Minimal moves to form a string by adding characters or appending string itself

Given a string S, we need to write a program to check if it is possible to construct the given string S by performing any of the below operations any number of times. In each step, we can:

• Add any character at the end of the string.
• or, append the string to the string itself.

The above steps can be applied any number of times. We need to write a program to print the minimum steps required to form the string. Examples:

Input : aaaaaaaa
Output : 4 
Explanation: move 1: add 'a' to form "a"
 move 2: add 'a' to form "aa"
 move 3: append "aa" to form "aaaa" 
 move 4: append "aaaa" to form "aaaaaaaa" 

Input: aaaaaa
Output: 4 
Explanation: move 1: add 'a' to form "a"
 move 2: add 'a' to form "aa"
 move 3: add 'a' to form "aaa" 
 move 4: append "aaa" to form "aaaaaa" 

Input: abcabca
Output: 5 

The idea to solve this problem is to use Dynamic Programming to count the minimum number of moves. Create an array named dp of size n, where n is the length of the input string. dp[i] stores the minimum number of moves that are required to make substring (0...i). According to the question there are two moves that are possible:

1. dp[i] = min(dp[i], dp[i-1] + 1) which signifies addition of characters.
2. dp[i*2+1] = min(dp[i]+1, dp[i*2+1]), appending of string is done if s[0...i]==s[i+1..i*2+1]

The answer will be stored in dp[n-1] as we need to form the string(0..n-1) index-wise. 

Below is the implementation of the above idea:

4

Time Complexity: O(n²), where n is the length of the input string.
Auxiliary Space: O(n)