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.
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).
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