// Convex Hull Trick - General Container
/* documents
https://usaco.guide/adv/line-container?lang=cpp
https://usaco.guide/plat/convex-hull-trick
https://cp-algorithms.com/geometry/convex_hull_trick.html
https://wiki.vnoi.info/algo/dp/cht
*/
#include <bits/stdc++.h>
using namespace std;
#define all(v) v.begin(),v.end()
#define pb push_back
#define ll long long
#define ld long double
#define fi first
#define se second
#define mk make_pair
typedef pair<int,int> pii;
const int maxn = 1e5 + 10;
// Code Convex Hull Trick - General Container
struct Line // y = ax + b
{
ll a,b;
mutable ld p; // position where this line intersects with next line in the container (aka where this line ends)
Line():a(0),b(0),p(LLONG_MAX){}
Line(ll _a,ll _b):a(_a),b(_b),p(LLONG_MAX){}
ll cal(ll x) const {return a * x + b;}
// Comparison operator for querying
bool operator < (const ll & x) const { return p < x;}
// Comparison operator for set ordering
bool operator < (const Line &o) const
{
if(a != o.a)return a > o.a;
return b < o.b;
}
};
// Line Container using multiset
struct CHT_General : multiset<Line, less<void>>
{
// Calculate intersection (cut) point between two lines
ld xcut(iterator a,iterator b) const { return (ld)(b->b - a->b) / (a->a - b->a);}
// Checks if line a should remain before line b
bool check(iterator a,iterator b)
{
if(b == end())
{
a->p = LLONG_MAX; // Set p as infinity for the last line
return 1;
}
if(a->a == b->a)return 0; // Parallel lines
a->p = xcut(a,b); // recalculate x intersection
return a->p < b->p;
}
// Add a new line to the container, maintaining optimal structure
void add(Line A)
{
iterator a = insert(A);
iterator b = next(a);
while(!check(a,b)) b = erase(b); // Remove useless lines after a
if(a != begin())
{
b = prev(a);
if(!check(b,a)) check(b,erase(a));
}
else return;
// Check previous lines to maintain container integrity
while(b != begin())
{
a = prev(b);
if(a->p >= b->p)
{
check(a,erase(b)); // erase b and recalculate a->p
b = a;
}
else break;
}
}
// Query the minimum value at x
ll cal(ll x) { return lower_bound(x)->cal(x); }
};
// End Code Convex Hull Trick - General Container
ll a[maxn],pre[maxn],dp[maxn];
int main()
{
ios_base::sync_with_stdio(false);
cin.tie(0);
// https://oj.uz/problem/view/CEOI17_building?locale=en
int n;
cin>>n;
for(int i = 1;i<=n;i++)cin>>a[i];
for(int i = 1;i<=n;i++)
{
cin>>pre[i];
pre[i] += pre[i-1];
}
CHT_General container;
for(int i = 1;i<=n;i++)
{
if(i == 1) dp[i] = 0;
else dp[i] = container.cal(a[i]) + a[i] * a[i] + pre[i-1];
// Add new line for future DP queries
container.add(Line( a[i] * -2, a[i] * a[i] - pre[i] + dp[i] ));
}
cout<<dp[n];
return 0;
}
