[pull] master from TheAlgorithms:master

DIRECTORY.md

@@ -22,6 +22,7 @@
   * [Rat In Maze](backtracking/rat_in_maze.py)
   * [Sudoku](backtracking/sudoku.py)
   * [Sum Of Subsets](backtracking/sum_of_subsets.py)
+  * [Word Ladder](backtracking/word_ladder.py)
   * [Word Search](backtracking/word_search.py)
 
 ## Bit Manipulation

backtracking/word_ladder.py

@@ -0,0 +1,100 @@
+"""
+Word Ladder is a classic problem in computer science.
+The problem is to transform a start word into an end word
+by changing one letter at a time.
+Each intermediate word must be a valid word from a given list of words.
+The goal is to find a transformation sequence
+from the start word to the end word.
+
+Wikipedia: https://en.wikipedia.org/wiki/Word_ladder
+"""
+
+import string
+
+
+def backtrack(
+    current_word: str, path: list[str], end_word: str, word_set: set[str]
+) -> list[str]:
+    """
+    Helper function to perform backtracking to find the transformation
+    from the current_word to the end_word.
+
+    Parameters:
+    current_word (str): The current word in the transformation sequence.
+    path (list[str]): The list of transformations from begin_word to current_word.
+    end_word (str): The target word for transformation.
+    word_set (set[str]): The set of valid words for transformation.
+
+    Returns:
+    list[str]: The list of transformations from begin_word to end_word.
+               Returns an empty list if there is no valid
+                transformation from current_word to end_word.
+
+    Example:
+    >>> backtrack("hit", ["hit"], "cog", {"hot", "dot", "dog", "lot", "log", "cog"})
+    ['hit', 'hot', 'dot', 'lot', 'log', 'cog']
+
+    >>> backtrack("hit", ["hit"], "cog", {"hot", "dot", "dog", "lot", "log"})
+    []
+
+    >>> backtrack("lead", ["lead"], "gold", {"load", "goad", "gold", "lead", "lord"})
+    ['lead', 'lead', 'load', 'goad', 'gold']
+
+    >>> backtrack("game", ["game"], "code", {"came", "cage", "code", "cade", "gave"})
+    ['game', 'came', 'cade', 'code']
+    """
+
+    # Base case: If the current word is the end word, return the path
+    if current_word == end_word:
+        return path
+
+    # Try all possible single-letter transformations
+    for i in range(len(current_word)):
+        for c in string.ascii_lowercase:  # Try changing each letter
+            transformed_word = current_word[:i] + c + current_word[i + 1 :]
+            if transformed_word in word_set:
+                word_set.remove(transformed_word)
+                # Recur with the new word added to the path
+                result = backtrack(
+                    transformed_word, [*path, transformed_word], end_word, word_set
+                )
+                if result:  # valid transformation found
+                    return result
+                word_set.add(transformed_word)  # backtrack
+
+    return []  # No valid transformation found
+
+
+def word_ladder(begin_word: str, end_word: str, word_set: set[str]) -> list[str]:
+    """
+    Solve the Word Ladder problem using Backtracking and return
+    the list of transformations from begin_word to end_word.
+
+    Parameters:
+    begin_word (str): The word from which the transformation starts.
+    end_word (str): The target word for transformation.
+    word_list (list[str]): The list of valid words for transformation.
+
+    Returns:
+    list[str]: The list of transformations from begin_word to end_word.
+               Returns an empty list if there is no valid transformation.
+
+    Example:
+    >>> word_ladder("hit", "cog", ["hot", "dot", "dog", "lot", "log", "cog"])
+    ['hit', 'hot', 'dot', 'lot', 'log', 'cog']
+
+    >>> word_ladder("hit", "cog", ["hot", "dot", "dog", "lot", "log"])
+    []
+
+    >>> word_ladder("lead", "gold", ["load", "goad", "gold", "lead", "lord"])
+    ['lead', 'lead', 'load', 'goad', 'gold']
+
+    >>> word_ladder("game", "code", ["came", "cage", "code", "cade", "gave"])
+    ['game', 'came', 'cade', 'code']
+    """
+
+    if end_word not in word_set:  # no valid transformation possible
+        return []
+
+    # Perform backtracking starting from the begin_word
+    return backtrack(begin_word, [begin_word], end_word, word_set)

data_structures/binary_tree/symmetric_tree.py

@@ -13,7 +13,21 @@
 @dataclass
 class Node:
     """
-    A Node has data variable and pointers to Nodes to its left and right.
+    A Node represents an element of a binary tree, which contains:
+
+    Attributes:
+    data: The value stored in the node (int).
+    left: Pointer to the left child node (Node or None).
+    right: Pointer to the right child node (Node or None).
+
+    Example:
+    >>> node = Node(1, Node(2), Node(3))
+    >>> node.data
+    1
+    >>> node.left.data
+    2
+    >>> node.right.data
+    3
     """
 
     data: int
@@ -24,12 +38,25 @@ class Node:
 def make_symmetric_tree() -> Node:
     r"""
     Create a symmetric tree for testing.
+
     The tree looks like this:
            1
          /   \
         2     2
       / \    / \
      3   4   4  3
+
+    Returns:
+    Node: Root node of a symmetric tree.
+
+    Example:
+    >>> tree = make_symmetric_tree()
+    >>> tree.data
+    1
+    >>> tree.left.data == tree.right.data
+    True
+    >>> tree.left.left.data == tree.right.right.data
+    True
     """
     root = Node(1)
     root.left = Node(2)
@@ -43,13 +70,26 @@ def make_symmetric_tree() -> Node:
 
 def make_asymmetric_tree() -> Node:
     r"""
-    Create a asymmetric tree for testing.
+    Create an asymmetric tree for testing.
+
     The tree looks like this:
            1
          /   \
         2     2
       / \    / \
      3   4   3  4
+
+    Returns:
+    Node: Root node of an asymmetric tree.
+
+    Example:
+    >>> tree = make_asymmetric_tree()
+    >>> tree.data
+    1
+    >>> tree.left.data == tree.right.data
+    True
+    >>> tree.left.left.data == tree.right.right.data
+    False
     """
     root = Node(1)
     root.left = Node(2)
@@ -63,7 +103,15 @@ def make_asymmetric_tree() -> Node:
 
 def is_symmetric_tree(tree: Node) -> bool:
     """
-    Test cases for is_symmetric_tree function
+    Check if a binary tree is symmetric (i.e., a mirror of itself).
+
+    Parameters:
+    tree: The root node of the binary tree.
+
+    Returns:
+    bool: True if the tree is symmetric, False otherwise.
+
+    Example:
     >>> is_symmetric_tree(make_symmetric_tree())
     True
     >>> is_symmetric_tree(make_asymmetric_tree())
@@ -76,8 +124,17 @@ def is_symmetric_tree(tree: Node) -> bool:
 
 def is_mirror(left: Node | None, right: Node | None) -> bool:
     """
+    Check if two subtrees are mirror images of each other.
+
+    Parameters:
+    left: The root node of the left subtree.
+    right: The root node of the right subtree.
+
+    Returns:
+    bool: True if the two subtrees are mirrors of each other, False otherwise.
+
+    Example:
     >>> tree1 = make_symmetric_tree()
-    >>> tree1.right.right = Node(3)
     >>> is_mirror(tree1.left, tree1.right)
     True
     >>> tree2 = make_asymmetric_tree()
@@ -91,7 +148,7 @@ def is_mirror(left: Node | None, right: Node | None) -> bool:
         # One side is empty while the other is not, which is not symmetric.
         return False
     if left.data == right.data:
-        # The values match, so check the subtree
+        # The values match, so check the subtrees recursively.
         return is_mirror(left.left, right.right) and is_mirror(left.right, right.left)
     return False