Word Ladder Solver

Use the solver below to find the shortest word ladder from one word to another, if one exists. Select the "Use Common Words" option to find a word ladder solution using more common dictionary words.

Shortest Path Solver

To solve the problem of finding the shortest path from the Start Word to the Target Word, we can model the problem as a graph traversal. Each word can be considered a node, and an edge exists between two nodes if one word can be transformed into the other by changing exactly one letter.

The breadth-first search algorithm (BFS) is ideal for this type of problem since it explores all nodes at the present depth level before moving on to nodes at the next depth level, guaranteeing that we find the shortest path.

The breadth-first search works as follows:

• We initialize a queue with the starting word, and each element in the queue is an array representing a path from the starting word to a current word.
• We use a set to track the words we've already visited to avoid revisiting them.
• At each step, we dequeue the current path, get the last word in that path, and explore all words in the word list that differ by exactly one letter.
• We check to see if the word is in the visited set. If not, we add a new array to the back of the queue that contains the updated path.
• If we find the target word, we return the path that leads to it. If we exhaust the queue without finding the target word, then no path exists.

The time complexity is O(N * M), where N is the number of words in the word list and M is the length of each word. This is because, for each word in the list, we are comparing it to the current word (which takes O(M) time).

The space complexity is O(N * M), as we store the entire word list and the paths in the queue.

Common Words Solver

The Common Words Solver additionaly takes word frequency into account so that the solver produces a path using words most people know and use everyday. Each word is assigned a frequency rating, where the lower the frequency rating, the more common the word. To modify the other search tree to find the path with the lowest sum of frequency ratings, or the shortest path using only common words, we can use a modified version of Dijkstra's algorithm. Instead of finding the shortest path based on the number of steps (as in BFS), we will now find the path with the lowest total frequency rating, where the frequency rating of each word is a weight on the node.

Here are the main differences from the Shortest Path Solver:

• Graph representation: Each word is a node, and an edge exists between two words if they differ by exactly one letter. Each word has a frequency rating, and we want to minimize the sum of these ratings along the path from the starting word to the target word.
• Modified Dijkstra's algorithm: Instead of using a simple queue like BFS, we will use a priority queue (min-heap) to always process the path with the lowest cumulative frequency score first.

The modified search tree using Dijkstra's algorithm works as follows:

• Begin at the starting word with a frequency rating of 0.
• For each word, look at all words that differ by one letter.
• If a word hasn't been visited or if visiting it via the current word results in a lower cumulative frequency rating, update its rating and add it to the priority queue.
• The process ends when we reach the target word or exhaust all possibilities.

The time complexity is O(N * M * log(N)), where N is the number of words in the word list and M is the length of each word. Sorting the priority queue takes O(log(N)), and for each word, we check all the words in the list, leading to the total time complexity.

The space complexity is O(N), where N is the number of words in the word list, because we store each word and its associated path and frequency sum in the priority queue and visited set.