Clone Graph
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: Clone Graph
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given a reference to a node in a connected undirected graph, return a deep copy (clone)
* of the graph. Each node contains a value (int) and a list of its neighbors.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* A "Deep Copy" means we must create entirely new instances of every node and replicate
* all adjacency connections. Simply copying references would result in a shallow copy.
*
* Algorithm (BFS with Hash Map):
* 1. Mapping Strategy:
* Use a Hash Map map<OriginalNode, ClonedNode> to keep track of already created
* clones. This prevents infinite loops in graphs with cycles.
* * 2. Traversal:
* - Initialize a Queue for BFS and add the starting node.
* - Create a clone of the starting node and store it in the map.
* * 3. Processing:
* While the Queue is not empty:
* - Pop the original node u.
* - For each neighbor v of u:
* a. If v has not been cloned yet (not in map):
* - Create a clone of v.
* - Store the mapping {v: clone_v} in the map.
* - Enqueue v to process its neighbors later.
* b. Add the clone of v to the neighbors list of the clone of u.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* - The number of nodes is between 1 and 100.
* - Node values are unique.
*
* Time Complexity:
* O(V + E) - We visit every node and every edge exactly once.
*
* Space Complexity:
* O(V) - To store the mapping of original nodes to cloned nodes and the BFS queue.
*
* ============================================================================================
/
class UndirectedGraphNode {
int label;
List
public UndirectedGraphNode cloneGraph(UndirectedGraphNode node1) {
return new UndirectedGraphNode(1);
}
}