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ConvexHullTrick-max.test.cpp
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ConvexHullTrick-max.test.cpp
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#define PROBLEM "https://yukicoder.me/problems/no/409"
#include <iostream>
#include <vector>
#include <queue>
using namespace std;
#include "../../lib/12-binary-search-tree/RandomizedBinarySearchTreeSet.cpp"
#include "../../lib/99-operator/operator/ValueMax.cpp"
#include "../../lib/16-convex-hull-trick/ConvexHullTrick.cpp"
using ll = long long;
int main(void){
ll N,A,B,W; cin >> N >> A >> B >> W;
vector<ll> D(N+2,0);
for(int i = 1; i <= N; ++i) cin >> D[i];
// dp[i]=min{j:[0,i)} -> dp[j]+B*k*(k+1)/2-k*A+D[i] (k=i-j-1)
// -> dp[j]+B*(i-j-1)*(i-j)/2-(i-j-1)*A+D[i]
// -> dp[j]+B/2*(i*i-2*i*j+j*j-i+j)-A*(i-j-1)+D[i]
// -> (-B*j)*i + dp[j]+B/2*(j*j+j)+A*j + B/2*(i*i-i)-A*(i-1)+D[i]
// dp[i]=-max{j:[0,i)}-> (B*j)*i + -{dp[j]+B/2*(j*j+j)+A*j}
// ->
vector<ll> dp(N+2,1e15);
dp[0]=W;
ConvexHullTrick<ValueMax<ll>> cht;
cht.insert(0,-dp[0]);
for(ll i=1;i<=N+1;++i){
dp[i]=-cht.get(i) + B*(i*i-i)/2-A*(i-1)+D[i];
cht.insert(B*i,-(dp[i]+B*(i*i+i)/2+A*i));
}
cout << dp[N+1] << endl;
return 0;
}