Graph representations (adjacency matrix vs list) in Data Structures Theory - Practice Questions

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding adjacency matrix size

If a graph has 7 nodes, how many elements does its adjacency matrix contain?

A 14

B 49

C 21

D 7

💡 Hint

An adjacency matrix is a square matrix with size equal to the number of nodes squared.

An adjacency matrix for a graph with n nodes is an n x n matrix. For 7 nodes, it has 7 rows and 7 columns, so 7 × 7 = 49 elements.

📋 Factual

intermediate

Memory efficiency of adjacency list

Which statement about adjacency lists is true for a graph with many nodes but few edges?

A Adjacency lists use less memory than adjacency matrices.

B Adjacency lists always use more memory than adjacency matrices.

C Adjacency lists and matrices use the same memory regardless of edges.

D Adjacency lists cannot represent graphs with few edges.

💡 Hint

Think about how adjacency lists store only existing edges.

Adjacency lists store only the edges that exist, so for graphs with few edges, they use less memory than adjacency matrices, which store all possible connections.

🔍 Analysis

advanced

Time complexity for checking edge existence

Which graph representation allows checking if an edge exists between two nodes in constant time?

A Edge list

B Adjacency list

C Adjacency matrix

D Incidence matrix

💡 Hint

Consider how data is accessed in each representation.

An adjacency matrix stores edges in a 2D array, so checking if an edge exists between two nodes is a simple lookup, which takes constant time.

❓ Comparison

advanced

Best representation for dense graphs

For a graph where most nodes are connected to many others, which representation is generally more efficient?

A Adjacency matrix

B Adjacency list

C Edge list

D None of the above

💡 Hint

Dense graphs have many edges close to the maximum possible.

Adjacency matrices are efficient for dense graphs because they provide quick access and the memory cost is justified by many edges.

❓ Reasoning

expert

Choosing representation for dynamic edge updates

Which graph representation is generally better for a graph where edges are frequently added and removed?

A Adjacency matrix

B Incidence matrix

C Edge list

D Adjacency list

💡 Hint

Think about how easy it is to add or remove edges in each structure.

Adjacency lists allow easy addition and removal of edges by updating lists, while adjacency matrices require changing matrix entries, which can be less efficient for frequent updates.

Practice

1. Which graph representation uses a 2D grid to show connections between nodes?

easy

A. Incidence matrix

B. Adjacency matrix

C. Edge list

D. Adjacency list

Solution

1. Step 1: Understand adjacency matrix structure

   An adjacency matrix is a 2D grid where rows and columns represent nodes, and cells show if an edge exists.

2. Step 2: Compare with other representations

   Adjacency lists store neighbors in lists, not grids. Edge lists and incidence matrices differ in format.

3. Final Answer:

   Adjacency matrix -> Option B

4. Quick Check:

   2D grid = adjacency matrix [OK]

Hint: Matrix means grid; list means neighbors [OK]

Common Mistakes:

• Confusing adjacency list with matrix
• Thinking edge list is a grid
• Mixing incidence matrix with adjacency matrix

2. Which of the following is the correct way to represent an adjacency list in Python?

easy

A. graph = [[1, 2], 0, [0, 1]]

B. graph = [[0,1,0],[1,0,1],[0,1,0]]

C. graph = [(0,1), (1,2), (2,0)]

D. graph = {0: [1, 2], 1: [0], 2: [0, 1]}

Solution

1. Step 1: Identify adjacency list format

   An adjacency list maps each node to a list of its neighbors, often using a dictionary in Python.

2. Step 2: Check each option

   graph = {0: [1, 2], 1: [0], 2: [0, 1]} uses a dictionary with keys as nodes and values as neighbor lists, which is correct. graph = [[0,1,0],[1,0,1],[0,1,0]] is a matrix, C is an edge list, D incorrectly uses an integer 0 for node 1 instead of a list.

3. Final Answer:

   graph = {0: [1, 2], 1: [0], 2: [0, 1]} -> Option D

4. Quick Check:

   Dict with neighbors = adjacency list [OK]

Hint: Adjacency list uses dict with node keys [OK]

Common Mistakes:

• Choosing matrix format as list
• Confusing edge list with adjacency list
• Using integer instead of list for neighbors

3. Given the adjacency matrix below, which nodes are connected to node 1?
graph = [[0, 1, 0], [1, 0, 1], [0, 1, 0]]

medium

A. Nodes 0 and 2

B. Nodes 1 and 2

C. Nodes 0 and 1

D. Nodes 2 only

Solution

1. Step 1: Locate row for node 1

   Row 1 in the matrix is [1, 0, 1], representing edges from node 1 to nodes 0, 1, and 2.

2. Step 2: Identify connected nodes

   Values 1 indicate connection. Here, positions 0 and 2 have 1, so node 1 connects to nodes 0 and 2.

3. Final Answer:

   Nodes 0 and 2 -> Option A

4. Quick Check:

   Row 1 has 1s at 0 and 2 [OK]

Hint: Check row for node, 1 means connected [OK]

Common Mistakes:

• Confusing row and column indices
• Including node itself as connected
• Misreading zeros as edges

4. What is wrong with this adjacency list representation?
graph = {0: [1, 2], 1: [0, 3], 2: [0], 3: 1}

medium

A. Node 3's neighbors should be in a list

B. Node 1 has an invalid neighbor

C. Node 0 should not have neighbors

D. The graph should be an adjacency matrix

Solution

1. Step 1: Check format of neighbors for each node

   Nodes 0, 1, and 2 have neighbor lists. Node 3 has a single integer instead of a list.

2. Step 2: Identify correct adjacency list format

   Neighbors must always be in a list, even if only one neighbor exists, to keep consistent structure.

3. Final Answer:

   Node 3's neighbors should be in a list -> Option A

4. Quick Check:

   Neighbors must be lists [OK]

Hint: Neighbors always in lists, never single values [OK]

Common Mistakes:

• Ignoring single neighbor format
• Thinking adjacency list must be matrix
• Assuming neighbors can be integers

5. For a graph with 1000 nodes and only 10,000 edges, which representation is more memory efficient and why?

hard

A. Adjacency matrix, because it allows faster edge checks

B. Adjacency matrix, because it uses fixed size memory

C. Adjacency list, because it stores only existing edges

D. Adjacency list, because it stores all possible edges

Solution

1. Step 1: Calculate memory use for adjacency matrix

   An adjacency matrix for 1000 nodes uses 1000x1000 = 1,000,000 cells, regardless of edges.

2. Step 2: Calculate memory use for adjacency list

   An adjacency list stores only the 10,000 edges, so memory use is proportional to edges, much less than matrix.

3. Step 3: Compare efficiency

   Since edges are sparse compared to possible connections, adjacency list is more memory efficient.

4. Final Answer:

   Adjacency list, because it stores only existing edges -> Option C

5. Quick Check:

   Sparse graph = adjacency list efficient [OK]

Hint: Sparse graph? Use adjacency list for less memory [OK]

Common Mistakes:

• Choosing matrix for sparse graphs
• Confusing speed with memory use
• Thinking adjacency list stores all edges