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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)


import java.util.Arrays;

class DisjointSet {
  public int[] upper;
  
  public DisjointSet(int n)
  {
      upper = new int[n];
      Arrays.fill(upper, -1);
  }
  
  public DisjointSet(DisjointSet ds)
  {
      this.upper = Arrays.copyOf(ds.upper, ds.upper.length);
  }       
  
  public int root(int x)
  {
      return upper[x] < 0 ? x : (upper[x] = root(upper[x]));
  }
  
  public boolean equiv(int x, int y)
  {
      return root(x) == root(y);
  }
  
  public boolean union(int x, int y)
  {
      x = root(x);
      y = root(y);
      if(x != y) {
          if(upper[y] < upper[x]) {
              int d = x; x = y; y = d;
          }
          upper[x] += upper[y];
          upper[y] = x;
      }
      return x == y;
  }
  
  public int count()
  {
      int ct = 0;
      for(int u : upper){
          if(u < 0)ct++;
      }
      return ct;
  }
}

class SegmentTreeRMQPos {
  public int M, H, N;
  public long[] st;
  public int[] pos;

  public SegmentTreeRMQPos(long[] a)
  {
      N = a.length;
      M = Integer.highestOneBit(Math.max(N-1, 1))<<2;
      H = M>>>1;
      st = new long[M];
      pos = new int[M];
      for(int i = 0;i < N;i++){
          st[H+i] = a[i];
          pos[H+i] = i;
      }
      Arrays.fill(st, H+N, M, Long.MAX_VALUE);
      for(int i = H-1;i >= 1;i--)propagate(i);
  }
  
  public void update(int pos, int x)
  {
      st[H+pos] = x;
      for(int i = (H+pos)>>>1;i >= 1;i >>>= 1)propagate(i);
  }
  
  private void propagate(int i)
  {
      if(st[2*i] <= st[2*i+1]){
          st[i] = st[2*i];
          pos[i] = pos[2*i];
      }else{
          st[i] = st[2*i+1];
          pos[i] = pos[2*i+1];
      }
  }
  
  public int minpos;
  public long minval;
    
  public long min(int l, int r){ 
      minpos = -1;
      minval = Long.MAX_VALUE;
      min(l, r, 0, H, 1);
      return minval;
  }
  
  private void min(int l, int r, int cl, int cr, int cur)
  {
      if(l <= cl && cr <= r){
          if(st[cur] < minval){
              minval = st[cur];
              minpos = pos[cur];
          }
      }else{
          int mid = cl+cr>>>1;
          if(cl < r && l < mid)min(l, r, cl, mid, 2*cur);
          if(mid < r && l < cr)min(l, r, mid, cr, 2*cur+1);
      }
  }
}


public class Main {
  private static void solve() {
    int n = ni();
    long d = ni();
    long[] a = nal(n);
    
    long[] left = new long[n];
    long[] right = new long[n];
    for (int i = 0; i < n; i ++) {
      left[i] = -i * d + a[i];
      right[i] = i * d + a[i];
    }
    
    long[][] table = new long[n][3];
    for (int i = 0; i < n; i ++) {
      table[i][0] = i;
      table[i][1] = a[i];
    }
    Arrays.sort(table, (o1, o2) -> Long.compare(o2[1], o1[1]));
    int[] idx = Arrays.stream(table).mapToInt(v -> (int)v[0]).toArray();
    
    SegmentTreeRMQPos stL = new SegmentTreeRMQPos(left);
    SegmentTreeRMQPos stR = new SegmentTreeRMQPos(right);
    
    int[] from = new int[n * 3];
    int[] to = new int[n * 3];
    long[] w = new long[n * 3];
    int ptr = 0;
    for (int i : idx) {
      stL.min(0, i);
      stR.min(i + 1, n);
      
      if (stL.minpos >= 0) {
        from[ptr] = i;
        to[ptr] = stL.minpos;
        w[ptr] = i * d + a[i] + stL.minval;
        ptr ++;
      }
      if (stR.minpos >= 0) {
        from[ptr] = i;
        to[ptr] = stR.minpos;
        w[ptr] = -i * d + a[i] + stR.minval;
        ptr ++;
      }
    }
    long ret = kruskal(from, to, w, ptr, n);
    System.out.println(ret);
  }
  
  public static long kruskal(int[] one, int[] other, long[] w, int m, int n)
  {
      DisjointSet ds = new DisjointSet(n);
      int p = m;
      long[] ord = new long[p];
      for(int i = 0;i < p;i++)ord[i] = (long)w[i]<<32 | i;
      Arrays.sort(ord);
      
      long d = 0;
      for(int i = 0;i < m;i++){
          int cur = (int)ord[i];
          if(!ds.equiv(one[cur], other[cur])){
              ds.union(one[cur], other[cur]);
              d += w[cur];
          }
      }
      
      return d;
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}