CSES Solutions – Minimal Rotation

A rotation of a string can be generated by moving characters one after another from beginning to end. For example, the rotations of acab are acab, caba, abac, and baca.

Your task is to determine the lexicographically minimal rotation of a string.

Examples:

Input: s = “acab”
Output: abac
Explanation: All possible rotations are: “acab”, “caba”, “abac” and “baca”. The lexicographically smallest string is “abac”.

Input: s = “GeeksforGeeks”
Output: GeeksGeeksfor
Explanation: Out of possible rotations “GeeksGeeksfor” is lexicographically smallest.

Approach: To solve the problem, follow the below idea:

The idea is to uses Booth’s algorithm, which is a well-known algorithm for this problem. The main idea of Booth’s algorithm is to construct a failure function (similar to the one used in the KMP pattern matching algorithm) for the given string concatenated with itself. This failure function is used to efficiently compare prefixes of the string to find the smallest rotation.

Let’s break our idea in some steps:

• Concatenate the string to itself: This allows us to easily handle the wrap-around nature of rotations without having to use modular arithmetic.
  
• Initialize the failure function and the minimum rotation index: The failure function f is an array that stores the length of the longest proper suffix of the substring S[0..j] which is also a proper prefix of S. The minimum rotation index k keeps track of the starting index of the smallest rotation found so far.
  
• Iterate over the concatenated string: For each character S[j] in the string, we compare it with the character at the corresponding position in the current smallest rotation. If S[j] is smaller, we update the minimum rotation index k.
  
• Update the failure function: The failure function is updated based on whether S[j] is equal to S[k + f[j – k – 1] + 1] . If they are equal, f[j – k] is set to f[j – k – 1] + 1. Otherwise, f[j – k] is set to -1.
  
• Return the minimum rotation index: After iterating over the entire string, the minimum rotation index k gives the starting index of the lexicographically smallest rotation.

Step-by-step algorithm:

• Create a function findMinRotation() takes a string s as input and finds the lexicographically minimal rotation of the string.
• It concatenates the input string with itself to avoid modular arithmetic.
• Create a failure function vector with -1 values and sets the initial minimum rotation index to 0.
• Iterates over the concatenated string to find the lexicographically minimal rotation.
• Updates the failure function and minimum rotation index based on comparisons.
• Returns the index of the lexicographically minimal rotation.

Below is the implementation of the algorithm:

C++

#include <bits/stdc++.h>
using namespace std;

// Function to find the lexicographically minimal rotation
// of a string
int findMinRotation(string s)
{
    // Concatenate string to itself to avoid modular
    // arithmetic
    s += s;

    // Initialize failure function
    vector<int> failureFunc(s.size(), -1);

    // Initialize least rotation of string found so far
    int minRotationIdx = 0;

    // Iterate over the concatenated string
    for (int currentIdx = 1; currentIdx < s.size();
         ++currentIdx) {
        char currentChar = s[currentIdx];
        int failureIdx
            = failureFunc[currentIdx - minRotationIdx - 1];

        // Find the failure function value
        while (
            failureIdx != -1
            && currentChar
                   != s[minRotationIdx + failureIdx + 1]) {
            if (currentChar
                < s[minRotationIdx + failureIdx + 1]) {
                minRotationIdx
                    = currentIdx - failureIdx - 1;
            }
            failureIdx = failureFunc[failureIdx];
        }

        // Update the failure function and the minimum
        // rotation index
        if (currentChar
            != s[minRotationIdx + failureIdx + 1]) {
            if (currentChar < s[minRotationIdx]) {
                minRotationIdx = currentIdx;
            }
            failureFunc[currentIdx - minRotationIdx] = -1;
        }
        else {
            failureFunc[currentIdx - minRotationIdx]
                = failureIdx + 1;
        }
    }

    // Return the index of the lexicographically minimal
    // rotation
    return minRotationIdx;
}

int main()
{
    // input string
    string s = "acab";

    // Find the lexicographically minimal rotation
    int minRotationIdx = findMinRotation(s);

    // Print the lexicographically minimal rotation
    cout << s.substr(minRotationIdx)
                + s.substr(0, minRotationIdx)
         << endl;

    return 0;
}

Java

import java.util.Arrays;

public class Main {
    // Function to find the lexicographically minimal rotation of a string
    static int findMinRotation(String s) {
        // Concatenate string to itself to avoid modular arithmetic
        s += s;

        // Initialize failure function
        int[] failureFunc = new int[s.length()];
        Arrays.fill(failureFunc, -1);

        // Initialize least rotation of string found so far
        int minRotationIdx = 0;

        // Iterate over the concatenated string
        for (int currentIdx = 1; currentIdx < s.length(); ++currentIdx) {
            char currentChar = s.charAt(currentIdx);
            int failureIdx = failureFunc[currentIdx - minRotationIdx - 1];

            // Find the failure function value
            while (failureIdx != -1 && currentChar != s.charAt(minRotationIdx + failureIdx + 1)) {
                if (currentChar < s.charAt(minRotationIdx + failureIdx + 1)) {
                    minRotationIdx = currentIdx - failureIdx - 1;
                }
                failureIdx = failureFunc[failureIdx];
            }

            // Update the failure function and the minimum rotation index
            if (currentChar != s.charAt(minRotationIdx + failureIdx + 1)) {
                if (currentChar < s.charAt(minRotationIdx)) {
                    minRotationIdx = currentIdx;
                }
                failureFunc[currentIdx - minRotationIdx] = -1;
            } else {
                failureFunc[currentIdx - minRotationIdx] = failureIdx + 1;
            }
        }

        // Return the index of the lexicographically minimal rotation
        return minRotationIdx;
    }

    public static void main(String[] args) {
        // Input string
        String s = "acab";

        // Find the lexicographically minimal rotation
        int minRotationIdx = findMinRotation(s);

        // Print the lexicographically minimal rotation
        System.out.println(s.substring(minRotationIdx) + s.substring(0, minRotationIdx));
    }
}

// This code is contributed by shivamgupta310570

Python

def find_min_rotation(s):
    # Concatenate string to itself to avoid modular arithmetic
    s += s
    
    # Initialize failure function
    failure_func = [-1] * len(s)
    
    # Initialize least rotation of string found so far
    min_rotation_idx = 0
    
    # Iterate over the concatenated string
    for current_idx in range(1, len(s)):
        current_char = s[current_idx]
        failure_idx = failure_func[current_idx - min_rotation_idx - 1]
        
        # Find the failure function value
        while (failure_idx != -1 and 
               current_char != s[min_rotation_idx + failure_idx + 1]):
            if current_char < s[min_rotation_idx + failure_idx + 1]:
                min_rotation_idx = current_idx - failure_idx - 1
            failure_idx = failure_func[failure_idx]
        
        # Update the failure function and the minimum rotation index
        if current_char != s[min_rotation_idx + failure_idx + 1]:
            if current_char < s[min_rotation_idx]:
                min_rotation_idx = current_idx
            failure_func[current_idx - min_rotation_idx] = -1
        else:
            failure_func[current_idx - min_rotation_idx] = failure_idx + 1
    
    # Return the index of the lexicographically minimal rotation
    return min_rotation_idx

def main():
    # Input string
    s = "acab"
    
    # Find the lexicographically minimal rotation
    min_rotation_idx = find_min_rotation(s)
    
    # Print the lexicographically minimal rotation
    print(s[min_rotation_idx:] + s[:min_rotation_idx])

if __name__ == "__main__":
    main()

JavaScript

// Function to find the lexicographically minimal rotation
// of a string
function findMinRotation(s) {
    // Concatenate string to itself to avoid modular
    // arithmetic
    s += s;

    // Initialize failure function
    const failureFunc = new Array(s.length).fill(-1);

    // Initialize least rotation of string found so far
    let minRotationIdx = 0;

    // Iterate over the concatenated string
    for (let currentIdx = 1; currentIdx < s.length; ++currentIdx) {
        const currentChar = s.charAt(currentIdx);
        let failureIdx = failureFunc[currentIdx - minRotationIdx - 1];

        // Find the failure function value
        while (failureIdx !== -1 && currentChar !== s.charAt(minRotationIdx + failureIdx + 1)) {
            if (currentChar < s.charAt(minRotationIdx + failureIdx + 1)) {
                minRotationIdx = currentIdx - failureIdx - 1;
            }
            failureIdx = failureFunc[failureIdx];
        }

        // Update the failure function and the minimum
        // rotation index
        if (currentChar !== s.charAt(minRotationIdx + failureIdx + 1)) {
            if (currentChar < s.charAt(minRotationIdx)) {
                minRotationIdx = currentIdx;
            }
            failureFunc[currentIdx - minRotationIdx] = -1;
        } else {
            failureFunc[currentIdx - minRotationIdx] = failureIdx + 1;
        }
    }

    // Return the index of the lexicographically minimal
    // rotation
    return minRotationIdx;
}

// Main function
function main() {
    // input string
    const s = "acab";

    // Find the lexicographically minimal rotation
    const minRotationIdx = findMinRotation(s);

    // Print the lexicographically minimal rotation
    console.log(s.substring(minRotationIdx) + s.substring(0, minRotationIdx));
}

// Call the main function to execute the program
main();

Output

abac

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