The easiest way to think about this problem is, what is the longest subsequence of the input array that we can use to form an alphabet? Then we can calculate how many missing letters we need to add.

Such a subsequence must be strictly increasing, so this is just a slight restatement of the longest increasing subsequence problem.

This can be solved in quadratic time using linear programming; we calculate in order the longest increasing subsequence for every prefix of the input string:

for (int i = 0; i < n; i++) {
  int best = 1;
  for (int j = 0; j < i; j++)
    if (s[i] > s[j] && 1 + bestv[j] > best)
      best = 1 + bestv[j];
  bestv[i] = best; // best subsequence of length i
}

It is not important for the small input given, but the problem can also be solved in ~\mathcal O(n \log n)~ time by keeping track only of the lowest ending value for subsequences of a given length, locating the correct subsequence to extend by binary search:

for (char c: s) {
  auto it = lower_bound(ec.begin(), ec.end(), c);
  if (it == ec.end())
    ec.push_back(c);
  else
    *it = c;
}

Because of the limited range of the input (only ~26~ different characters), this is actually a linear-time solution in this particular case.