Minimum time to write characters using insert, delete and copy operation

We need to write N same characters on a screen and each time we can insert a character, delete the last character and copy and paste all written characters i.e. after copy operation count of total written character will become twice. Now we are given time for insertion, deletion and copying. We need to output minimum time to write N characters on the screen using these operations.

Examples:

Input : N = 9    
        insert time = 1    
        removal time = 2    
        copy time = 1
Output : 5
N character can be written on screen in 5 time units as shown below,
insert a character    
characters = 1  total time = 1
again insert character      
characters = 2  total time = 2
copy characters             
characters = 4  total time = 3
copy characters             
characters = 8  total time = 4
insert character           
characters = 9  total time = 5



We can solve this problem using dynamic programming. We can observe a pattern after solving some examples by hand that for writing each character we have two choices either get it by inserting or get it by copying, whichever takes less time. Now writing relation accordingly,
Let dp[i] be the optimal time to write i characters on screen then,

If i is even then,
   dp[i] = min((dp[i-1] + insert_time), 
               (dp[i/2] + copy_time))
Else (If i is odd)
   dp[i] = min(dp[i-1] + insert_time),
              (dp[(i+1)/2] + copy_time + removal_time) 

In the case of odd, removal time is added because when (i+1)/2 characters will be copied one extra character will be on the screen which needs to be removed. Total time complexity of solution will be O(N) and auxiliary space needed will be O(N).

// C++ program to write characters in
// minimum time by inserting, removing
// and copying operation
#include <bits/stdc++.h>
using namespace std;
  
//  method returns minimum time to write
// 'N' characters
int minTimeForWritingChars(int N, int insert,
                       int remove, int copy)
{
    if (N == 0)
       return 0;
    if (N == 1)
       return insert;
  
    //  declare dp array and initialize with zero
    int dp[N + 1];
    memset(dp, 0, sizeof(dp));
  
    //  loop for 'N' number of times
    for (int i = 1; i <= N; i++)
    {
        /*  if current char count is even then
            choose minimum from result for (i-1)
            chars and time for insertion and
            result for half of chars and time
            for copy  */
        if (i % 2 == 0)
            dp[i] = min(dp[i-1] + insert,
                        dp[i/2] + copy);
  
        /*  if current char count is odd then
            choose minimum from
            result for (i-1) chars and time for
            insertion and
            result for half of chars and time for
            copy and one extra character deletion*/
        else
            dp[i] = min(dp[i-1] + insert,
                        dp[(i+1)/2] + copy + remove);
    }
    return dp[N];
}
  
// Driver code to test above methods
int main()
{
    int N = 9;
    int insert = 1, remove = 2, copy = 1;
    cout << minTimeForWritingChars(N, insert,
                                remove, copy);
    return 0;
}

Output:



5

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