_ARGS__)
#else
#define dbg(...) 0
#endif
// FILE I/O
void setIn(string s) { freopen(s.c_str(),"r",stdin); }
void setOut(string s) { freopen(s.c_str(),"w",stdout); }
void unsyncIO() { ios_base::sync_with_stdio(0); cin.tie(0); }
void setIO(string s = "") {
unsyncIO();
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) { setIn(s+".in"), setOut(s+".out"); } // for USACO
}
/**
* Description: LineContainer; add lines of the form $kx+m$,
* compute greatest $y$-coordinate for any $x$.
* Time: O(\log N)
* Source: KACTL
* https://github.com/kth-competitive-programming/kactl/commit/165807e28402c9be906f6e6a09452431787bb70d?diff=unified
* Verification:
* CSA Squared Ends not working :(
* https://codeforces.com/contest/1083/problem/E
* https://atcoder.jp/contests/arc066/tasks/arc066_d
*/
bool Q;
struct Line {
mutable ll k, m, p, lnum; // slope, y-intercept, last optimal x, LAMBDAS used
pair<ll, int> eval (ll x) { return mp(k*x+m, lnum); }
bool operator<(const Line& o) const { return Q?p<o.p:k<o.k; }
};
// for doubles, use inf = 1/.0, divi(a,b) = a/b
const ll inf = LLONG_MAX;
// floored div
ll divi(ll a, ll b) { return a/b-((a^b) < 0 && a%b); }
// last x such that first line is better
ll bet(const Line& x, const Line& y) {
if (x.k == y.k){
if(x.m > y.m) return inf;
if(x.m < y.m) return -inf;
if(x.lnum <= y.lnum) return inf;
return -inf;
}
if((y.m-x.m) % (x.k-y.k) == 0){
if(x.lnum <= y.lnum){
return (y.m-x.m)/(x.k-y.k)-1;
}
return (y.m-x.m)/(x.k-y.k);
}
return divi(y.m-x.m,x.k-y.k); }
/**
* Description: LineContainer assuming both slopes and queries monotonic.
* Time: O(1)
* Source: Own
* Verification: http://acm.hdu.edu.cn/diy/contest_showproblem.php?cid=36005&pid=1009
*/
struct LCdeque : deque<Line> {
void addBack(Line L) { // assume nonempty
while (1) {
auto a = bk; pop_back(); a.p = bet(a,L);
if (size() && bk.p >= a.p) continue;
pb(a); break;
}
L.p = inf; pb(L);
}
void addFront(Line L) {
while (1) {
if (!size()) { L.p = inf; break; }
if ((L.p = bet(L,ft)) >= ft.p) pop_front();
else break;
}
push_front(L);
}
void add(ll k, ll m, ll lnum) { // line goes to one end of deque
k = -k;
m = -m;
lnum = -lnum;
if (!size() || k <= ft.k) addFront({k,m,0,lnum});
else assert(k >= bk.k), addBack({k,m,0,lnum});
}
int ord = 1; // 1 = increasing, -1 = decreasing
pair<ll, int> query(ll x) {
assert(ord);
pair<ll, int> rval;
if (ord == 1) {
while (ft.p < x) pop_front();
rval = ft.eval(x);
} else {
while(size()>1&&prev(prev(end()))->p>=x)pop_back();
rval = bk.eval(x);
}
rval.f = -rval.f;
rval.s = -rval.s;
return rval;
}
};
vpl points;
pair<ll, int> dp[100005];
LCdeque lins;
pair<ll, int> testLambda(ll lamb){
//clear memory
for(int i = 0; i < sz(points); i++){
dp[i] = mp(INF, -1);
}
lins.clear();
lins.add(2*(-points[0].f+1), (-points[0].f+1)*(-points[0].f+1)+lamb, 1);
for(int i = 0; i < sz(points); i++){
//query to get dp[i]
dp[i] = lins.query(points[i].s);
dp[i].f+=ll(points[i].s)*points[i].s;
//add to deque if it's not sz(points)-1
if(i == sz(points)-1) break;
if(points[i].s < points[i+1].f){
lins.add(2*(-points[i+1].f+1), dp[i].f+(-points[i+1].f+1)*(-points[i+1].f+1)+lamb, dp[i].s+1);
}
else{
lins.add(2*(-points[i+1].f+1), dp[i].f+(-points[i+1].f+1)*(-points[i+1].f+1)-(points[i].s-points[i+1].f+1)*(points[i].s-points[i+1].f+1)+lamb, dp[i].s+1);
}
}
//ps(lamb, dp[sz(points)-1]);
return dp[sz(points)-1];
}
ll take_photos(int n, int m, int k, vi r, vi c) {
for(int i = 0; i < n; i++){
if(r[i] > c[i]) swap(r[i], c[i]);
}
vector<pair<pi, int>> sortorig;
for(int i = 0; i < n; i++){
sortorig.pb(mp(mp(r[i], c[i]), i));
}
sort(all(sortorig));
//ps(sortorig);
int curr = -1;
for(int i = 0; i < sz(sortorig); i++){
pair<pi, int> u = sortorig[i];
if(i+1 < sz(sortorig) && sortorig[i].f.f == sortorig[i+1].f.f) continue;
if(u.f.s <= curr) continue;
curr = u.f.s;
points.pb(mp(u.f.f, u.f.s));
}
//ps(points);
ll lo = 0;
ll hi = 1000000000005LL;
while(lo < hi){
ll mid = (lo+hi)/2;
pair<ll, int> tans = testLambda(mid);
if(tans.s <= k){
hi = mid;
}
else{
lo = mid+1;
}
}
pair<ll, int> fans = testLambda(lo);
ll ans = fans.f;
ans-=lo*ll(k);
return ans;
}
