Maximum Depth
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
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.graphs;
import java.util.*;
/*
* ============================================================================================
* PROBLEM: Maximum Depth
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given a tree with A nodes rooted at 1. Each node has a specific value.
* We need to answer Q queries (L, X):
* - Find a node at level: targetLevel = L % (MaxDepth + 1)
* - The node must have value >= X.
* - If multiple such nodes exist, return the SMALLEST value.
* - If no such node exists, return -1.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* Step 1: Level Mapping (Preprocessing)
* - Use DFS or BFS to traverse the tree and calculate the level of each node.
* - Maintain a List of Lists (or Map): levels[d] contains the values of all nodes
* at depth d.
* - Calculate MaxDepth during this traversal.
*
* Step 2: Sorting for Binary Search
* - For each depth d in the levels structure, sort the list of values.
* - Sorting allows us to find the smallest value >= X using Binary Search (Lower Bound).
*
* Step 3: Query Processing
* - For each query (L, X):
* a. Calculate depth = L % (MaxDepth + 1).
* b. Access the sorted list at levels[depth].
* c. Use Collections.binarySearch or a custom lowerBound function:
* - If found, return the value.
* - If not found (target > all elements), return -1.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Time Complexity:
* - Preprocessing: O(A) for DFS + O(A log A) for sorting all level lists.
* - Queries: O(Q log A) - binary search on the specific level list.
* Total: O((A + Q) log A).
*
* Space Complexity:
* O(A) to store the tree and the level-wise value lists.
*
* ============================================================================================
/
public class MaximumDepth {
public int solve(int A) {
// TODO: Implement solution
return 0;
}
public int[] solve(int a, int[] b, int[] c, int[] d, int[] l, int[] x) {
return new int[0];
}
}