What is the optimal approach and complexity for Clone Graph?

Create each copy the first time its node is seen; the map finds an existing copy in one lookup with optimal time complexity O(V + E) and space complexity O(V).

How do you identify the pattern for Clone Graph?

Look for key signals in the prompt: "deep copy of the whole graph" indicates Every node is copied once and the links are rebuilt between the copies. A traversal visits each node once.; "connected, undirected graph" indicates The traversal reaches every node from the start, so one search from one node finds all of them.; "node may be reached again" indicates Cycles exist, so the search needs a memory of which nodes it has seen, here a map..

Clone Graph

Medium·LeetCode #133 Pro OnlyMedium

LeetCode

Problem Statement

You are given a reference to one node of a connected, undirected graph. Every node has an integer value and a list of neighbours.

Return a deep copy of the whole graph. Every node must be copied exactly once. The copy must have the same values and the same links as the original, but it must share no node with the original. Editing the copy must never change the input.

Input

A reference node to one node of the graph, or null for an empty graph. Each node has val and a list neighbors.

Output

A reference to the copy of node in the new graph, or null if the input is null.

Example 1

Input: adjList = [[2,4],[1,3],[2,4],[1,3]]

Output: the same graph, built from new nodes

Node 1 is linked to 2 and 4, node 2 to 1 and 3, node 3 to 2 and 4, and node 4 to 1 and 3. The copy has the same four links, on four new nodes.

Example 2

Input: adjList = [[]]

Output: one new node with no links

A single node with no neighbours. The copy is one new node with an empty neighbour list.

Constraints

• The number of nodes is in the range [0, 100].
• 1 ≤ Node.val ≤ 100
• Node.val is unique for each node.
• There are no repeated edges and no self-loops.
• The graph is connected, and every node can be reached from the given node.

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“deep copy of the whole graph”

→ Every node is copied once and the links are rebuilt between the copies. A traversal visits each node once.

“connected, undirected graph”

→ The traversal reaches every node from the start, so one search from one node finds all of them.

“node may be reached again”

→ Cycles exist, so the search needs a memory of which nodes it has seen, here a map.

Approach Complexity Ladder

Brute Force — search a list for each neighbour's copyO(V · E)O(V)

Collect all nodes in a list, make one copy per node, then find each neighbour's copy by a linear search

Optimal — one BFS with a map from original to copyO(V + E)O(V)

Create each copy the first time its node is seen; the map finds an existing copy in one lookup

Brute Force Intuition

First collect every original node in one list with a BFS. Make one new node for each entry. To link the copies, look at each neighbour of an original node and find its copy by scanning the list from the front, then add that copy to the matching copy's neighbour list.

Recognition rule for this sub-pattern:

Adjacency list + visited set. Count how many times you must start a new traversal.

Remember this · Graph BFS / DFS

Trigger

Grid of cells, islands, regions, "minimum steps", transformations between states, connected components.

Think

What is a node, what is an edge, and do I need the shortest path or just reachability?

Key question

How do I mark visited so I never process a node twice?

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Beyond The LeetCode Problem

How this concept connects to System Design & C++ Systems

Bridge DSA with production engineering

Top engineering interviews test whether you understand how algorithms like Graph BFS / DFS translate into production architectures and low-level memory mechanics.

LLD Real-World Story

Food Delivery System (Zomato / DoorDash)

Geospatial indexing and shortest dispatch routing for delivery partners.

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C++ Internals

Virtual Method Table (vtable) & Dynamic Dispatch

Polymorphic graph nodes and state machine execution overhead.

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