Deep Dive: Graph Cloning

Graphs are fundamental data structures used to represent relationships between objects. Sometimes, you might need to create an exact copy of a graph, preserving its structure and all connections. This is crucial for tasks like creating snapshots for debugging, implementing undo/redo functionality, or when working with algorithms that modify graphs destructively. This challenge focuses on creating a deep copy of an undirected graph.

Problem Description

You are tasked with implementing a function that takes a graph as input and returns a deep copy of that graph. A deep copy means that not only are the nodes duplicated, but all the connections (edges) between these duplicated nodes are also replicated. You should not return a reference to the original graph or any of its nodes.

Key Requirements:

• Node Duplication: Each node in the original graph must have a corresponding, newly created node in the cloned graph.
• Edge Replication: If there's an edge between node A and node B in the original graph, there must be an identical edge between the cloned node A' and cloned node B' in the new graph.
• Undirected Graph: The graph is undirected, meaning if A is connected to B, then B is also connected to A. Your cloning process must respect this bidirectionality.
• No Shared References: The cloned graph should be entirely independent of the original graph. Modifying the cloned graph should not affect the original, and vice-versa.

Expected Behavior:

The function should return the starting node of the newly created, cloned graph.

Edge Cases to Consider:

• An empty graph (no nodes).
• A graph with a single node and no edges.
• A graph with cycles.

Examples

Example 1:

Input: A graph with two nodes, '1' and '2', connected by an edge.
       Node '1' has a neighbor '2'.
       Node '2' has a neighbor '1'.

Original Graph Structure:
Node 1 <---> Node 2

Output: A new graph with two nodes, '1'' and '2'', connected by an edge.
        Node '1'' has a neighbor '2''.
        Node '2'' has a neighbor '1''.

Cloned Graph Structure:
Node 1' <---> Node 2'

Explanation: The input graph has two nodes and one edge. The output is a new graph with two new nodes and a new edge connecting them, mirroring the original structure.

Example 2:

Input: A graph with three nodes: 'A', 'B', 'C'.
       'A' is connected to 'B'.
       'B' is connected to 'A' and 'C'.
       'C' is connected to 'B'.

Original Graph Structure:
   A <---> B <---> C

Output: A new graph with three nodes: 'A'', 'B'', 'C''.
        'A'' is connected to 'B''.
        'B'' is connected to 'A'' and 'C''.
        'C'' is connected to 'B''.

Cloned Graph Structure:
   A' <---> B' <---> C'

Explanation: The input graph shows a linear structure. The output is a deep copy of this structure.

Example 3: (Graph with a cycle)

Input: A graph with three nodes: 'X', 'Y', 'Z'.
       'X' is connected to 'Y' and 'Z'.
       'Y' is connected to 'X' and 'Z'.
       'Z' is connected to 'X' and 'Y'.

Original Graph Structure:
   X <-----> Y
   ^ \     / ^
   |  \   /  |
   |   \ /   |
   |    Z    |
   +---------+

Output: A new graph with three nodes: 'X'', 'Y'', 'Z''.
        'X'' is connected to 'Y'' and 'Z''.
        'Y'' is connected to 'X'' and 'Z''.
        'Z'' is connected to 'X'' and 'Y''.

Cloned Graph Structure:
   X' <-----> Y'
   ^ \     / ^
   |  \   /  |
   |   \ /   |
   |    Z'   |
   +---------+

Explanation: This example demonstrates cloning a graph with a cycle. The output correctly replicates all connections, including those forming the cycle.

Constraints

• The number of nodes in the graph will be between 0 and 100.
• Each node will have a unique identifier (e.g., integer, string).
• The graph can contain cycles.
• The input will be a reference to one of the nodes in the graph (or null if the graph is empty).
• Your solution should aim for an efficient time complexity, ideally proportional to the number of nodes and edges in the graph.

Notes

• You'll likely need a way to keep track of nodes that have already been cloned to avoid infinite recursion in the presence of cycles. A hash map (dictionary) is often a good choice for this.
• Consider the data structure representing the graph. Typically, a graph is represented by a collection of nodes, where each node contains its own value and a list of references to its neighbors.
• Think about how to represent the cloned nodes and their connections. You'll need to create new node objects and populate their neighbor lists with references to the newly created neighbor nodes.