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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Convert;
using static System.Math;
//using static System.Globalization.CultureInfo;
using System.Text;
class Program
{
    public static bool chmin<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == 1) { num = val; return true; } return false; }
    public static bool chmax<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == -1) { num = val; return true; } return false; }
    public static void swap<T>(ref T v1, ref T v2)
    { var t = v2; v2 = v1; v1 = t; }
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        new Program().Solve();
        Console.Out.Flush();
    }
    void Solve()
    {
        int num;long d;
        Input.Make(out num, out d);
        var ar = Input.ar;
        var so = Enumerable.Range(0, num).Select(v => new Pair<int, int>(ar[v], v)).ToArray();
      	Array.Sort(so);
        var edge = new List<Pair<long, int, int>>();
        Func<Pair<long, int>, Pair<long, int>, Pair<long, int>> func =
            (a, b) => a.v1 > b.v1 ? b : a;
        var leftseg = new SegmentTree<Pair<long, int>>(num, new Pair<long, int>(long.MaxValue, -1), func);
        var rightseg = new SegmentTree<Pair<long, int>>(num, new Pair<long, int>(long.MaxValue, -1), func);
        for (var i = 0; i < num; i++)
        {
            leftseg.Update(i, new Pair<long, int>(ar[i] - i * d, i));
            rightseg.Update(i, new Pair<long, int>(ar[i] + i *d, i));
        }
        for (var i = num - 1; i >= 0; i--)
        {
            if (so[i].v2 != 0)
            {
                var min = leftseg.Query(0, so[i].v2);
                edge.Add(new Pair<long, int, int>(min.v1 + so[i].v2 * d + so[i].v1, so[i].v2, min.v2));
                leftseg.Update(so[i].v2, new Pair<long, int>(long.MaxValue, -1));
            }
            if (so[i].v2 != num - 1)
            {
                var min = rightseg.Query(so[i].v2 + 1,num);
                edge.Add(new Pair<long, int, int>(min.v1 - so[i].v2 * d + so[i].v1, so[i].v2, min.v2));
                rightseg.Update(so[i].v2, new Pair<long, int>(long.MaxValue, -1));
            }
        }
        edge.Sort();
        var uf = new UnionFind(num);
        var res = 0L;
        foreach (var e in edge)
            if (!uf.IsSame(e.v2, e.v3))
            {
                res += e.v1;
                uf.Union(e.v2, e.v3);
            }
        WriteLine(res);
    }
}

public class UnionFind
{
    private int[] _parent, _size, _rank;
    public UnionFind(int num)
    {
        IndependenceNum = num;
        _parent = new int[num]; _size = new int[num]; _rank = new int[num];
        for (var i = 0; i < num; i++) { _parent[i] = i; _size[i] = 1; }
    }
    //独立な集合の数
    public int IndependenceNum { get; private set; }
    private int Find(int i)
        => i == _parent[i] ? i : (_parent[i] = Find(_parent[i]));
    public int Parent(int i)
        => Find(i);
    public int Size(int i)
        => _size[Find(i)];
    public bool Union(int v1, int v2)
    {
        v1 = Find(v1); v2 = Find(v2);
        if (v1 == v2) return false;
        IndependenceNum--;
        if (_rank[v1] > _rank[v2])
        {
            _parent[v2] = v1;
            _size[v1] += _size[v2];
        }
        else
        {
            _parent[v1] = v2;
            _size[v2] += _size[v1];
            if (_rank[v1] == _rank[v2])
                _rank[v2]++;
        }
        return true;
    }
    public bool IsSame(int v1, int v2)
        => Find(v1) == Find(v2);
}

public class SegmentTree<T>
{
    protected readonly T[] item;
    protected readonly int num;
    protected readonly Func<T, T, T> func;
    protected readonly Func<T, T, T> updateFunc;
    protected readonly T init;

    protected int Parent(int index)
        => (index - 1) >> 1;
    protected int Left(int index)
        => (index << 1) + 1;
    protected int Right(int index)
        => (index + 1) << 1;
    public T this[int i]
    {
        get { return item[i + num - 1]; }
        set { item[i + num - 1] = value; }
    }

    public SegmentTree(int num, T init, Func<T, T, T> func, Func<T, T, T> updateFunc = null)
    {
        this.func = func;
        this.num = 1;
        this.init = init;
        this.updateFunc = updateFunc ?? ((T val1, T val2) => val2);
        while (this.num <= num)
            this.num *= 2;
        item = new T[2 * this.num - 1];
        for (var i = 0; i < 2 * this.num - 1; i++)
            item[i] = init;
    }
    public void Update(int index, T value)
    {
        index += num - 1;
        item[index] = updateFunc(item[index], value);
        while (index > 0)
        {
            index = Parent(index);
            item[index] = func(item[Left(index)], item[Right(index)]);
        }
    }
    public virtual void Update(int left, int right, T value)
        => Update(left, right, 0, 0, num, value);
    protected virtual void Update(int left, int right, int k, int l, int r, T value)
    {
        if (r <= left || right <= l) return;
        if (left <= l && r <= right) item[k] = updateFunc(item[k], value);
        else
        {
            Update(left, right, Left(k), l, (l + r) >> 1, value);
            Update(left, right, Right(k), (l + r) >> 1, r, value);
        }
    }
    public void All_Update()
    {
        for (int i = num - 2; i >= 0; i--)
            item[i] = func(item[Left(i)], item[Right(i)]);
    }
    public T Query(int index)
    {
        index += num - 1;
        var value = func(init, item[index]);
        while (index > 0)
        {
            index = Parent(index);
            value = func(value, item[index]);
        }
        return value;
    }
    //[left,right)
    public virtual T Query(int left, int right)
        => Query(left, right, 0, 0, num);
    protected virtual T Query(int left, int right, int k, int l, int r)
    {
        if (r <= left || right <= l) return init;
        if (left <= l && r <= right) return item[k];
        else
            return func(Query(left, right, Left(k), l, (l + r) >> 1), Query(left, right, Right(k), (l + r) >> 1, r));
    }
}

public class Input
{
    public static string read => ReadLine().Trim();
    public static int[] ar => read.Split(' ').Select(int.Parse).ToArray();
    public static int num => ToInt32(read);
    public static long[] arL => read.Split(' ').Select(long.Parse).ToArray();
    public static long numL => ToInt64(read);
    public static T[] create<T>(int n, Func<int, T> f)
        => Enumerable.Repeat(0, n).Select(f).ToArray();
    public static char[][] grid(int h)
        => create(h, _ => read.ToCharArray());
    public static int[] ar1D(int n)
        => create(n, _ => num);
    public static long[] arL1D(int n)
        => create(n, _ => numL);
    public static string[] strs(int n)
        => create(n, _ => read);
    public static int[][] ar2D(int n)
        => create(n, _ => ar);
    public static long[][] arL2D(int n)
        => create(n, _ => arL);
    public static List<T>[] edge<T>(int n)
        => create(n, _ => new List<T>());
    public static T GetValue<T>(string g)
    {
        var t = typeof(T);
        if (t == typeof(int))
            return (T)(object)int.Parse(g);
        if (t == typeof(long))
            return (T)(object)long.Parse(g);
        if (t == typeof(string))
            return (T)(object)g;
        if (t == typeof(char))
            return (T)(object)char.Parse(g);
        if (t == typeof(double))
            return (T)(object)double.Parse(g);
        if (t == typeof(bool))
            return (T)(object)bool.Parse(g);
        return default(T);
    }
    public static void Make<T1, T2>(out T1 v1, out T2 v2)
    {
        v1 = Next<T1>();
        v2 = Next<T2>();
    }
    public static void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3)
    {
        Make(out v1, out v2);
        v3 = Next<T3>();
    }
    public static void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4)
    {
        Make(out v1, out v2, out v3);
        v4 = Next<T4>();
    }
    public static void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5)
    {
        Make(out v1, out v2, out v3, out v4);
        v5 = Next<T5>();
    }
    public static void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6)
    {
        Make(out v1, out v2, out v3, out v4, out v5);
        v6 = Next<T6>();
    }
    private static Queue<string> sc;
    public static T Next<T>() { sc = sc ?? new Queue<string>(); if (sc.Count == 0) foreach (var item in read.Split(' ')) sc.Enqueue(item); return GetValue<T>(sc.Dequeue()); }
    public static void Next<T>(ref T val) => val = Next<T>();
    public const long Inf = (long)1e18;
    public const double eps = 1e-6;
    public const string Alfa = "abcdefghijklmnopqrstuvwxyz";
    public const int MOD = 1000000007;
}

public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1 { get; set; }
    public T2 v2 { get; set; }
    public Pair() { v1 = Input.Next<T1>(); v2 = Input.Next<T2>(); }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString()
        => $"{v1.ToString()} {v2.ToString()}";
    public override bool Equals(object obj)
        => this == (Pair<T1, T2>)obj;
    public override int GetHashCode()
        => v1.GetHashCode() ^ v2.GetHashCode();
    public static bool operator ==(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == 0;
    public static bool operator !=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != 0;
    public static bool operator >(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == 1;
    public static bool operator >=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != -1;
    public static bool operator <(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == -1;
    public static bool operator <=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != 1;
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3 { get; set; }
    public Pair() : base() { v3 = Input.Next<T3>(); }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString()
        => $"{base.ToString()} {v3.ToString()}";
}