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>  // clang-format off
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define each(x,v) for(auto& x : v)
#define all(v) (v).begin(),(v).end()
#define sz(v) ((int)(v).size())
#define ini(...) int __VA_ARGS__; in(__VA_ARGS__)
#define inl(...) long long __VA_ARGS__; in(__VA_ARGS__)
#define ins(...) string __VA_ARGS__; in(__VA_ARGS__)

#ifdef ONLINE_JUDGE
  #define rep(i,N) for(int i = 0; i < (int)(N); i++)
  #define repr(i,N) for(int i = (int)(N) - 1; i >= 0; i--)
  #define rep1(i,N) for(int i = 1; i <= (int)(N) ; i++)
  #define repr1(i,N) for(int i = (N) ; (int)(i) > 0 ; i--)
#else
  #define rep(i,N) for(long long i = 0; i < (long long)(N); i++)
  #define repr(i,N) for(long long i = (long long)(N) - 1; i >= 0; i--)
  #define rep1(i,N) for(long long i = 1; i <= (long long)(N) ; i++)
  #define repr1(i,N) for(long long i = (N) ; (long long)(i) > 0 ; i--)
#endif

using namespace std; void solve();
using ll = long long; template<class T = ll> using V = vector<T>;
using vi = V<int>; using vl = V<>; using vvi = V< V<int> >;
constexpr int inf = 1001001001; constexpr ll infLL = (1LL << 61) - 1;
struct IoSetupNya {IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7);} } iosetupnya;
template<typename T, typename U> inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template<typename T, typename U> inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; }
template<typename T, typename U> ostream& operator <<(ostream& os, const pair<T, U> &p) { os << p.first << " " << p.second; return os; }
template<typename T, typename U> istream& operator >>(istream& is, pair<T, U> &p) { is >> p.first >> p.second; return is; }
template<typename T> ostream& operator <<(ostream& os, const vector<T> &v) { int s = (int)v.size(); for(int i=0;i<s;i++) os << (i ? " " : "") << v[i]; return os; }
template<typename T> istream& operator >>(istream& is, vector<T> &v) { for(auto &x : v) is >> x; return is; }
void in(){} template <typename T,class... U> void in(T &t,U &...u){ cin >> t; in(u...);}
void out(){cout << "\n";} template <typename T,class... U> void out(const T &t,const U &...u){ cout << t; if(sizeof...(u)) cout << " "; out(u...);}
template<typename T>void die(T x){out(x); exit(0);}

#ifdef NyaanDebug
  #include "NyaanDebug.h"
  #define trc(...) do { cerr << #__VA_ARGS__ << " = "; dbg_out(__VA_ARGS__);} while(0)
  #define trca(v,N) do { cerr << #v << " = "; array_out(v , N);} while(0)
  #define trcc(v) do { cerr << "name : " << #v << "\n"; int cnt = 0; each(x , v){cerr << (cnt++) << " : "; trc(x); } } while(0)
#else
  #define trc(...)
  #define trca(...)
  #define trcc(...)
  int main(){solve();}
#endif

#define inc(...) char __VA_ARGS__; in(__VA_ARGS__)
#define in2(s,t)   rep(i,sz(s)){in(s[i] , t[i]);}
#define in3(s,t,u) rep(i,sz(s)){in(s[i] , t[i] , u[i]);}
using vd = V<double>; using vs = V<string>; using vvl = V< V<> >;
template<typename T,typename U>ll ceil(T a,U b){return (a + b - 1) / b;}

using P = pair<ll,ll>; using vp = V<P>;
constexpr int MOD = /**/ 1000000007; //*/ 998244353;
// clang-format on
///////////////////////////////////////////////////////////


template< typename T >
struct edge {
  int src, to;
  T cost;

  edge(int to, T cost) : src(-1), to(to), cost(cost) {}
  edge(int src, int to, T cost) : src(src), to(to), cost(cost) {}

  edge &operator=(const int &x) {
    to = x;
    return *this;
  }

  operator int() const { return to; }
};
template< typename T >
using Edges = vector< edge< T > >;
template< typename T >
using WeightedGraph = vector< Edges< T > >;
using UnweightedGraph = vector< vector <int> >;

// グラフの入力
UnweightedGraph graph(
  int N , int M = -1 , bool is_directed = false, bool is_1origin = true){
  UnweightedGraph g(N);
  if(M == -1) M = N - 1;
  for(int _ = 0 ; _ < M ; _++){
    int x , y; cin >> x >> y;
    if(is_1origin) x-- , y--;
    g[x].pb(y);
    if(!is_directed) g[y].pb(x);
  }
  return g;
}

// 重み付きグラフの入力
template<typename T>
WeightedGraph<T> wgraph(
  int N , int M = -1 , bool is_directed = false , bool is_1origin = true){
  WeightedGraph<T> g(N);
  if(M == -1) M = N - 1;
  for(int _ = 0 ; _ < M ; _++){
    int x , y; cin >> x >> y;
    T c; cin >> c;
    if(is_1origin) x-- , y--;
    g[x].eb(y , c);
    if(!is_directed) g[y].eb(x , c);
  }
  return g;
}

struct UF{
  vector<int> par,sizes;
  int cnt;
  UF(int N) : par(N),sizes(N, 1),cnt(N){ for(int i=0; i<N; i++) par[i]=i; }

  int find(int x){ return par[x]==x ? x : par[x] = find(par[x]); }

  bool same(int x,int y){ return find(x) == find(y); }

  int size(int x){ return sizes[find(x)]; }

  void unite(int x, int y){
    x = find(x); y = find(y);
    if(x == y) return;
    if(sizes[x] < sizes[y]) swap(x,y);
    par[y] = x;
    sizes[x] += sizes[y];
    cnt--;
  }
  
  int count() const{return cnt;}
};

template<typename T>
T kruskal(int N,Edges<T> &es){
  sort(begin(es) , end(es) , [&](edge<T> a,edge<T> b){
    return a.cost < b.cost;
  });
  UF uf(N);
  T ret = 0;
  for(edge<T> &e : es){
    if(uf.same(e.src , e.to)) continue;
    ret += e.cost;
    uf.unite(e.src , e.to);
  }
  return ret;
}

void solve(){
  inl(N,D);
  vl A(N);in(A);
  set<P> left,right;

  // left A - D * i
  // right A + D * i
  rep1(i , N - 1){
    right.emplace(A[i] + D * i , i);
  }

  Edges<ll> es;
  auto add = [&](int i,int j){
    ll c = A[i] + A[j] + ll(abs(i - j)) * D;
    es.eb(i , j , c);
  };

  rep(i , N){
    assert(sz(left) + sz(right) == N - 1);
    if(!left.empty()){
      auto it = left.begin();
      add(i , it->second);
    }
    if(!right.empty()){
      auto it = right.begin();
      add(i , it->second);
    }

    if(i != N - 1){
      right.erase(P(A[i + 1] + D * (i + 1), i + 1));
      left.emplace(A[i] - D * i , i);
    }
  }
  each(x,es)trc(x.src,x.to,x.cost);
  out(kruskal(N , es));
}