1483 - Kth Ancestor of a Tree Node.md

Author: vivekvardhan2810

dsa-solutions/lc-solutions/1400-1500/1483 - Kth Ancestor of a Tree Node.md

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+---
+id: kth-ancestor-of-a-tree node
+title: Kth Ancestor of a Tree Node
+sidebar_label: 1483 Kth Ancestor of a Tree Node
+tags:
+- Java
+- Binary Search
+- Dynamic Programming
+- Tree
+- Depth-First Search
+- Breadth-First Search
+- Design
+description: "This document provides a solution where we Find the kth ancestor of a given node."
+---
+
+## Problem
+
+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 $i^{\text{th}}$ node. The root of the tree is node $0$. Find the $k^{\text{th}}$ ancestor of a given node.
+
+The $k^{\text{th}}$ ancestor of a tree node is the $k^{\text{th}}$ 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 $k^{\text{th}}$ ancestor of the given node node. If there is no such ancestor, return $-1$.
+
+### Examples
+
+**Example 1:**
+
+![image](https://github.com/vivekvardhan2810/codeharborhub.github.io/assets/91594529/c96383bd-e0bb-494d-8112-226686ab1832)
+
+**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
+
+- $1 <= k <= n <= 5 * 10^4$
+- $parent.length == n$
+- $parent[0] == -1$
+- $0 <= parent[i] < n$ for all $0 < i < n$
+- $0 <= node < n$
+- There will be at most $5 * 10^4$ queries.
+---
+
+## Approach
+
+To solve the problem, we need to understand the nature of the allowed moves:
+
+1. **Binary Lifting Setup**:
+
+   - Calculate the maximum power of 2 needed, which is $\log_2(n)$. This represents the maximum number of jumps we need to consider.
+  
+   - Create a 2D array **'dp'** where **'dp[i][j]'** represents the $2^j$-th ancestor of node i.
+     
+2. **Preprocessing**:
+
+   - Initialize **'dp[i][0]'** with the parent of node i.
+     
+   - Fill in the rest of the **'dp'** table using dynamic programming. For each power j, compute **'dp[i][j]'** using **'dp[i][j-1]'**. Specifically, **'dp[i][j]'** is $2^j$-th ancestor of node i, which is the $2^j-1$-th ancestor of node i.
+   
+3. **Query Processing**:
+
+   - For each query to find the 𝑘-th ancestor of a node, use the binary representation of 𝑘. For each bit that is set in 𝑘, jump accordingly using the **'dp'** table.
+     
+   - If at any point the ancestor becomes $'-1'$, return $'-1'$.
+     
+## Solution for Kth Ancestor of a Tree Node
+
+- The goal is to find the $k^{\text{th}}$ ancestor of a node efficiently. A naive approach would involve iterating up the tree one node at a time, which would be too slow for large values of k. Instead, we use a technique called "binary lifting".
+ 
+- Binary lifting allows us to jump in powers of two, which significantly speeds up the process. By preprocessing the tree, we can answer each query in logarithmic time.
+
+#### Code in Java
+
+```java
+import java.util.*;
+
+class TreeAncestor {
+    private int[][] dp;
+    private int maxPower;
+
+    public TreeAncestor(int n, int[] parent) {
+        maxPower = (int) (Math.log(n) / Math.log(2)) + 1;
+        dp = new int[n][maxPower];
+        
+        for (int i = 0; i < n; i++) {
+            dp[i][0] = parent[i];
+        }
+        
+        for (int j = 1; j < maxPower; j++) {
+            for (int i = 0; i < n; i++) {
+                int midAncestor = dp[i][j - 1];
+                if (midAncestor != -1) {
+                    dp[i][j] = dp[midAncestor][j - 1];
+                } else {
+                    dp[i][j] = -1;
+                }
+            }
+        }
+    }
+
+    public int getKthAncestor(int node, int k) {
+        for (int j = 0; j < maxPower; j++) {
+            if ((k & (1 << j)) > 0) {
+                node = dp[node][j];
+                if (node == -1) {
+                    return -1;
+                }
+            }
+        }
+        return node;
+    }
+
+    public static void main(String[] args) {
+        TreeAncestor treeAncestor = new TreeAncestor(7, new int[]{-1, 0, 0, 1, 1, 2, 2});
+        System.out.println(treeAncestor.getKthAncestor(3, 1)); // returns 1
+        System.out.println(treeAncestor.getKthAncestor(5, 2)); // returns 0
+        System.out.println(treeAncestor.getKthAncestor(6, 3)); // returns -1
+    }
+}    
+```
+
+### Complexity Analysis
+
+#### Time Complexity: $O(n log n)$
+
+> **Reason**: Initializing the dp array requires iterating through all nodes for each power of 2 up to log𝑛, where as coming to query it involves checking up to log 𝑛 bits in the binary representation of 𝑘 and making jumps accordingly.
+
+#### Space Complexity: $O(log n)$
+
+> **Reason**: The space complexity is $O(log n)$, The **'dp'** table requires 𝑛 × log𝑛, where 𝑛 is the number of nodes and log𝑛 is the maximum power of 2 we consider.
+
+# References
+
+- **LeetCode Problem:** [Kth Ancestor of a Tree Node](https://leetcode.com/problems/kth-ancestor-of-a-tree-node/description/)
+- **Solution Link:** [Kth Ancestor of a Tree Node Solution on LeetCode](https://leetcode.com/problems/kth-ancestor-of-a-tree-node/solutions/)
+- **Authors LeetCode Profile:** [Vivek Vardhan](https://leetcode.com/u/vivekvardhan43862/)