python coding test course, least common ancestor

Hello! In this lecture, we will discuss one of the algorithm problems known as ‘Lowest Common Ancestor (LCA)’. The LCA problem involves finding the closest common ancestor of two given nodes in a tree structure. This problem is very important in various fields and is often asked in many interviews.

Problem Description

Given the value of two nodes in a binary tree (including binary search trees), find their lowest common ancestor.

Input

• Number of nodes N (1 ≤ N ≤ 1000)
• Values of N nodes
• Two node values A, B (A and B must exist in the tree)

Output

Output the value of the lowest common ancestor of the two nodes.

Understanding the Problem

The Lowest Common Ancestor problem helps in understanding the characteristics of the data structure that makes up the tree and the relationships between nodes. A common ancestor refers to a node that two nodes meet at while moving up. For example, let’s consider the tree below.

              3
            /   \
          5      1
         / \    / \
        6   2  0   8
           / \
          7   4
    

In this tree, the lowest common ancestor of nodes 5 and 1 is 3, and the lowest common ancestor of nodes 6 and 4 is 5.

Solution Strategy

The basic method to find the lowest common ancestor is as follows:

1. Traverse the tree and store the paths of the given two nodes.
2. Find the last common node in both paths.

This method is intuitive and straightforward, but it can be inefficient in some cases. Instead, a more efficient method can be used as described below.

Efficient Method

An efficient way to find the LCA in a binary tree is to use Depth First Search (DFS). This method recursively explores nodes with the given values, and when both nodes are found, it returns that node.

Code Implementation

Now we will implement this algorithm in Python. Below is the code to find the LCA:

class TreeNode:
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None

def lowest_common_ancestor(root, p, q):
    # Base condition
    if not root or root == p or root == q:
        return root

    # Recursively look for LCA in left and right
    left = lowest_common_ancestor(root.left, p, q)
    right = lowest_common_ancestor(root.right, p, q)

    # If both child nodes exist
    if left and right:
        return root
    return left if left else right

# Test case
if __name__ == "__main__":
    # Constructing the tree
    root = TreeNode(3)
    root.left = TreeNode(5)
    root.right = TreeNode(1)
    root.left.left = TreeNode(6)
    root.left.right = TreeNode(2)
    root.right.left = TreeNode(0)
    root.right.right = TreeNode(8)
    root.left.right.left = TreeNode(7)
    root.left.right.right = TreeNode(4)

    # Setting nodes p and q
    p = root.left  # 5
    q = root.right  # 1

    # Calculating LCA
    lca = lowest_common_ancestor(root, p, q)
    print(f"Lowest Common Ancestor: {lca.val}")
    

Algorithm Analysis

This algorithm visits each node in the tree only once, therefore the time complexity is O(N), and the space complexity is O(H), where N is the number of nodes and H is the height of the tree. In the worst case, H can be equal to N, resulting in an overall time and space complexity of O(N).

Conclusion

In this lecture, we learned about the Lowest Common Ancestor problem. Although this problem requires considering many cases, it can be solved simply with the correct algorithm. When solving actual problems, it is essential to have a solid understanding of the tree structure and the relationships between nodes. In the next lecture, we will cover another algorithm problem. Thank you!