# 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:

## C++

 `// CPP program to print the` `// Minimal moves to form a string` `// by appending string and adding characters` `#include ` `using` `namespace` `std;`   `// function to return the minimal number of moves` `int` `minimalSteps(string s, ``int` `n)` `{`   `    ``int` `dp[n];`   `    ``// initializing dp[i] to INT_MAX` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``dp[i] = INT_MAX;`   `    ``// initialize both strings to null` `    ``string s1 = ``""``, s2 = ``""``;` `    `  `    ``// base case` `    ``dp[0] = 1;`   `    ``s1 += s[0];`   `    ``for` `(``int` `i = 1; i < n; i++) {` `        ``s1 += s[i];`   `        ``// check if it can be appended` `        ``s2 = s.substr(i + 1, i + 1);`   `        ``// addition of character takes one step` `        ``dp[i] = min(dp[i], dp[i - 1] + 1);`   `        ``// appending takes 1 step, and we directly` `        ``// reach index i*2+1 after appending` `        ``// so the number of steps is stored in i*2+1` `        ``if` `(s1 == s2)` `            ``dp[i * 2 + 1] = min(dp[i] + 1, dp[i * 2 + 1]);` `    ``}`   `    ``return` `dp[n - 1];` `}`   `// Driver Code` `int` `main()` `{`   `    ``string s = ``"aaaaaaaa"``;` `    ``int` `n = s.length();`   `    ``// function call to return minimal number of moves` `    ``cout << minimalSteps(s, n);`   `    ``return` `0;` `}`

## Java

 `// Java program to print the` `// Minimal moves to form a string` `// by appending string and adding characters` `import` `java.util.*;`   `class` `GFG` `{`   `// function to return the minimal number of moves` `static` `int` `minimalSteps(String s, ``int` `n)` `{`   `    ``int` `[]dp = ``new` `int``[n];` `    `  `    ``// initializing dp[i] to INT_MAX` `    ``for` `(``int` `i = ``0``; i < n; i++)` `        ``dp[i] = Integer.MAX_VALUE;`   `    ``// initialize both strings to null` `    ``String s1 = ``""``, s2 = ``""``;` `    `  `    ``// base case` `    ``dp[``0``] = ``1``;`   `    ``s1 += s.charAt(``0``);`   `    ``for` `(``int` `i = ``1``; i < n; i++)` `    ``{` `        ``s1 += s.charAt(i);`   `        ``// check if it can be appended` `        ``s2 = s.substring(i + ``1``, i + ``1``);`   `        ``// addition of character takes one step` `        ``dp[i] = Math.min(dp[i], dp[i - ``1``] + ``1``);`   `        ``// appending takes 1 step, and we directly` `        ``// reach index i*2+1 after appending` `        ``// so the number of steps is stored in i*2+1` `        ``if` `(s1 == s2)` `            ``dp[i * ``2` `+ ``1``] = Math.min(dp[i] + ``1``, ` `                                   ``dp[i * ``2` `+ ``1``]);` `    ``}` `    ``return` `dp[n - ``1``];` `}`   `// Driver Code` `public` `static` `void` `main(String args[])` `{`   `    ``String s = ``"aaaaaaaa"``;` `    ``int` `n = s.length();`   `    ``// function call to return minimal number of moves` `    ``System.out.println(minimalSteps(s, n)/``2``);` `}` `}`   `// This code is contributed by ` `// Shashank_Sharma`

## Python3

 `# Python program to print the ` `# Minimal moves to form a string ` `# by appending string and adding characters `   `INT_MAX ``=` `100000000`   `# function to return the ` `# minimal number of moves ` `def` `minimalSteps(s, n):` `    ``dp ``=` `[INT_MAX ``for` `i ``in` `range``(n)] ` `    `  `    ``# initialize both strings to null ` `    ``s1 ``=` `""` `    ``s2 ``=` `""` `    `  `    ``# base case ` `    ``dp[``0``] ``=` `1` `    ``s1 ``+``=` `s[``0``]` `    `  `    ``for` `i ``in` `range``(``1``, n):` `        ``s1 ``+``=` `s[i]` `    `  `        ``# check if it can be appended ` `        ``s2 ``=` `s[i ``+` `1``: i ``+` `1` `+` `i ``+` `1``]` `    `  `        ``# addition of character ` `        ``# takes one step ` `        ``dp[i] ``=` `min``(dp[i], dp[i ``-` `1``] ``+` `1``)` `    `  `        ``# appending takes 1 step, and ` `        ``# we directly reach index i*2+1 ` `        ``# after appending so the number` `        ``# of steps is stored in i*2+1 ` `        ``if` `(s1 ``=``=` `s2): ` `            ``dp[i ``*` `2` `+` `1``] ``=` `min``(dp[i] ``+` `1``, ` `                                ``dp[i ``*` `2` `+` `1``])` `    `  `    ``return` `dp[n ``-` `1``]`   `# Driver Code ` `s ``=` `"aaaaaaaa"` `n ``=``len``(s)`   `# function call to return ` `# minimal number of moves ` `print``( minimalSteps(s, n) ) `   `# This code is contributed ` `# by sahilshelangia`

## C#

 `// C# program to print the` `// Minimal moves to form a string` `// by appending string and adding characters` `using` `System;` `    `  `class` `GFG` `{`   `// function to return the minimal number of moves` `static` `int` `minimalSteps(String s, ``int` `n)` `{`   `    ``int` `[]dp = ``new` `int``[n];` `    `  `    ``// initializing dp[i] to INT_MAX` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``dp[i] = ``int``.MaxValue;`   `    ``// initialize both strings to null` `    ``String s1 = ``""``, s2 = ``""``;` `    `  `    ``// base case` `    ``dp[0] = 1;`   `    ``s1 += s[0];`   `    ``for` `(``int` `i = 1; i < n; i++)` `    ``{` `        ``s1 += s[i];`   `        ``// check if it can be appended` `        ``s2 = s.Substring(i , 1);`   `        ``// addition of character takes one step` `        ``dp[i] = Math.Min(dp[i], dp[i - 1] + 1);`   `        ``// appending takes 1 step, and we directly` `        ``// reach index i*2+1 after appending` `        ``// so the number of steps is stored in i*2+1` `        ``if` `(s1 == s2)` `            ``dp[i * 2 + 1] = Math.Min(dp[i] + 1, ` `                                ``dp[i * 2 + 1]);` `    ``}` `    ``return` `dp[n - 1];` `}`   `// Driver Code` `public` `static` `void` `Main(String []args)` `{`   `    ``String s = ``"aaaaaaaa"``;` `    ``int` `n = s.Length;`   `    ``// function call to return minimal number of moves` `    ``Console.Write(minimalSteps(s, n)/2);` `}` `}`   `// This code has been contributed by 29AjayKumar`

## PHP

 ``

## Javascript

 `// Javascript program to print the` `// Minimal moves to form a string` `// by appending string and adding characters`   `// function to return the minimal number of moves` `function` `minimalSteps(s, n)` `{` `    ``let dp = [n]`   `    ``// initializing dp[i] to INT_MAX` `    ``for` `(let i = 0; i < n; i++)` `        ``dp[i] = Number.MAX_SAFE_INTEGER`   `    ``// initialize both strings to null` `    ``let s1 = ``""` `    ``let s2 = ``""` `    `  `    ``// base case` `    ``dp[0] = 1`   `    ``s1 += s[0]`   `    ``for` `(let i = 1; i < n; i++) {` `        ``s1 += s[i]`   `        ``// check if it can be appended` `        ``s2 = s.substr(i + 1, i + 1);`   `        ``// addition of character takes one step` `        ``dp[i] = Math.min(dp[i], dp[i - 1] + 1);`   `        ``// appending takes 1 step, and we directly` `        ``// reach index i*2+1 after appending` `        ``// so the number of steps is stored in i*2+1` `        ``if` `(s1 == s2)` `            ``dp[i * 2 + 1] = Math.min(dp[i] + 1, dp[i * 2 + 1])` `    ``}`   `    ``return` `dp[n - 1]` `}`   `// Driver Code` `let s = ``"aaaaaaaa"` `let n = s.length`   `// function call to return minimal number of moves` `console.log(minimalSteps(s, n))`   `// This code is contributed by Samim Hossain Mondal.`

Output

`4`

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

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