Problem:
Score : 600 points

Problem Statement

There are N cities in Republic of AtCoder. The size of the i-th city is A_{i}.
Takahashi would like to build N-1 bidirectional roads connecting two cities so that any city can be reached from any other city by using these roads.

Assume that the cost of building a road connecting the i-th city and the j-th city is |i-j| \times D + A_{i} + A_{j}.
For Takahashi, find the minimum possible total cost to achieve the objective.

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq D \leq 10^9

1 \leq A_{i} \leq 10^9

A_{i} and D are integers.

Input

Input is given from Standard Input in the following format:

N D
A_1 A_2 ... A_N

Output

Print the minimum possible total cost.

Sample Input 1

3 1
1 100 1

Sample Output 1

106

This cost can be achieved by, for example, building roads connecting City 1, 2 and City 1, 3.

Sample Input 2

3 1000
1 100 1

Sample Output 2

2202

Sample Input 3

6 14
25 171 7 1 17 162

Sample Output 3

497

Sample Input 4

12 5
43 94 27 3 69 99 56 25 8 15 46 8

Sample Output 4

658

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
using namespace std::chrono;
using ll=long long;
using ull=unsigned long long;
using uint=unsigned int;
using pcc=pair<char,char>;
using pii=pair<int,int>;
using pll=pair<ll,ll>;
using pdd=pair<double,double>;
using tuplis=pair<ll,pll>;
const ll LINF=0x1fffffffffffffff;
const ll MOD=1000000007;
const ll MODD=0x3b800001;
const int INF=0x3fffffff;
const double DINF=numeric_limits<double>::infinity();
const vector<pii> four={{-1,0},{0,1},{1,0},{0,-1}};
#define _overload4(_1,_2,_3,_4,name,...) name
#define _overload3(_1,_2,_3,name,...) name
#define _rep1(n) _rep2(i,n)
#define _rep2(i,n) _rep3(i,0,n)
#define _rep3(i,a,b) for(ll i=a;i<b;++i)
#define _rep4(i,a,b,c) for(ll i=a;i<b;i+=c)
#define rep(...) _overload4(__VA_ARGS__,_rep4,_rep3,_rep2,_rep1)(__VA_ARGS__)
#define _rrep1(n) _rrep2(i,n)
#define _rrep2(i,n) _rrep3(i,0,n)
#define _rrep3(i,a,b) for(ll i=b-1;i>=a;i--)
#define _rrep4(i,a,b,c) for(ll i=a+(b-a-1)/c*c;i>=a;i-=c)
#define rrep(...) _overload4(__VA_ARGS__,_rrep4,_rrep3,_rrep2,_rrep1)(__VA_ARGS__)
#define each(i,a) for(auto&& i:a)
#define sum(...) accumulate(range(__VA_ARGS__),0LL)
#define dsum(...) accumulate(range(__VA_ARGS__),double(0))
#define _range(i) (i).begin(),(i).end()
#define _range2(i,k) (i).begin(),(i).begin()+k
#define _range3(i,a,b) (i).begin()+a,(i).begin()+b
#define range(...) _overload3(__VA_ARGS__,_range3,_range2,_range)(__VA_ARGS__)
#define _rrange(i) (i).rbegin(),(i).rend()
#define _rrange2(i,k) (i).rbegin(),(i).rbegin()+k
#define _rrange3(i,a,b) (i).rbegin()+a,(i).rbegin()+b
#define rrange(...) _overload3(__VA_ARGS__,_rrange3,_rrange2,_rrange)(__VA_ARGS__)
#define yes(i) out(i?"yes":"no")
#define Yes(i) out(i?"Yes":"No")
#define YES(i) out(i?"YES":"NO")
#define Possible(i) out(i?"Possible":"Impossible")
#define elif else if
#define unless(a) if(!(a))
#define mp make_pair
#define INT(...) int __VA_ARGS__;in(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;in(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;in(__VA_ARGS__)
#define CHR(...) char __VA_ARGS__;in(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__;in(__VA_ARGS__)
#define vec(type,name,...) vector<type> name(__VA_ARGS__)
#define VEC(type,name,size) vector<type> name(size);in(name)
#define vv(type,name,h,...) vector<vector<type>>name(h,vector<type>(__VA_ARGS__))
#define VV(type,name,h,...) vector<vector<type>>name(h,vector<type>(__VA_ARGS__));in(name)
__attribute__((constructor)) void SETTINGS(){cin.tie(0); cout.tie(0); ios::sync_with_stdio(0); cout<<fixed<<setprecision(20);};
template<class T>
inline constexpr T gcd (T a,T b) {if(!a||!b)return 0;return a%b?gcd(b,a%b):b;}
template<class T>
inline constexpr T lcm (T a,T b) {return a*b/gcd(a,b);}
template<class T>
inline constexpr T min(vector<T>& v){return *min_element(range(v));}
inline char min(string& v){return *min_element(range(v));}
template<class T>
inline constexpr T max(vector<T>& v){return *max_element(range(v));}
inline char max(string& v){return *max_element(range(v));}
inline ll pow(ll a,ll b){if(b==0)return 1;ll ans=pow(a,b/2);return ans*ans*(b&1?a:1);}
inline ll modpow(ll a,ll b,ll mod){if(b==0)return 1;ll ans=pow(a,b/2);return ans*ans*(b&1?a:1)%mod;}
template<typename T>
inline bool update_min(T& mn,const T& cnt){if(mn>cnt){mn=cnt;return 1;}else return 0;}
template<typename T>
inline bool update_max(T& mx,const T& cnt){if(mx<cnt){mx=cnt;return 1;}else return 0;}
inline void in() {}
template<class T>
istream& operator >> (istream& is, vector<T>& vec);
template<class T,size_t size>
istream& operator >> (istream& is, array<T,size>& vec);
template<class T,class L>
istream& operator >> (istream& is, pair<T,L>& p);
template<class T>
ostream& operator << (ostream& os, vector<T>& vec);
template<class T,class L>
ostream& operator << (ostream& os, pair<T,L>& p);
template<class T>
istream& operator >> (istream& is, vector<T>& vec){for(T& x: vec) is >> x;return is;}
template<class T,class L>
istream& operator >> (istream& is, pair<T,L>& p){is >> p.first;is >> p.second;return is;}
template<class T>
ostream& operator << (ostream& os, vector<T>& vec){os << vec[0];rep(i,1,vec.size()){os << ' ' << vec[i];}return os;}
template<class T>
ostream& operator << (ostream& os, deque<T>& deq){os << deq[0];rep(i,1,deq.size()){os << ' ' << deq[i];}return os;}
template<class T,class L>
ostream& operator << (ostream& os, pair<T,L>& p){os << p.first << " " << p.second;return os;}
template<class T,class L>
pair<T,L> operator + (pair<T,L> a, pair<T,L> b){return {a.first + b.first, a.second + b.second};}
template <class Head, class... Tail>
inline void in(Head&& head,Tail&&... tail){cin>>head;in(move(tail)...);}
template <class T>
inline bool out(T t){cout<<t<<'\n';return 0;}
inline bool out(){cout<<'\n';return 0;}
template <class Head, class... Tail>
inline bool out(Head head,Tail... tail){cout<<head<<' ';out(move(tail)...);return 0;}
template <class T>
inline void err(T t){cerr<<t<<'\n';}
inline void err(){cerr<<'\n';}
template <class Head, class... Tail>
inline void err(Head head,Tail... tail){cerr<<head<<' ';out(move(tail)...);}


#define int ll
class UnionFind{
public:
    vector<int>data;
    UnionFind(int n):data(n,-1){}
    bool Union(int a,int b){
        a=root(a);b=root(b);
        if(a==b)return 1;
        if(data[a]>data[b])swap(a,b);
        data[a]+=data[b];
        data[b]=a;
        return 0;
    }
    bool Find(int a,int b){return root(a)==root(b);}
    int root(int a){return data[a]<0?a:root(data[a]);}
};
signed main(){
    LL(n,d);
    VEC(ll,a,n);
    vector<pll>s(n),t(n);
    rep(n)s[i]={a[i]+d*i,i};
    rep(n)t[i]={a[i]+d*(n-1-i),i};
    rrep(n-1)update_min(s[i],s[i+1]);
    rep(n-1)update_min(t[i+1],t[i]);
    vector<unordered_map<ll,ll>>g(n);
    rep(n-1){
        g[i][s[i+1].second]=g[s[i+1].second][i]=a[s[i+1].second]+a[i]+abs(s[i+1].second-i)*d;
        g[i+1][t[i].second]=g[t[i].second][i+1]=a[t[i].second]+a[i+1]+abs(t[i].second-(i+1))*d;
    }
    ll ans=0;
    vec(bool,used,n);
    priority_queue<tuplis,vector<tuplis>,greater<tuplis>>q;
    UnionFind uf(n);
    used[0]=1;
    each(i,g[0])q.push({i.second,{0,i.first}});
    while (!q.empty()) {
        auto cost=q.top().first,x=q.top().second.first,y=q.top().second.second;
        q.pop();
        if(uf.Find(x,y))continue;
        uf.Union(x,y);
        ans+=cost;
        each(i,g[y])if(!uf.Find(y,i.first))q.push({i.second,{y,i.first}});
    }
    out(ans);
}