What is the time complexity of Rotate Image?

The optimal solution has a time complexity of O(n^2).

What data structures are used to solve Rotate Image?

This problem uses Array, Math, Matrix. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Rotate Image?

Insight 1: Rotating a matrix 90 degrees clockwise can be achieved by first transposing the matrix (swapping rows and columns) and then reversing each row. Insight 2: The in-place constraint means you must manipulate the original matrix directly. This limits your options but also simplifies the space complexity.

Which companies ask LeetCode 48. Rotate Image in interviews?

This problem is frequently asked at AMD, Accenture, Adobe, Amazon, Apple and 28 more companies.

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Rotate Image - LeetCode 48 Solution

Rotate Image - Complete Solution Guide

Rotate Image is LeetCode problem 48, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an n x n 2D matrix representing an image, rotate the image by 90 degrees (clockwise). You have to rotate the image in-place , which means you have to modify the input 2D matrix directly. DO NOT allocate another 2D matrix and do the rotation. Example 1: Input: matrix = [[1,2,3],[4,5,6],[7,8,9]] Output: [[7,4,1],[8,5,2],[9,6,3]] Example 2: Input: matrix = [[5,1,9,11],[2,4,8,10],[13,3,6,7],[15,14,12,16]] Output: [[15,13,2,5],[14,3,4,1],[12,6,8,9],[16,7,10,11]] Constraints: n == matrix

Detailed Explanation

The problem requires rotating an `n x n` 2D matrix representing an image by 90 degrees clockwise in-place. This means modifying the original matrix directly without using extra space for a new matrix. The input is a 2D array (matrix) of integers, and the output is the modified input matrix after the rotation.

Solution Approach

The provided solution approach involves two main steps: transposing the matrix and then reversing each row. Transposing involves swapping elements across the main diagonal (i.e., swapping matrix[i][j] with matrix[j][i]). Reversing each row involves swapping the elements from the start and end of the row, moving towards the middle.

Step-by-Step Algorithm

Step 1: Iterate through the matrix to transpose it. This involves iterating over the upper triangle of the matrix (i.e., i from 0 to n-1, and j from i+1 to n-1) and swapping matrix[i][j] with matrix[j][i]. This avoids redundant swaps.
Step 2: Iterate through each row of the matrix.
Step 3: Reverse each row by using two pointers (left and right) initialized to the start and end of the row. Swap the elements at the left and right pointers, and then move the left pointer one step to the right and the right pointer one step to the left. Continue until the left and right pointers meet.
Step 4: The matrix is now rotated 90 degrees clockwise.

Key Insights

Insight 1: Rotating a matrix 90 degrees clockwise can be achieved by first transposing the matrix (swapping rows and columns) and then reversing each row.
Insight 2: The in-place constraint means you must manipulate the original matrix directly. This limits your options but also simplifies the space complexity.
Insight 3: Understanding the index transformations during transpose and row reversal is crucial for correct implementation.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(1)

Topics

This problem involves: Array, Math, Matrix.

Companies

Asked at: AMD, Accenture, Adobe, Amazon, Apple, Bloomberg, Capital One, Cisco, ConsultAdd, DE Shaw, Goldman Sachs, Google, IBM, Infosys, Intel, J.P. Morgan, Mastercard, Meta, Microsoft, Netflix, Nvidia, Oracle, PayPal, Qualcomm, Rakuten, Roblox, Tinkoff, Uber, WatchGuard, Yahoo, Yandex, ZScaler, Zoho.

Rotate Image | LeetCode 48

Medium

Array Math Matrix

Time: O(n^2)Space: O(1)

Solution Code

Available in:

class Solution {
    public void rotate(int[][] matrix) {
        int n = matrix.length;
        
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                int temp = matrix[i][j];
                matrix[i][j] = matrix[j][i];
                matrix[j][i] = temp;
            }
        }
        
        for (int i = 0; i < n; i++) {
            int left = 0;
            int right = n - 1;
            while (left < right) {
                int temp = matrix[i][left];
                matrix[i][left] = matrix[i][right];
                matrix[i][right] = temp;
                left++;
                right--;
            }
        }
    }
}

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Problem Info

Description

Problem 48: Rotate Image - Detailed explanation and solution approaches.

Topics

Array Math Matrix

Companies

AMDAccentureAdobeAmazonAppleBloombergCapital OneCisco

Stats

Difficulty:Medium

Languages:4

Companies:33

Problem #:48

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Step 1: Iterate through the matrix to transpose it. This involves iterating over the upper triangle of the matrix (i.e., i from 0 to n-1, and j from i+1 to n-1) and swapping matrix[i][j] with matrix[j][i]. This avoids redundant swaps.
2
Step 2: Iterate through each row of the matrix.
3
Step 3: Reverse each row by using two pointers (left and right) initialized to the start and end of the row. Swap the elements at the left and right pointers, and then move the left pointer one step to the right and the right pointer one step to the left. Continue until the left and right pointers meet.
4
Step 4: The matrix is now rotated 90 degrees clockwise.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is determined by the nested loops used for transposing and reversing. The space complexity is constant because we don't create any additional data structures that scale with the input size.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Rotation by Layers

Rotate the matrix layer by layer. For each layer, move the elements from top to right, right to bottom, bottom to left, and left to top. This approach is more intuitive to understand.

Time: O(n^2)Space: O(1)

Trade-offs: While conceptually simpler, the implementation can be more verbose and potentially harder to get correct due to managing indices and temporary variables for each layer. The provided solution is generally preferred for its conciseness and clarity.

✨

Key Insights

1Insight 1: Rotating a matrix 90 degrees clockwise can be achieved by first transposing the matrix (swapping rows and columns) and then reversing each row.
2Insight 2: The in-place constraint means you must manipulate the original matrix directly. This limits your options but also simplifies the space complexity.
3Insight 3: Understanding the index transformations during transpose and row reversal is crucial for correct implementation.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: Empty matrix or a matrix with only one element. The code will still work correctly as the loops will not execute or execute minimally, respectively. The transpose and reverse steps will effectively do nothing, and the input matrix will remain unchanged, which is technically a valid 90-degree rotation.
Edge case 2: Non-square matrix: The problem statement specifies an n x n matrix, so a non-square matrix is not a valid input according to the constraints. However, if a non-square matrix were allowed, this approach would need significant modifications as the transpose step wouldn't work as intended and result in incorrect output.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Transposing the entire matrix (i.e., iterating with `j` from `0` to `n-1` instead of `i+1` to `n-1`). This will cause elements to be swapped twice, effectively undoing the transpose and leading to an incorrect rotation.

✗Common mistake 2: Incorrectly handling indices when reversing the rows. Off-by-one errors can easily occur when setting the initial values of the `left` and `right` pointers or when updating them inside the `while` loop.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Image processing. Rotating images is a fundamental operation in many image editing applications and computer vision systems. This algorithm can be applied directly to rotate image pixel data stored in a matrix.

🏦Real-world scenario 2: Game development. Rotating game boards or textures can be achieved using matrix rotation. This is particularly relevant in tile-based games or 3D environments where objects need to be oriented in different directions.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: July 19, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 13

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 6

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Rotate Image"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether math techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?