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Largest Distance Between Nodes Of A Tree

Logical Breakdown

  • [ ] Subproblem Identification:
  • [ ] Optimal Substructure:
  • [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
  • [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

\[ dp[i] = ... \]

Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

Complexity Analysis

Approach Time Complexity Space Complexity
Iterative DP \(O(N)\) \(O(1)\)

Code Reference

package com.dsa.graphs;

/ * ============================================================================================ * PROBLEM: Largest Distance between nodes of a Tree (Tree Diameter) * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a tree of N nodes. Find the maximum distance (number of edges) between any * two nodes in the tree. * - Input: Array A where A[i] is the parent of node i. * - A[i] = -1 signifies the root. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * The largest distance between any two nodes in a tree is known as its Diameter*. * * Algorithm (Double BFS/DFS Property): * 1. Step 1: Pick an arbitrary node 'u' (e.g., node 0 or the root). * 2. Step 2: Perform a BFS/DFS from 'u' to find the node 'v' that is furthest from it. * - Mathematical Property: In a tree, node 'v' is guaranteed to be one of the * endpoints of the tree's diameter. * 3. Step 3: Perform a second BFS/DFS starting from node 'v' to find the node * furthest from 'v'. Let this furthest distance be 'D'. * 4. Result: 'D' is the diameter of the tree. * * * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 2 <= N <= 40,000 * * Time Complexity: * O(N) - We perform two linear traversals of the tree. * - Building adjacency list: O(N) * - Two BFS/DFS runs: O(N) * * Space Complexity: * O(N) - To store the adjacency list, visited array, and recursion/queue stack. * * ============================================================================================ / public class LargestDistanceBetweenNodesOfATree { public int solve(int[] A) { // TODO: Implement solution return 0; } }