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;

public class Main {
  private static void solve() {
    int n = ni();
    long d = ni();
    int[] a = na(n);

    int[] right = new int[n];
    int[] left = new int[n];

    left[1] = 0;
    for (int i = 2; i < n; i++) {
      long d1 = d * (i - left[i - 1]) + a[left[i - 1]];
      long d2 = d + a[i - 1];
      left[i] = d1 > d2 ? i - 1 : left[i - 1];
    }

    right[n - 2] = n - 1;
    for (int i = n - 3; i >= 0; i--) {
      long d1 = d * (right[i + 1] - i) + a[right[i + 1]];
      long d2 = d + a[i + 1];
      right[i] = d1 > d2 ? i + 1 : right[i + 1];
    }

    int[] from = new int[n * 2 - 2];
    int[] to = new int[n * 2 - 2];
    long[] w = new long[n * 2 - 2];
    int ptr = 0;
    for (int i = 1; i < n; i++, ptr++) {
      from[ptr] = i;
      to[ptr] = left[i];
      w[ptr] = Math.abs(i - left[i]) * d + a[i] + a[left[i]];
    }
    for (int i = 0; i < n - 1; i++, ptr++) {
      from[ptr] = i;
      to[ptr] = right[i];
      w[ptr] = Math.abs(i - right[i]) * d + a[i] + a[right[i]];
    }
    System.out.println(kruskal(from, to, w, n));
  }

  public static long kruskal(int[] one, int[] other, long[] w, int n) {
    DisjointSet ds = new DisjointSet(n);
    int p = one.length;
    Integer[] ord = new Integer[p];
    for (int i = 0; i < p; i++)
      ord[i] = i;
    Arrays.sort(ord, (o1, o2) -> Long.compare(w[o1], w[o2]));

    long d = 0;
    for (int i = 0; i < ord.length; 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 class DisjointSet {
    public int[] upper;

    public DisjointSet(int n) {
      upper = new int[n];
      Arrays.fill(upper, -1);
    }

    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[][] toBucket() {
      int n = upper.length;
      int[][] ret = new int[n][];
      int[] rp = new int[n];
      for (int i = 0; i < n; i++) {
        if (upper[i] < 0)
          ret[i] = new int[-upper[i]];
      }
      for (int i = 0; i < n; i++) {
        int r = root(i);
        ret[r][rp[r]++] = i;
      }
      return ret;
    }
  }

  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));
  }
}