// Line Container - MultiSet
/* 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 Line Container - MultiSet
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;
}
};
// UNLIMITED POWER
struct LC_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 'b' is valid
bool check(iterator a,iterator b) // O( 1 )
{
if(b == end())
{
a->p = LLONG_MAX; // the last line ends at infinity
return 1;
}
if(a->a == b->a)return 0; // Parallel lines, and since our comparator ensured that a->b <= b->b , line 'b' is useless
a->p = xcut(a,b); // recalculate intersection
return a->p < b->p;
}
// Add a new line to the container & maintain structure
void add(Line A) // O( log(n) ) amortized
{
iterator a = insert(A);
iterator b = next(a);
// Remove invalid lines behind the new line
while(!check(a,b)) b = erase(b);
// Same, but infront
bool forced = 1; // we must check the new line's validity AND atleast one of the lines infront
while(a != begin())
{
b = a;
a = prev(a);
if(!check(a,b)) check(a,erase(b)); // erase 'b' and recalculate a->p
else if(!forced) break;
forced = 0;
}
}
// Query the minimum value at x
ll cal(ll x) // O( log(n) )
{
return lower_bound(x)->cal(x);
}
};
//End Code Line Container - MultiSet
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];
}
LC_General CHT;
for(int i = 1;i<=n;i++)
{
if(i == 1) dp[i] = 0;
else dp[i] = CHT.cal(a[i]) + a[i] * a[i] + pre[i-1];
// Add new line for future DP queries
CHT.add(Line( a[i] * -2, a[i] * a[i] - pre[i] + dp[i] ));
}
cout<<dp[n];
return 0;
}
