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)

//author: camypaper
using System;
using System.Linq;
using System.Collections.Generic;
using System.Text.RegularExpressions;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using static System.Math;

namespace Program {
    public class Solver {
        Random rnd = new Random(0);
        public void Solve() {
            var n = ri;
            var d = rl;
            var a = Enumerate(n, x => rl);
            var ans = 0L;
            var s = new DisjointSet(n);
            var cost = new long[n];
            var to = new int[n];
            for (int T = 0; T < 18; T++) {
                for (int i = 0; i < n; i++) {
                    cost[i] = INF;
                    to[i] = -1;
                }
                var mi = new long[2] { INF, -1 };
                var se = new long[2] { INF, -1 };
                for (int i = 0; i < n; i++) {
                    var r = s[i];
                    if (mi[1] != r && cost[r] > mi[0] + a[i] + i * d) {
                        cost[r] = mi[0] + a[i] + i * d;
                        to[r] = (int)mi[1];
                    }
                    if (se[1] != r && cost[r] > se[0] + a[i] + i * d) {
                        cost[r] = se[0] + a[i] + i * d;
                        to[r] = (int)se[1];
                    }
                    var u = new long[] { -i * d + a[i], r };
                    if (mi[0] > u[0]) {
                        if (mi[1] != u[1]) { se = mi; mi = u; }
                        else mi = u;
                    }
                    else if (se[0] > u[0] && mi[1] != u[1]) se = u;
                }
                mi = new long[2] { INF, -1 };
                se = new long[2] { INF, -1 };
                for (int i = n - 1; i >= 0; i--) {
                    var r = s[i];
                    if (mi[1] != r && cost[r] > mi[0] + a[i] - i * d) {
                        cost[r] = mi[0] + a[i] - i * d;
                        to[r] = (int)mi[1];
                    }
                    if (se[1] != r && cost[r] > se[0] + a[i] - i * d) {
                        cost[r] = se[0] + a[i] - i * d;
                        to[r] = (int)se[1];
                    }
                    var u = new long[] { i * d + a[i], r };
                    if (mi[0] > u[0]) {
                        if (mi[1] != u[1]) { se = mi; mi = u; }
                        else mi = u;
                    }
                    else if (se[0] > u[0] && mi[1] != u[1]) se = u;
                }
                for (int i = 0; i < n; i++)
                    if (s[i] == i && to[i] != -1 && !s.IsUnited(i, to[i])) {
                        s.Unite(i, to[i]);
                        ans += cost[i];
                    }
            }
            Console.WriteLine(ans);
        }
        const long INF = 1L << 60;
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static IEnumerable<int> Rep(int n) => Enumerable.Range(0, n);
        static IEnumerable<int> RRep(int n) => Enumerable.Range(0, n).Reverse();
        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < a.Length; ++i) a[i] = f(i);
            return a;
        }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex {
    static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
        return string.Join(st, ie);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        solver.Solve();
        Console.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO {
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer : StreamWriter {
        public override IFormatProvider FormatProvider => CultureInfo.InvariantCulture;
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;

        private byte read() {
            if (isEof) return 0;
            if (ptr >= len) {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0) {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char() {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan() {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine() {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
}

#endregion

#region DisjointSet
public class DisjointSet {
    int[] par;
    byte[] rank;
    public DisjointSet(int n) {
        par = new int[n];
        for (int i = 0; i < n; i++)
            par[i] = -1;
        rank = new byte[n];
    }
    public int this[int id] {
        get {
            if ((par[id] < 0)) return id;
            return par[id] = this[par[id]];
        }
    }
    public bool Unite(int x, int y) {
        x = this[x]; y = this[y];
        if (x == y) return false;
        if (rank[x] < rank[y]) { var z = x; x = y; y = z; }
        par[x] += par[y];
        par[y] = x;
        if (rank[x] == rank[y])
            rank[x]++;
        return true;
    }
    public int Size(int x) { return -par[this[x]]; }
    public bool IsUnited(int x, int y) { return this[x] == this[y]; }

}
#endregion