Maxsubsequence - What are the key insights for this problem?

Question

Below is the problem assignment using tree recursion approach:

Maximum Subsequence A subsequence of a number is a series of (not necessarily contiguous) digits of the number. For example, 12345 has subsequences that include 123, 234, 124, 245, etc. Your task is to get the maximum subsequence below a certain length.

def max_subseq(n, l):
    """
    Return the maximum subsequence of length at most l that can be found in the given number n.
    For example, for n = 20125 and l = 3, we have that the subsequences are
        2
        0
        1
        2
        5
        20
        21
        22
        25
        01
        02
        05
        12
        15
        25
        201
        202
        205
        212
        215
        225
        012
        015
        025
        125
    and of these, the maxumum number is 225, so our answer is 225.

    >>> max_subseq(20125, 3)
    225
    >>> max_subseq(20125, 5)
    20125
    >>> max_subseq(20125, 6) # note that 20125 == 020125
    20125
    >>> max_subseq(12345, 3)
    345
    >>> max_subseq(12345, 0) # 0 is of length 0
    0
    >>> max_subseq(12345, 1)
    5
    """
    "*** YOUR CODE HERE ***"

---

There are two key insights for this problem

• You need to split into the cases where the ones digit is used and the one where it is not. In the case where it is, we want to reduce l since we used one of the digits, and in the case where it isn't we do not.
• In the case where we are using the ones digit, you need to put the digit back onto the end, and the way to attach a digit d to the end of a number n is 10 * n + d.

---

I could not understand the insights of this problem, mentioned below 2 points:

1. split into the cases where the ones digit is used and the one where it is not

2. In the case where we are using the ones digit, you need to put the digit back onto the end

---

My understanding of this problem:

Solution to this problem looks to generate all subsequences upto l, pseudo code looks like:

digitSequence := strconv.Itoa(n) // "20125"

printSubSequence = func(digitSequence string, currenSubSequenceSize int) { // digitSequence is "20125" and currenSubSequenceSize is say 3
        printNthSubSequence(digitSequence, currenSubSequenceSize) + printSubSequence(digitSequence, currenSubSequenceSize-1)
    }

where printNthSubSequence prints subsequences for (20125, 3) or (20125, 2) etc...

Finding max_subseq among all these sequences then becomes easy

---

Can you help me understand the insights given in this problem, with an example(say 20125, 1)? here is the complete question

Answer 1

Something like this? As the instructions suggest, try it with and without the current digit:

function f(s, i, l){
  if (i + 1 <= l)
    return Number(s.substr(0, l));

  if (!l)
    return 0;
    
  return Math.max(
    // With
    Number(s[i]) + 10 * f(s, i - 1, l - 1),
    // Without
    f(s, i - 1, l)
  );
}

var input = [
  ['20125', 3],
  ['20125', 5],
  ['20125', 6],
  ['12345', 3],
  ['12345', 0],
  ['12345', 1]
];

for (let [s, l] of input){
  console.log(s + ', l: ' + l);
  console.log(f(s, s.length-1, l));
  console.log('');
}