# Solution For Word Ladder Ii

Problem Statement:

Given two words (beginWord and endWord), and a dictionary’s word list, find all shortest transformation sequence(s) 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 that beginWord is not a transformed word.

Note:

Return an empty list if there is no such transformation sequence.

All words have the same length.

All words contain only lowercase alphabetic characters.

You may assume no duplicates in the word list.

You may assume beginWord and endWord are non-empty and are not the same.

Example 1:

Input:

beginWord = “hit”,

endWord = “cog”,

wordList = [“hot”,”dot”,”dog”,”lot”,”log”,”cog”]

Output:

[

[“hit”,”hot”,”dot”,”dog”,”cog”],

[“hit”,”hot”,”lot”,”log”,”cog”]
]

Explanation:

There are 2 shortest transformation sequences:

“hit” -> “hot” -> “dot” -> “dog” -> “cog”

“hit” -> “hot” -> “lot” -> “log” -> “cog”

Example 2:

Input:

beginWord = “hit”

endWord = “cog”

wordList = [“hot”,”dot”,”dog”,”lot”,”log”]

Output: []

Explanation: The endWord “cog” is not in wordList, so there is no valid transformation sequence.

Approach:

- Begin by building a dictionary in order to retrieve the list of neighbors for each word from the list of words.
- In order to find all possible word ladders, we need to use a BFS approach in order to generate all possible word ladders from the beginWord to the endWord.
- The idea is to start from the beginWord and adding to the queue with the first word in the ladder. Then, we add each of its neighbors to the queue as the second word in the ladder. We continue this process until we reach the endWord.
- Along with each word, we add its ladder to the queue.
- We use a visited set to ensure that we don’t visit the same word again.
- Finally, we return all the possible word ladders.

Solution:

from collections import defaultdict, deque

def findLadders(beginWord: str, endWord: str, wordList: List[str]) -> List[List[str]]:

```
# build a dictionary of all possible combinations of words
word_dict = defaultdict(list)
for word in wordList:
for i in range(len(word)):
word_dict[word[:i] + "*" + word[i+1:]].append(word)
# perform a BFS to get all possible ladder
visited = set([beginWord])
queue = deque([(beginWord, [beginWord])])
res = []
while queue:
curr_word, curr_ladder = queue.popleft()
for i in range(len(curr_word)):
# find all the words that could be created with the current word
next_words = word_dict[curr_word[:i] + "*" + curr_word[i+1:]]
for next_word in next_words:
if next_word in visited:
continue
# if we find the endWord, append the whole ladder to res
# otherwise, create a new ladder with the current ladder + next_word
if next_word == endWord:
res.append(curr_ladder + [next_word])
else:
queue.append((next_word, curr_ladder + [next_word]))
visited.add(next_word)
return res
```

Time Complexity:

Building the dictionary takes O(N*L^2), where N is the number of words and L is the length of the words. This is because we iterate through each character in each word and create a key with the empty space and then we append the word to the value of that key. To find all possible neighbors for a word, it takes O(L^2*N), as for each word we iterate over its length and then over the number of words in the dictionary.

Performing the BFS algorithm takes O(N*L^2) time in the worst case, where we have to process each word in the dictionary once. In each iteration of the algorithm, we generate all possible neighbors of the current word with wildcard queries. A typical query in the dictionary takes O(L) time and we have to query each word, hence O(NL) total query time. Therefore, the time complexity of the algorithm is O(N*L^2+N

*L^2) ~ O(N*L^2).

Space Complexity:

Creating the dictionary takes up O(N*L^2) space in the worst case. In the BFS approach we need to store the visited words in a set, which takes up O(N) space. We also need to store the whole ladder of words in each step of the BFS. Therefore, the space complexity of the algorithm is O(N*L^2).

Overall, we have discussed a detailed solution for Word Ladder II problem on LeetCode and analyzed its time and space complexities.

## Step by Step Implementation For Word Ladder Ii

This is a classic graph problem that can be solved using a Breadth First Search algorithm. The basic idea is to start from the beginning word and explore all the possible paths (ladders) to the end word. To keep track of the paths, we can use a data structure called a queue. We can also use a HashSet to keep track of the words that have been visited. This will ensure that we don't explore the same word more than once. Here is a pseudocode outline of the algorithm: 1. Create a queue and add the starting word to it 2. Create a HashSet to keep track of the words that have been visited 3. While the queue is not empty: 4. Remove the first word from the queue 5. If the removed word is the end word, then return the path 6. Otherwise, for each possible transformation of the removed word: 7. If the transformation is not in the HashSet, then add it to the HashSet and add it to the queue 8. Otherwise, continue to the next transformation

from collections import defaultdict, deque class Solution: def findLadders(self, beginWord: str, endWord: str, wordList: List[str]) -> List[List[str]]: # If end word not in word list, then there are no ladders if endWord not in wordList: return [] # Initialize variables length = len(beginWord) all_combo_dict = defaultdict(list) result = [] min_length = float('inf') # Initialize queue for BFS queue = deque([[beginWord, 1]]) # Initialize visited set for BFS visited = set() visited.add(beginWord) # Initialize unvisited set for BFS unvisited = set(wordList) # Perform BFS while queue: current_word, level = queue.popleft() # If current word is the end word, check length if current_word == endWord: if level > min_length: break else: result.append([current_word]) min_length = level # Get all possible combinations of current word for i in range(length): for c in 'abcdefghijklmnopqrstuvwxyz': new_word = current_word[:i] + c + current_word[i+1:] # If new word is unvisited, add to queue if new_word in unvisited: queue.append([new_word, level + 1]) visited.add(new_word) # Add all possible combinations of current word to dictionary if new_word in all_combo_dict: all_combo_dict[new_word].append(current_word) else: all_combo_dict[new_word] = [current_word] # Perform DFS to find all possible ladders def dfs(word, path): if word == beginWord: result.append(path) return for next_word in all_combo_dict[word]: if next_word not in path: dfs(next_word, path + [next_word]) # Perform DFS on all words in result for word in result: dfs(word[0], word) return result

Given two words (beginWord and endWord), and a dictionary's word list, find all shortest transformation sequence(s) 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 that beginWord is not a transformed word. Note: Return an empty list if there is no such transformation sequence. All words have the same length. All words contain only lowercase alphabetic characters. You may assume no duplicates in the word list. You may assume beginWord and endWord are non-empty and are not the same. Example 1: Input: beginWord = "hit", endWord = "cog", wordList = ["hot","dot","dog","lot","log","cog"] Output: [ ["hit","hot","dot","dog","cog"], ["hit","hot","lot","log","cog"] ] Example 2: Input: beginWord = "hit" endWord = "cog" wordList = ["hot","dot","dog","lot","log"] Output: [] Explanation: The endWord "cog" is not in wordList, therefore no possible transformation.

There are many possible solutions to this problem. One approach would be to use a breadth-first search algorithm to find all the shortest paths from the start word to the end word. Then, you could backtrack through the paths to find all the words in between. Another approach would be to use a depth-first search algorithm to find all the paths from the start word to the end word. You could then keep track of the words used in each path, and output all the paths that use the fewest words.

public class Solution { public IList> FindLadders(string beginWord, string endWord, IList wordList) { // TODO: Implement your solution here } }