Adjacency List vs Matrix When to Choose Which in DSA Typescript - Complexity Comparison

Time Complexity: Adjacency List vs Matrix When to Choose Which

O(1) for adjacency matrix, O(k) for adjacency list

Understanding Time Complexity

Choosing between adjacency list and adjacency matrix affects how fast graph operations run.

We want to know how the time to check or add edges changes as the graph grows.

Scenario Under Consideration

Analyze the time complexity of checking if an edge exists in two graph representations.

// Using adjacency matrix
function hasEdgeMatrix(matrix: number[][], u: number, v: number): boolean {
  return matrix[u][v] === 1;
}

// Using adjacency list
function hasEdgeList(list: number[][], u: number, v: number): boolean {
  return list[u].includes(v);
}
    

This code checks if there is an edge from node u to node v using two different graph storage methods.

Identify Repeating Operations

Look at what repeats when checking edges.

• Primary operation: For matrix, direct access; for list, searching through neighbors.
• How many times: Matrix access is once; list search depends on number of neighbors of u.

How Execution Grows With Input

How does checking edges grow as graph size grows?

Input Size (n nodes)	Matrix Accesses	List Searches (neighbors)
10	1	Up to 9
100	1	Up to 99
1000	1	Up to 999

Matrix access stays constant; list search grows with number of neighbors, which depends on graph density.

Final Time Complexity

Time Complexity: O(1) for adjacency matrix, O(k) for adjacency list

This means matrix checks edges instantly, while list checks take time proportional to neighbors count.

Common Mistake

[X] Wrong: "Adjacency list is always faster than matrix because it uses less space."

[OK] Correct: List uses less space but edge checks can be slower if many neighbors exist, unlike constant-time matrix access.

Interview Connect

Understanding these differences helps you pick the right graph storage for fast operations in real problems.

Self-Check

"What if the graph is very sparse? How does that affect the choice between adjacency list and matrix for edge checks?"

Practice

1. Which data structure is generally more space-efficient for representing a graph with very few edges compared to the number of vertices?

easy

A. Adjacency List

B. Adjacency Matrix

C. Edge List

D. Incidence Matrix

Solution

1. Step 1: Understand graph density

   Sparse graphs have few edges compared to vertices, so storing all possible edges wastes space.

2. Step 2: Compare space usage of adjacency list and matrix

   Adjacency list stores only existing edges, using less space for sparse graphs. Adjacency matrix always uses space for all possible edges.

3. Final Answer:

   Adjacency List -> Option A

4. Quick Check:

   Sparse graph = Adjacency List [OK]

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

Common Mistakes:

• Choosing adjacency matrix for sparse graphs
• Confusing edge count with vertex count
• Ignoring space complexity differences

2. Which of the following is the correct TypeScript syntax to declare an adjacency matrix for a graph with 3 vertices initialized with zeros?

easy

A. const matrix: number[][] = new Array(3).fill(new Array(3).fill(0));

B. const matrix: number[] = [0,0,0];

C. const matrix: number[][] = [[0,0,0],[0,0,0],[0,0,0]];

D. const matrix: number[][] = Array(3).map(() => Array(3).fill(0));

Solution

1. Step 1: Understand adjacency matrix type

   An adjacency matrix is a 2D array (array of arrays) of numbers representing edges.

2. Step 2: Check initialization correctness

   const matrix: number[][] = [[0,0,0],[0,0,0],[0,0,0]]; explicitly creates a 3x3 matrix with zeros. const matrix: number[][] = new Array(3).fill(new Array(3).fill(0)); uses fill with the same inner array reference, causing bugs. const matrix: number[][] = Array(3).map(() => Array(3).fill(0)); uses map on empty array, which won't work as expected. const matrix: number[] = [0,0,0]; is 1D array, not matrix.

3. Final Answer:

   const matrix: number[][] = [[0,0,0],[0,0,0],[0,0,0]]; -> Option C

4. Quick Check:

   Explicit 2D array with zeros = const matrix: number[][] = [[0,0,0],[0,0,0],[0,0,0]]; [OK]

Hint: Use explicit 2D array for adjacency matrix initialization [OK]

Common Mistakes:

• Using fill with nested arrays causing shared references
• Declaring 1D array instead of 2D
• Using map on empty array without initialization

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

const graph = {
  0: [1, 2],
  1: [2],
  2: [0, 3],
  3: []
};
console.log(graph[2]);

What will be the output?

medium

A. [2]

B. [1, 2]

C. undefined

D. [0, 3]

Solution

1. Step 1: Identify the adjacency list for vertex 2

   In the graph object, key '2' maps to the array [0, 3].

2. Step 2: Understand console.log output

   Logging graph[2] prints the array [0, 3] representing edges from vertex 2.

3. Final Answer:

   [0, 3] -> Option D

4. Quick Check:

   graph[2] = [0, 3] [OK]

Hint: Access adjacency list by vertex key to get connected nodes [OK]

Common Mistakes:

• Confusing keys and values in adjacency list
• Expecting adjacency matrix output
• Assuming undefined for existing keys

4. Consider this TypeScript code snippet for creating an adjacency matrix for a graph with 3 vertices:

const size = 3;
const matrix = new Array(size).fill(new Array(size).fill(0));
matrix[0][1] = 1;
console.log(matrix);

What is the problem with this code?

medium

A. All rows refer to the same inner array, so all rows change when one element changes

B. The matrix is not initialized with zeros

C. The matrix is not a 2D array

D. The size variable is not used correctly

Solution

1. Step 1: Analyze the fill method usage

   Using fill with new Array(size).fill(0) creates one inner array and fills outer array with references to the same inner array.

2. Step 2: Understand effect of modifying one element

   Changing matrix[0][1] changes that element in all rows because they share the same inner array reference.

3. Final Answer:

   All rows refer to the same inner array, so all rows change when one element changes -> Option A

4. Quick Check:

   Shared inner array reference causes unexpected changes [OK]

Hint: Avoid fill with nested arrays; use map to create separate arrays [OK]

Common Mistakes:

• Thinking fill creates separate inner arrays
• Ignoring shared reference problem
• Assuming matrix is 1D array

5. You need to represent a dense graph with 1000 vertices where you frequently check if an edge exists between any two vertices. Which representation is best and why?

hard

A. Adjacency List, because it uses less space for dense graphs

B. Adjacency Matrix, because it allows O(1) edge existence checks

C. Edge List, because it is fastest for dense graphs

D. Incidence Matrix, because it uses less space than adjacency matrix

Solution

1. Step 1: Understand graph density and operation needs

   Dense graph means many edges; checking edge existence frequently requires fast lookup.

2. Step 2: Compare adjacency list and matrix for edge checks

   Adjacency matrix allows direct O(1) access to check if edge exists between two vertices. Adjacency list requires searching neighbors, which can be slower in dense graphs.

3. Step 3: Consider space tradeoff

   Though adjacency matrix uses more space, for dense graphs this is acceptable and beneficial for speed.

4. Final Answer:

   Adjacency Matrix, because it allows O(1) edge existence checks -> Option B

5. Quick Check:

   Dense graph + frequent edge check = Adjacency Matrix [OK]

Hint: Dense graph + frequent edge check? Use adjacency matrix [OK]

Common Mistakes:

• Choosing adjacency list for dense graphs needing fast edge checks
• Ignoring time complexity of edge existence queries
• Confusing edge list with adjacency matrix benefits