The Word Ladder problem is a classic algorithmic challenge that tests one’s ability to navigate transformations within a given set of constraints. Utilizing the Breadth-First Search (BFS) algorithm, we can efficiently find the shortest transformation sequence from a start word to an end word. This article provides a comprehensive guide to implementing a BFS-based solution to the Word Ladder problem in Python, complete with detailed explanations and documentation.

Understanding the Word Ladder Problem

The Word Ladder problem, originally posed by Lewis Carroll, involves transforming one word into another by altering a single letter at a time. Each intermediate word must exist within a provided dictionary of valid words. The objective is to determine the minimum number of steps required to achieve this transformation.

Problem Statement

Given two words, beginWord and endWord, and a dictionary’s word list, find the length of the shortest transformation sequence from beginWord to endWord, such that:

• Only one letter can be changed at a time.
• Each transformed word must exist in the word list.

Note:

• Return 0 if there is no such transformation sequence.
• All words have the same length.
• All words contain only lowercase alphabetic characters.

Approach to Solving the Word Ladder Problem

To solve the Word Ladder problem efficiently, we employ the Breadth-First Search (BFS) algorithm. BFS is ideal for finding the shortest path in unweighted graphs, which aligns with our goal of determining the minimal number of transformations.

Why BFS?

BFS explores all nodes (in this case, words) at the present depth before moving on to nodes at the next depth level. This ensures that the first time we reach the endWord, we have done so via the shortest possible path.

Algorithm Steps

1. Initialization: Start by initializing a queue for BFS and a set for the word list to allow O(1) lookups.
2. Level Tracking: Use a variable to keep track of the number of transformation steps.
3. BFS Traversal:
   • Dequeue the current word from the queue.
   • For each character in the current word, try changing it to every other possible lowercase letter.
   • If the new word is the endWord, return the current transformation step count plus one.
   • If the new word exists in the word list, enqueue it and remove it from the word list to prevent revisiting.
4. No Transformation Found: If the queue is exhausted without finding the endWord, return 0.

Python Program: Word Ladder Using BFS

# Word Ladder Problem Solution using BFS
# This program finds the length of the shortest transformation sequence from beginWord to endWord.

from collections import deque

def ladder_length(beginWord, endWord, wordList):
    """
    Finds the length of the shortest transformation sequence from beginWord to endWord.

    Parameters:
    beginWord (str): The starting word.
    endWord (str): The target word.
    wordList (List[str]): List of valid intermediate words.

    Returns:
    int: The number of words in the shortest transformation sequence, or 0 if no such sequence exists.
    """
    word_set = set(wordList)  # Convert wordList to a set for O(1) lookups
    if endWord not in word_set:
        return 0

    queue = deque([(beginWord, 1)])  # Initialize BFS queue with tuple(word, level)
    while queue:
        current_word, level = queue.popleft()
        if current_word == endWord:
            return level

        # Try changing each character in the current word to find all possible transformations
        for i in range(len(current_word)):
            for c in 'abcdefghijklmnopqrstuvwxyz':
                if c == current_word[i]:
                    continue  # Skip same character replacement
                next_word = current_word[:i] + c + current_word[i+1:]
                if next_word in word_set:
                    queue.append((next_word, level + 1))
                    word_set.remove(next_word)  # Remove to prevent revisiting

    return 0  # Transformation not possible

def main():
    """
    Main function to execute the Word Ladder program.
    """
    # Example 1
    beginWord1 = "hit"
    endWord1 = "cog"
    wordList1 = ["hot","dot","dog","lot","log","cog"]
    print("Example 1:")
    print(f"Begin Word: {beginWord1}")
    print(f"End Word: {endWord1}")
    print(f"Word List: {wordList1}")
    result1 = ladder_length(beginWord1, endWord1, wordList1)
    print(f"Shortest Transformation Sequence Length: {result1}\n")

    # Example 2
    beginWord2 = "hit"
    endWord2 = "cog"
    wordList2 = ["hot","dot","dog","lot","log"]
    print("Example 2:")
    print(f"Begin Word: {beginWord2}")
    print(f"End Word: {endWord2}")
    print(f"Word List: {wordList2}")
    result2 = ladder_length(beginWord2, endWord2, wordList2)
    print(f"Shortest Transformation Sequence Length: {result2}\n")

if __name__ == "__main__":
    main()
    

Program Documentation

Function: ladder_length

This function computes the length of the shortest transformation sequence from beginWord to endWord using BFS.

• Parameters:
  • beginWord (str): The starting word for the transformation.
  • endWord (str): The target word for the transformation.
  • wordList (List[str]): A list containing all valid intermediate words.
• Returns:
  • int: The number of words in the shortest transformation sequence, or 0 if no such sequence exists.

Function: main

The entry point of the program. It defines example scenarios, invokes the ladder_length function, and displays the results.

How to Run the Program

1. 1. Ensure you have Python installed on your system (Python 3.x recommended).
   2. Copy the provided Python code into a file named word_ladder.py.
   3. Open a terminal or command prompt and navigate to the directory containing the word_ladder.py file.
   4. Run the program using the command:

python word_ladder.py

1. The program will execute the defined examples and output the length of the shortest transformation sequences.

Example Output

Example 1:
Begin Word: hit
End Word: cog
Word List: ['hot', 'dot', 'dog', 'lot', 'log', 'cog']
Shortest Transformation Sequence Length: 5

Example 2:
Begin Word: hit
End Word: cog
Word List: ['hot', 'dot', 'dog', 'lot', 'log']
Shortest Transformation Sequence Length: 0
    

Customization

To use the program with different words or word lists, modify the beginWord, endWord, and wordList variables within the main function.

For example, to solve a new word ladder problem:

def main():
    # New Example
    beginWord = "game"
    endWord = "math"
    wordList = ["fame", "fate", "gate", "mate", "math"]
    print("New Example:")
    print(f"Begin Word: {beginWord}")
    print(f"End Word: {endWord}")
    print(f"Word List: {wordList}")
    result = ladder_length(beginWord, endWord, wordList)
    print(f"Shortest Transformation Sequence Length: {result}")
    

Running the program will now evaluate the new example accordingly.

Conclusion

The Word Ladder problem serves as an excellent example of applying BFS to solve real-world challenges involving state transitions and pathfinding. By leveraging BFS, we can efficiently navigate through possible transformations to find the shortest path from a starting word to a target word. This Python implementation demonstrates the effectiveness of BFS in addressing such algorithmic problems, providing a foundation for tackling more complex variations and related challenges.

Explore how to solve the Word Ladder problem using BFS in Python. This guide includes a detailed program structure, comprehensive documentation, and example usage.