Adjacency Matrix Representation in DSA Typescript - Time & Space Complexity

Time Complexity: Adjacency Matrix Representation

O(n²)

Understanding Time Complexity

We want to understand how the time to check or build connections in a graph grows as the graph gets bigger.

How does the number of nodes affect the work done using an adjacency matrix?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function printAdjacencyMatrix(matrix: number[][]): void {
  const n = matrix.length;
  for (let i = 0; i < n; i++) {
    for (let j = 0; j < n; j++) {
      console.log(`Edge from ${i} to ${j}: ${matrix[i][j]}`);
    }
  }
}
    

This code prints all edges in a graph represented by an adjacency matrix of size n x n.

Identify Repeating Operations

• Primary operation: Nested loops over the matrix rows and columns.
• How many times: Outer loop runs n times, inner loop runs n times for each outer loop, total n x n = n² times.

How Execution Grows With Input

As the number of nodes n grows, the total operations grow by n squared because we check every possible pair of nodes.

Input Size (n)	Approx. Operations
10	100
100	10,000
1000	1,000,000

Pattern observation: Doubling the number of nodes roughly quadruples the work.

Final Time Complexity

Time Complexity: O(n²)

This means the time to process the adjacency matrix grows with the square of the number of nodes.

Common Mistake

[X] Wrong: "Checking all edges in an adjacency matrix is a quick, linear operation."

[OK] Correct: Because the matrix has n rows and n columns, you must check n x n entries, so the work grows much faster than just n.

Interview Connect

Understanding adjacency matrix time complexity helps you explain graph operations clearly and choose the right graph representation for different problems.

Self-Check

"What if we used an adjacency list instead of a matrix? How would the time complexity for checking edges change?"

Practice

1. What does an adjacency matrix represent in a graph?
Adjacency matrix is a way to show connections between nodes. What does a 1 in the matrix mean?

easy

A. There is no edge between two nodes

B. The graph is weighted

C. The nodes are the same

D. There is a direct edge between two nodes

Solution

1. Step 1: Understand adjacency matrix basics

   An adjacency matrix uses rows and columns to represent nodes. Each cell shows if two nodes connect.

2. Step 2: Interpret the value 1 in the matrix

   A value of 1 means there is a direct edge between the nodes represented by that row and column.

3. Final Answer:

   There is a direct edge between two nodes -> Option D

4. Quick Check:

   1 means edge exists [OK]

Hint: 1 means connected, 0 means no connection [OK]

Common Mistakes:

• Thinking 1 means no edge
• Confusing node identity with connection
• Assuming matrix shows weights always

2. Which of the following is the correct TypeScript syntax to declare a 3x3 adjacency matrix initialized with zeros without shared row references?

easy

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

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

C. const matrix = Array(3).map(Array(3).fill(0));

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

Solution

1. Step 1: Check TypeScript array declaration syntax

   const matrix: number[][] = [[0,0,0],[0,0,0],[0,0,0]]; uses explicit 2D array with nested arrays and zeros, which is valid syntax and creates independent rows.

2. Step 2: Identify issues in other options

   const matrix: number[3][3] = new Array(3).fill(new Array(3).fill(0)); uses invalid TypeScript syntax for fixed-size arrays. const matrix = Array(3).map(Array(3).fill(0)); passes an array to map() instead of a function, causing a runtime error. const matrix: number[][] = new Array(3).fill(new Array(3).fill(0)); uses fill with the same inner array reference, causing shared rows.

3. Final Answer:

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

4. Quick Check:

   Explicit nested arrays with zeros [OK]

Hint: Use explicit nested arrays to avoid shared references [OK]

Common Mistakes:

• Using fill with same inner array causing shared rows
• Passing non-function to map()
• Wrong type annotations

3. Given the following adjacency matrix for a directed graph:

const matrix = [
  [0, 1, 0],
  [0, 0, 1],
  [1, 0, 0]
];
console.log(matrix[2][0]);

What will be the output?

medium

A. 1

B. Error

C. undefined

D. 0

Solution

1. Step 1: Locate matrix[2][0]

   Row 2 is [1, 0, 0], so element at index 0 is 1.

2. Step 2: Confirm output of console.log

   console.log prints the value 1 found at matrix[2][0].

3. Final Answer:

   1 -> Option A

4. Quick Check:

   matrix[2][0] = 1 [OK]

Hint: Remember matrix[row][column] indexing [OK]

Common Mistakes:

• Confusing row and column indices
• Expecting zero instead of one
• Thinking index out of range

4. Consider this TypeScript code snippet to create an adjacency matrix for 3 nodes:

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. It creates a matrix with all zeros correctly

B. The matrix is not initialized at all

C. All rows point to the same inner array, so all rows change when one changes

D. It throws a runtime error

Solution

1. Step 1: Understand fill with reference types

   Using fill with new Array(size).fill(0) creates one inner array shared by all rows.

2. Step 2: Effect of modifying matrix[0][1]

   Changing matrix[0][1] changes the shared inner array, so all rows show the change.

3. Final Answer:

   All rows point to the same inner array, so all rows change when one changes -> Option C

4. Quick Check:

   Shared inner array causes unexpected updates [OK]

Hint: Avoid fill with objects/arrays; use map for unique rows [OK]

Common Mistakes:

• Assuming fill creates independent arrays
• Ignoring shared reference behavior
• Expecting only one row to change

5. You have an undirected graph with 4 nodes. You want to represent it using an adjacency matrix in TypeScript. Which of the following code snippets correctly initializes the matrix and sets an edge between node 1 and node 3?

hard

A. const size = 4;
const matrix = new Array(size).fill(new Array(size).fill(0));
matrix[1][3] = 1;
matrix[3][1] = 1;

B. const size = 4;
const matrix = Array.from({ length: size }, () => Array(size).fill(0));
matrix[1][3] = 1;
matrix[3][1] = 1;

C. const size = 4;
const matrix = Array(size).fill(Array(size).fill(0));
matrix[1][3] = 1;

D. const size = 4;
const matrix = Array(size).map(() => Array(size).fill(0));
matrix[1][3] = 1;

Solution

1. Step 1: Check matrix initialization for unique rows

   const size = 4; const matrix = Array.from({ length: size }, () => Array(size).fill(0)); matrix[1][3] = 1; matrix[3][1] = 1; uses Array.from with a function to create unique inner arrays, avoiding shared references.

2. Step 2: Verify setting edges for undirected graph

   Setting matrix[1][3] and matrix[3][1] to 1 correctly represents an undirected edge.

3. Step 3: Identify issues in other options

   Options B and C use fill causing shared inner arrays. const size = 4; const matrix = Array(size).map(() => Array(size).fill(0)); matrix[1][3] = 1; initializes correctly but only sets matrix[1][3] = 1, missing matrix[3][1] = 1 for undirected graph.

4. Final Answer:

   const size = 4; const matrix = Array.from({ length: size }, () => Array(size).fill(0)); matrix[1][3] = 1; matrix[3][1] = 1; -> Option B

5. Quick Check:

   Use Array.from with function for unique rows [OK]

Hint: Use Array.from with function to avoid shared rows [OK]

Common Mistakes:

• Using fill causing shared inner arrays
• Using map on empty array without initialization
• Setting only one direction for undirected edge