Kth Ancestor of a Tree Node

Here’s the implementation of the TreeAncestor class:

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class TreeAncestor:
    def __init__(self, n, parent):
        self.MAX_JUMP = 16  # Maximum jump steps for binary lifting
        self.jump = [[-1] * self.MAX_JUMP for _ in range(n)]

        # Filling in the first level of jumps (parent)
        for i in range(n):
            self.jump[i][0] = parent[i]

        # Filling in the rest of the jumps using dynamic programming
        for j in range(1, self.MAX_JUMP):
            for i in range(n):
                if self.jump[i][j - 1] != -1:
                    self.jump[i][j] = self.jump[self.jump[i][j - 1]][j - 1]

    def getKthAncestor(self, node, k):
        for i in range(self.MAX_JUMP):
            if k & (1 << i):  # Check if the i-th bit of k is set
                node = self.jump[node][i]
                if node == -1:
                    return -1
        return node

This implementation uses binary lifting to efficiently find the kth ancestor of a given node in a tree. The jump array is precomputed to store the ancestors for each node using powers of 2. The getKthAncestor method uses binary representation of k to determine the appropriate jumps to make.

Identifying Problem Isomorphism

“Lowest Common Ancestor of a Binary Tree” is an approximate isomorphism to “Kth Ancestor of a Tree Node”. Both entail a deep understanding of tree traversal and ancestor lookup.

In “Lowest Common Ancestor of a Binary Tree”, we have to find the lowest common ancestor of two given nodes in the binary tree. This problem requires traversing the tree and backtracking to discover the common ancestor.

In “Kth Ancestor of a Tree Node”, the objective is to find the Kth ancestor for a given node. This requires traversing up the tree from the given node to its Kth ancestor.

While both problems deal with ancestors in a tree, their requirements differ. “Lowest Common Ancestor of a Binary Tree” is about finding a common node for two given nodes, whereas “Kth Ancestor of a Tree Node” focuses on finding a specific ancestor of a single node.

“Lowest Common Ancestor of a Binary Tree” is simpler. It involves basic tree traversal and backtracking, while “Kth Ancestor of a Tree Node” is more complex, as it may require handling larger trees and performing multiple ancestor lookups efficiently.

10 Prerequisite LeetCode Problems

“1483. Kth Ancestor of a Tree Node” involves the concepts of trees, binary lifting, and sparse tables. Here are some problems to grasp these concepts:

236. Lowest Common Ancestor of a Binary Tree: Understanding how to find the lowest common ancestor is key to solving more complex tree problems.
543. Diameter of Binary Tree: This problem will help you understand tree traversal and the concept of depth in a tree.
104. Maximum Depth of Binary Tree: This problem gives an understanding of how to calculate depth in a tree, which is necessary for binary lifting.
114. Flatten Binary Tree to Linked List: Flattening a binary tree is a good exercise for manipulating tree structures.
105. Construct Binary Tree from Preorder and Inorder Traversal: Constructing a tree from given traversals will help you understand tree structure more deeply.
1123. Lowest Common Ancestor of Deepest Leaves: This problem can help in understanding the concept of LCA (Lowest Common Ancestor) in the context of binary trees.
199. Binary Tree Right Side View: A good problem to understand depth-first search in trees.
298. Binary Tree Longest Consecutive Sequence: This problem will help you understand how to traverse the tree with keeping track of some property (consecutive sequence in this case).
987. Vertical Order Traversal of a Binary Tree: This problem teaches traversal with a twist - this time you need to do it vertically.
863. All Nodes Distance K in Binary Tree: This problem is about finding nodes at a certain distance, which may be useful in understanding how to move up and down a tree.

Problem Classification

Problem Statement:You are given a tree with n nodes numbered from 0 to n - 1 in the form of a parent array parent where parent[i] is the parent of ith node. The root of the tree is node 0. Find the kth ancestor of a given node.

The kth ancestor of a tree node is the kth node in the path from that node to the root node.

Implement the TreeAncestor class:

TreeAncestor(int n, int[] parent) Initializes the object with the number of nodes in the tree and the parent array. int getKthAncestor(int node, int k) return the kth ancestor of the given node node. If there is no such ancestor, return -1.

Example 1:

Input [“TreeAncestor”, “getKthAncestor”, “getKthAncestor”, “getKthAncestor”] [[7, [-1, 0, 0, 1, 1, 2, 2]], [3, 1], [5, 2], [6, 3]] Output [null, 1, 0, -1]

Explanation TreeAncestor treeAncestor = new TreeAncestor(7, [-1, 0, 0, 1, 1, 2, 2]); treeAncestor.getKthAncestor(3, 1); // returns 1 which is the parent of 3 treeAncestor.getKthAncestor(5, 2); // returns 0 which is the grandparent of 5 treeAncestor.getKthAncestor(6, 3); // returns -1 because there is no such ancestor

Constraints:

1 <= k <= n <= 5 * 104 parent.length == n parent[0] == -1 0 <= parent[i] < n for all 0 < i < n 0 <= node < n There will be at most 5 * 104 queries.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.