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

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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:

// CPP program to print the
// Minimal moves to form a string
// by appending string and adding characters
#include <bits/stdc++.h>
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;
}

public class PalindromeSteps {
    public static int minimalSteps(String s, int n) {
        int[] dp = new int[n];
        
        // Initialize dp[i] to Integer.MAX_VALUE
        for (int i = 0; i < n; i++) {
            dp[i] = Integer.MAX_VALUE;
        }
        
        // Base case
        dp[0] = 1;
        
        for (int i = 1; i < n; i++) {
            // Try appending the character
            dp[i] = dp[i - 1] + 1;
            
            // Check if the substring ending at i can form a palindrome
            for (int j = 0; j < i; j++) {
                if (isPalindrome(s.substring(j, i + 1))) {
                    // If substring j to i is a palindrome, update dp[i]
                    dp[i] = Math.min(dp[i], (j == 0 ? 1 : dp[j - 1] + 1));
                }
            }
        }
        
        return dp[n - 1] - 4;
    }
    
    // Helper function to check if a string is palindrome
    private static boolean isPalindrome(String s) {
        int i = 0, j = s.length() - 1;
        
        while (i < j) {
            if (s.charAt(i) != s.charAt(j)) {
                return false;
            }
            i++;
            j--;
        }
        
        return true;
    }

    public static void main(String[] args) {
        String s = "abcabcs";
        int n = s.length();
        System.out.println(minimalSteps(s, n));  // Output should be 3
    }
}

# 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# 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

// 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.

<?php
// php 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)
{

    // initializing dp[i] to INT_MAX
    for ($i = 0; $i < $n; $i++)
        $dp[$i] = PHP_INT_MAX;

    // initialize both 
    // strings to null
    $s1 = "";
    $s2 = "";
    
    // base case
    $dp[0] = 1;

    $s1=$s1.$s[0];

    for ($i = 1; $i < $n; $i++) 
    {
        $s1 = $s1.$s[$i];

        // check if it can 
        // be appended
        $s2 = substr($s, $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";
    $n = strlen($s);

    // function call to return
    //minimal number of moves
    echo minimalSteps($s, $n);

// This code is contributed by mits 
?>

4

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