Group Anagrams in Python with O(n * log n * L * log L) Time Complexity

Given an array of strings, group the anagrams together. You can return the answer in any order.

An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, using all the original letters exactly once.

Example:

Input:   strings = ["eat","tea","tan","ate","nat","bat"]

Output:  [["bat"],["nat","tan"],["ate","eat","tea"]]

For this lesson, your algorithm should run in O(n * log n * L * log L) time and use O(n * L) extra space.
(There are faster solutions which we will discuss in future lessons)

Understanding the Problem

The core challenge of this problem is to identify and group words that are anagrams of each other. An anagram is a rearrangement of the letters of a word to form another word. For example, "eat" and "tea" are anagrams because they contain the same letters.

Common applications of this problem include text analysis, cryptography, and natural language processing. A potential pitfall is not recognizing that different words with the same letters should be grouped together.

Approach

To solve this problem, we need to identify a way to group words that are anagrams. A naive solution might involve comparing each word with every other word, but this would be inefficient.

Instead, we can use a more optimized approach:

• Sort each word alphabetically. Anagrams will result in the same sorted string.
• Use a dictionary to group words by their sorted string.

This approach is efficient because sorting each word takes O(L log L) time, and we do this for each of the n words, resulting in a total time complexity of O(n * L log L).

Algorithm

Here is a step-by-step breakdown of the algorithm:

1. Initialize an empty dictionary to store groups of anagrams.
2. Iterate through each word in the input list.
3. Sort the characters of the word to form a key.
4. Use the sorted word as a key in the dictionary and append the original word to the list of anagrams for that key.
5. Return the values of the dictionary as the grouped anagrams.

Code Implementation

def group_anagrams(strings):
    # Dictionary to hold the groups of anagrams
    anagrams = {}
    
    for word in strings:
        # Sort the word to form the key
        sorted_word = ''.join(sorted(word))
        
        # Add the word to the corresponding anagram group
        if sorted_word in anagrams:
            anagrams[sorted_word].append(word)
        else:
            anagrams[sorted_word] = [word]
    
    # Return the grouped anagrams
    return list(anagrams.values())

# Example usage
strings = ["eat", "tea", "tan", "ate", "nat", "bat"]
print(group_anagrams(strings))

Complexity Analysis

The time complexity of this approach is O(n * L log L), where n is the number of words and L is the maximum length of a word. Sorting each word takes O(L log L) time, and we do this for each of the n words.

The space complexity is O(n * L) because we store each word in the dictionary.

Edge Cases

Potential edge cases include:

• Empty input list: The function should return an empty list.
• Single word: The function should return a list containing a single list with that word.
• All words are anagrams: The function should return a single list containing all words.

Examples:

# Edge case: Empty input list
print(group_anagrams([]))  # Output: []

# Edge case: Single word
print(group_anagrams(["word"]))  # Output: [["word"]]

# Edge case: All words are anagrams
print(group_anagrams(["abc", "bca", "cab"]))  # Output: [["abc", "bca", "cab"]]

Testing

To test the solution comprehensively, we should include a variety of test cases:

• Simple cases with a few words.
• Cases with no anagrams.
• Cases with multiple groups of anagrams.
• Edge cases as discussed above.

Using a testing framework like unittest in Python can help automate and organize these tests.

Thinking and Problem-Solving Tips

When approaching such problems, it's important to:

• Understand the problem requirements and constraints.
• Think about different ways to group or categorize the input.
• Consider using data structures like dictionaries to efficiently group data.
• Practice solving similar problems to improve problem-solving skills.

Conclusion

Grouping anagrams is a common problem that can be efficiently solved using sorting and dictionaries. Understanding and implementing this solution helps improve problem-solving skills and prepares you for more complex challenges.

Practice and explore further to deepen your understanding and proficiency in solving such problems.

Additional Resources

• LeetCode: Group Anagrams
• GeeksforGeeks: Group Anagrams Together
• Real Python: Python Dictionaries