Let's discuss the last example which pretty much covers the other cases as well. The thing to note is we can only add, we can't delete. So, we need to insert the corresponding bracket whenever there is a violation. We can track the violation using stack similar way we check for valid parenthesis.

The algorithm will be like follow,

1. Create a stack to store characters, stack<char> st;
2. Initialize the count of minimum numbers to be added, count=0;

3. for i=0 to s.length()-1
       if(s[i]  is  opening bracket )
           push to stack
       else{ //s[i] is closing bracket
           if stack is empty{
               increment count as its a violation
               continue;
           }
           else if(s[i] is closing bracket but the stack top is not opening bracket){
               // it's a violation and increment count;               
               st.pop();
           }
       }
   

4. If there is still some opening brackets left in the stack they needs to be paired as well, so

   count=count+stack size;
   

   Stack size may be zero too
   Count gives the final result

Now let's execute for our example,

String s="()))(("
Initially count=0
Stack is empty

i=0
s[i]=('
push to stack
stack
--------
 (
--------
Count=0

i=1
s[i]=')'
s[i] correctly pairs with top of stack
pop from stack
stack is empty now
--------

--------
Count=0


i=2
s[i]=')'
stack is empty, nothing to pair, 
it's a violation and we need to add '(' here
--------

--------
increment count
Count=1

i=3
s[i]=')'
stack is empty, nothing to pair, 
it's a violation again and we need to add '(' here
--------

--------
increment count
Count=2


i=4
s[i]=('
push to stack
stack
--------
(
--------
Count=2

i=5
s[i]=('
push to stack
stack
--------
(
--------
(
--------
Count=2

That's end the loop

Out of the loop stack is not empty 
and we need to add two closing brackets 
to pair up those remaining open brackets

Hence, count = count + stack size = 4

That's the final result

C++ Implementation:

#include <bits/stdc++.h>
using namespace std;

int minAddToMakeValid(string s)
{
    stack<char> st;
    int count = 0;
    for (int i = 0; i < s.length(); i++) {
        if (s[i] == '(')
            st.push(s[i]);
        else {
            if (st.empty()) {
                count++;
                continue;
            }

            else if (s[i] == ')' && st.top() != '(')
                count++;

            st.pop();
        }
    }

    count += st.size();

    return count;
}

int main()
{
    string s;
    cout << "Enter the parenthesis string\n";
    cin >> s;
 
    cout << "Minimum brackets to add: " << minAddToMakeValid(s) << endl;

    return 0;
}

Output:

RUN 1:
Enter the parenthesis string
(())))((
Minimum brackets to add: 4

RUN 2:
Enter the parenthesis string
()(((((((((()
Minimum brackets to add: 9