#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;
// Optimization
#pragma GCC optimize("O3,unroll-loops,no-stack-protector")
#define endl '\n'
// Shortcut
#define int long long
#define eb emplace_back
#define pb push_back
#define pob pop_back
#define mp make_pair
#define upb upper_bound
#define lwb lower_bound
#define fi first
#define se second
#define For(i, l, r) for (int i = l; i < r; i++)
#define ForE(i, l, r) for (int i = l; i <= r; i++)
#define Ford(i, r, l) for (int i = r; i > l; i--)
#define FordE(i, r, l) for (int i = r; i >= l; i--)
#define Fora(i, a) for (auto i : a)
// I/O & Debug
#define PrintV(a) Fora(iiii, a) cout << iiii << ' '; cout << endl;
#define PrintVl(a) Fora(iiii, a) cout << iiii << endl;
#define PrintA(a, l, r) for (int iiii = l; iiii <= r; iiii++) cout << a[iiii] << ' '; cout << endl;
#define PrintAl(a, l, r) for (int iiii = l; iiii <= r; iiii++) cout << a[iiii] << endl;
#define Ptest(x) return cout << x, 0;
#define gl(s) getline(cin, s);
#define setpre(x) fixed << setprecision(x)
/*
#define debug(args...){ string _sDEB = #args; replace(_sDEB.begin(), _sDEB.end(), ',', ' '); stringstream _ssDEB(_sDEB); istream_iterator<string> _itDEB(_ssDEB); DEB(_itDEB, args); }
void DEB(istream_iterator<string> it) {}
template<typename T, typename... Args>
void DEB(istream_iterator<string> it, T a, Args... args){
cout << *it << " = " << a << endl;
DEB(++it, args...);
}
*/
// Functions
//#define isvowel(a) (a == 'a' || a == 'e' || a == 'i' || a == 'o' || a == 'u')
#define bend(a) a.begin(), a.end()
#define rbend(a) a.rbegin(), a.rend()
#define mset(a) memset(a, 0, sizeof(a))
#define mset1(a) memset(a, 1, sizeof(a))
#define msetn1(a) memset(a, -1, sizeof(a))
#define msetinf(a) memset(a, 0x3f, sizeof(a))
#define gcd __gcd
#define __builtin_popcount __builtin_popcountll
//mt19937 rando(chrono::steady_clock::now().time_since_epoch().count());
//int randt(int l, int r){ return (rando() % (r - l + 1) + l); }
// Data Structure
#define pque priority_queue
#define mts multiset
#define y0 _y0_
#define y1 _y1_
#define div divi
typedef long long ll;
typedef long double ld;
typedef vector <int> vi;
typedef vector <ll> vll;
typedef vector <ld> vld;
typedef vector <string> vs;
typedef pair <int, int> pii;
typedef pair <ll, ll> pll;
typedef vector <vi > vvi;
typedef vector <vll > vvll;
typedef vector <pii > vpii;
typedef vector <pll > vpll;
const int N = 1e5 + 5;
int mod = 1e9 + 7, mod1 = 998244353, mod2 = 1e9 + 9, inf = 1e9 + 7;
ll infll = 1e18 + 7;
/*
* Dynamic version of data structure
* to be used in dynamic programming optimisation
* called "Convex Hull trick"
* 'Dynamic' means that there is no restriction on adding lines order
*/
class ConvexHullDynamic
{
typedef long long coef_t;
typedef long long coord_t;
typedef long long val_t;
/*
* Line 'y=a*x+b' represented by 2 coefficients 'a' and 'b'
* and 'xLeft' which is intersection with previous line in hull(first line has -INF)
*/
private:
struct Line
{
coef_t a, b;
ld xLeft;
enum Type {line, maxQuery, minQuery} type;
coord_t val;
explicit Line(coef_t aa=0, coef_t bb=0) : a(aa), b(bb), xLeft(-infll), type(Type::line), val(0) {}
val_t valueAt(coord_t x) const { return a*x+b; }
friend bool areParallel(const Line& l1, const Line& l2) { return l1.a==l2.a; }
friend ld intersectX(const Line& l1, const Line& l2) { return areParallel(l1,l2)?infll:1.0*(l2.b-l1.b)/(l1.a-l2.a); }
bool operator<(const Line& l2) const
{
if (l2.type == line)
return this->a < l2.a;
if (l2.type == maxQuery)
return this->xLeft < l2.val;
if (l2.type == minQuery)
return this->xLeft > l2.val;
}
};
private:
bool isMax; //whether or not saved envelope is top(search of max value)
std::set<Line> hull; //envelope itself
private:
/*
* INFO: Check position in hull by iterator
* COMPLEXITY: O(1)
*/
bool hasPrev(std::set<Line>::iterator it) { return it!=hull.begin(); }
bool hasNext(std::set<Line>::iterator it) { return it!=hull.end() && std::next(it)!=hull.end(); }
/*
* INFO: Check whether line l2 is irrelevant
* NOTE: Following positioning in hull must be true
* l1 is next left to l2
* l2 is right between l1 and l3
* l3 is next right to l2
* COMPLEXITY: O(1)
*/
bool irrelevant(const Line& l1, const Line& l2, const Line& l3) { return intersectX(l1,l3) <= intersectX(l1,l2); }
bool irrelevant(std::set<Line>::iterator it)
{
return hasPrev(it) && hasNext(it)
&& ( isMax && irrelevant(*std::prev(it), *it, *std::next(it))
|| !isMax && irrelevant(*std::next(it), *it, *std::prev(it)) );
}
/*
* INFO: Updates 'xValue' of line pointed by iterator 'it'
* COMPLEXITY: O(1)
*/
std::set<Line>::iterator updateLeftBorder(std::set<Line>::iterator it)
{
if (isMax && !hasPrev(it) || !isMax && !hasNext(it))
return it;
double val = intersectX(*it, isMax?*std::prev(it):*std::next(it));
Line buf(*it);
it = hull.erase(it);
buf.xLeft = val;
it = hull.insert(it, buf);
return it;
}
public:
explicit ConvexHullDynamic(bool isMax): isMax(isMax) {}
/*
* INFO: Adding line to the envelope
* Line is of type 'y=a*x+b' represented by 2 coefficients 'a' and 'b'
* COMPLEXITY: Adding N lines(N calls of function) takes O(N*log N) time
*/
void addLine(coef_t a, coef_t b)
{
//find the place where line will be inserted in set
Line l3 = Line(a, b);
auto it = hull.lower_bound(l3);
//if parallel line is already in set, one of them becomes irrelevant
if (it!=hull.end() && areParallel(*it, l3))
{
if (isMax && it->b < b || !isMax && it->b > b)
it = hull.erase(it);
else
return;
}
//try to insert
it = hull.insert(it, l3);
if (irrelevant(it)) { hull.erase(it); return; }
//remove lines which became irrelevant after inserting line
while (hasPrev(it) && irrelevant(std::prev(it))) hull.erase(std::prev(it));
while (hasNext(it) && irrelevant(std::next(it))) hull.erase(std::next(it));
//refresh 'xLine'
it = updateLeftBorder(it);
if (hasPrev(it))
updateLeftBorder(std::prev(it));
if (hasNext(it))
updateLeftBorder(std::next(it));
}
/*
* INFO: Query, which returns max/min(depends on hull type - see more info above) value in point with abscissa 'x'
* COMPLEXITY: O(log N), N-amount of lines in hull
*/
val_t getBest(coord_t x) const
{
Line q;
q.val = x;
q.type = isMax ? Line::Type::maxQuery : Line::Type::minQuery;
auto bestLine = hull.lower_bound(q);
if (isMax) --bestLine;
return bestLine->valueAt(x);
}
} dch(0);
int n;
int h[N], w[N], b[N], dp[N];
signed main(){
ios_base::sync_with_stdio(0);
cin.tie(0); cout.tie(0);
cin >> n;
ForE(i, 1, n){
cin >> h[i];
}
ForE(i, 1, n){
cin >> w[i];
b[i] = b[i - 1] + w[i];
}
dp[1] = 0;
dch.addLine(-2 * h[1], dp[1] - b[1] + h[1] * h[1]);
// cout << dch.getBest(h[1]) << ' ';
ForE(i, 2, n){
dp[i] = b[i - 1] + h[i] * h[i] + dch.getBest(h[i]);
// cout << dch.getBest(h[i]) << ' ';
dch.addLine(-2 * h[i], dp[i] - b[i] + h[i] * h[i]);
}
// cout << endl;
// PrintA(dp, 1, n);
cout << dp[n];
}
/*
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INPUT: |
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OUTPUT: |
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*/
