Adjacency List vs Matrix When to Choose Which in DSA C# - Key Differences

Recall & Review

beginner

What is an adjacency list in graph representation?

An adjacency list stores each node with a list of its connected neighbors. It uses less space for sparse graphs and is efficient for traversing neighbors.

beginner

What is an adjacency matrix in graph representation?

An adjacency matrix is a 2D array where each cell [i][j] shows if there is an edge between node i and node j. It uses more space but allows quick edge checks.

intermediate

When is it better to use an adjacency list over an adjacency matrix?

Use an adjacency list when the graph is sparse (few edges) because it saves memory and is faster to iterate neighbors.

intermediate

When is an adjacency matrix preferred?

Use an adjacency matrix when the graph is dense (many edges) or when you need to quickly check if an edge exists between two nodes.

intermediate

What is the space complexity difference between adjacency list and matrix?

Adjacency list uses O(V + E) space, where V is vertices and E is edges. Adjacency matrix uses O(V²) space regardless of edges.

Which graph representation is more memory efficient for a graph with 1000 nodes and 10 edges?

A Adjacency List

B Adjacency Matrix

C Both use the same memory

D None of the above

Adjacency list uses less memory for sparse graphs like this one with few edges.

Which representation allows O(1) time to check if an edge exists between two nodes?

A Adjacency List

B Neither

C Both

D Adjacency Matrix

Adjacency matrix stores edges in a 2D array, so checking an edge is a direct lookup.

For a dense graph, which representation is usually better?

A Adjacency List

B Adjacency Matrix

C Linked List

D Stack

Dense graphs have many edges, so adjacency matrix is efficient for storage and edge checks.

What is the main disadvantage of adjacency matrix for large sparse graphs?

A Complex to implement

B Slow edge lookup

C High memory usage

D Cannot represent directed edges

Adjacency matrix uses O(V²) space, which is large for sparse graphs.

Which representation is better for iterating over all neighbors of a node?

A Adjacency List

B Adjacency Matrix

C Both are equal

D None

Adjacency list stores neighbors directly, making iteration faster and simpler.

Explain when to choose adjacency list versus adjacency matrix for graph representation.

Think about number of edges and how fast you want to check connections.

Describe the space complexity of adjacency list and adjacency matrix and why it matters.

Consider how many nodes and edges the graph has.

Practice

1. Which data structure is generally better for representing a graph with many nodes but very few edges?

easy

A. Adjacency List

B. Adjacency Matrix

C. Array

D. Stack

Solution

1. Step 1: Understand graph density

   A graph with many nodes but few edges is called a sparse graph.

2. Step 2: Choose data structure for sparse graph

   Adjacency List stores only existing edges, saving space for sparse graphs.

3. Final Answer:

   Adjacency List -> Option A

4. Quick Check:

   Sparse graph = Adjacency List [OK]

Hint: Sparse graph? Use adjacency list for space efficiency [OK]

Common Mistakes:

• Choosing adjacency matrix for sparse graphs wastes space
• Confusing sparse with dense graphs
• Thinking arrays or stacks represent graphs directly

2. Which of the following is the correct way to declare an adjacency matrix for a graph with 4 nodes in C#?

easy

A. int[][] adjacencyMatrix = new int[4][]; for(int i=0; i<4; i++) adjacencyMatrix[i] = new int[4];

B. int adjacencyMatrix = new int[4];

C. List<int>[] adjacencyMatrix = new List<int>[4];

D. int[,] adjacencyMatrix = new int[4,4];

Solution

1. Step 1: Identify adjacency matrix type

   An adjacency matrix is a 2D array representing edges between nodes.

2. Step 2: Check C# syntax for 2D arrays

   In C#, 2D arrays use syntax like int[,] array = new int[rows, columns];

3. Final Answer:

   int[,] adjacencyMatrix = new int[4,4]; -> Option D

4. Quick Check:

   2D array syntax = int[,] [OK]

Hint: Adjacency matrix uses 2D array syntax int[,] [OK]

Common Mistakes:

• Using jagged arrays instead of 2D arrays for matrix
• Declaring adjacencyMatrix as a single int
• Using List<int>[] which is adjacency list, not matrix

3. Given the following adjacency list representation of a graph in C#:

var graph = new List<int>[] {
  new List<int>{1, 2},
  new List<int>{2},
  new List<int>{0}
};
Console.WriteLine(graph[0][1]);

What will be the output?

medium

A. 0

B. 1

C. 2

D. IndexOutOfRangeException

Solution

1. Step 1: Understand the adjacency list structure

   graph[0] = List<int>{1, 2}, so graph[0][1] accesses the second element which is 2.

2. Step 2: Identify the output of Console.WriteLine(graph[0][1])

   It prints the value 2 from the list at index 1 of graph[0].

3. Final Answer:

   2 -> Option C

4. Quick Check:

   graph[0][1] = 2 [OK]

Hint: Indexing starts at 0; graph[0][1] is second element in first list [OK]

Common Mistakes:

• Confusing indices and printing 1 instead of 2
• Expecting an error due to nested lists
• Thinking graph[0][1] accesses graph[1]

4. Consider this adjacency matrix code snippet in C#:

int[,] matrix = new int[3,3];
matrix[0,1] = 1;
matrix[1,2] = 1;
matrix[2,0] = 1;
Console.WriteLine(matrix[1,0]);

What is the output and why?

medium

A. 1 because matrix is symmetric

B. 0 because matrix[1,0] was never set

C. Runtime error due to invalid index

D. 1 because of default initialization

Solution

1. Step 1: Analyze matrix assignments

   Only matrix[0,1], matrix[1,2], and matrix[2,0] are set to 1; matrix[1,0] remains 0.

2. Step 2: Understand default values in int[] arrays

   Unassigned int elements default to 0 in C#.

3. Final Answer:

   0 because matrix[1,0] was never set -> Option B

4. Quick Check:

   Unset matrix elements = 0 [OK]

Hint: Unset matrix cells default to 0, not symmetric [OK]

Common Mistakes:

• Assuming adjacency matrix is symmetric by default
• Expecting runtime error for valid indices
• Confusing default values with assigned values

5. You need to represent a social network graph with 10,000 users where each user has on average 10 friends. Which representation is best and why? Choose between adjacency list and adjacency matrix.

hard

A. Adjacency list, because it uses less memory for sparse graphs

B. Adjacency matrix, because it allows quick edge checks

C. Adjacency matrix, because it is simpler to implement

D. Adjacency list, because it stores all possible edges

Solution

1. Step 1: Calculate graph density

   10,000 users with 10 friends each means about 100,000 edges, which is sparse compared to 100 million possible edges.

2. Step 2: Choose data structure for sparse graph

   Adjacency list stores only existing edges, saving huge memory compared to adjacency matrix.

3. Final Answer:

   Adjacency list, because it uses less memory for sparse graphs -> Option A

4. Quick Check:

   Sparse large graph = adjacency list [OK]

Hint: Large sparse graph? Use adjacency list for memory efficiency [OK]

Common Mistakes:

• Choosing adjacency matrix despite huge memory waste
• Thinking adjacency matrix is always simpler
• Misunderstanding adjacency list stores only existing edges