What is the time complexity of Toeplitz Matrix?

The optimal solution has a time complexity of O(m*n).

What data structures are used to solve Toeplitz Matrix?

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

What are the key insights for solving Toeplitz Matrix?

Insight 1: The core concept is checking if matrix[i][j] == matrix[i+1][j+1] for every valid i and j. Insight 2: We can iterate through the matrix and compare adjacent diagonal elements efficiently.

Which companies ask LeetCode 766. Toeplitz Matrix in interviews?

This problem is frequently asked at tcs.

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Toeplitz Matrix - LeetCode 766 Solution

Toeplitz Matrix - Complete Solution Guide

Toeplitz Matrix is LeetCode problem 766, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given an m x n matrix , return true if the matrix is Toeplitz. Otherwise, return false . A matrix is Toeplitz if every diagonal from top-left to bottom-right has the same elements. Example 1: Input: matrix = [[1,2,3,4],[5,1,2,3],[9,5,1,2]] Output: true Explanation: In the above grid, the diagonals are: "[9]", "[5, 5]", "[1, 1, 1]", "[2, 2, 2]", "[3, 3]", "[4]". In each diagonal all elements are the same, so the answer is True. Example 2: Input: matrix = [[1,2],[2,2]] Output: false Explanation: T

Detailed Explanation

The problem asks us to determine if a given m x n matrix is a Toeplitz matrix. A matrix is Toeplitz if every diagonal from top-left to bottom-right has the same element. In other words, for any cell matrix[i][j], it should be equal to matrix[i+1][j+1] as long as both indices are within the bounds of the matrix. We need to return `true` if the matrix is Toeplitz, and `false` otherwise.

Solution Approach

The solution iterates through the matrix, comparing each element with its bottom-right neighbor. If any two such neighbors are not equal, the matrix is not Toeplitz, and we return `false`. If the entire matrix is traversed without finding any unequal neighbors, it means all diagonals have the same elements, and we return `true`.

Step-by-Step Algorithm

Step 1: Iterate through the rows of the matrix from index 0 to rows - 2 (exclusive of the last row).
Step 2: Iterate through the columns of the matrix from index 0 to cols - 2 (exclusive of the last column).
Step 3: Inside the inner loop, compare the current element matrix[i][j] with its bottom-right neighbor matrix[i+1][j+1].
Step 4: If the elements are not equal, immediately return `false` because the matrix is not Toeplitz.
Step 5: After the loops complete without finding any unequal neighbors, return `true` because the matrix is Toeplitz.

Key Insights

Insight 1: The core concept is checking if matrix[i][j] == matrix[i+1][j+1] for every valid i and j.
Insight 2: We can iterate through the matrix and compare adjacent diagonal elements efficiently.
Insight 3: No additional data structures are necessary; we can perform the check in-place.

Complexity Analysis

Time Complexity: O(m*n)

Space Complexity: O(1)

Topics

This problem involves: Array, Matrix.

Companies

Asked at: tcs.

Toeplitz Matrix | LeetCode 766

Easy

Array Matrix

Time: O(m*n)Space: O(1)

Solution Code

Available in:

class Solution {
    public boolean isToeplitzMatrix(int[][] matrix) {
        for (int i = 0; i < matrix.length - 1; i++) {
            for (int j = 0; j < matrix[0].length - 1; j++) {
                if (matrix[i][j] != matrix[i + 1][j + 1]) {
                    return false;
                }
            }
        }
        return true;
    }
}

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Problem Info

Description

Problem 766: Toeplitz Matrix - Detailed explanation and solution approaches.

Topics

Array Matrix

Companies

tcs

Stats

Difficulty:Easy

Languages:4

Companies:1

Problem #:766

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

5 steps

1
Step 1: Iterate through the rows of the matrix from index 0 to rows - 2 (exclusive of the last row).
2
Step 2: Iterate through the columns of the matrix from index 0 to cols - 2 (exclusive of the last column).
3
Step 3: Inside the inner loop, compare the current element matrix[i][j] with its bottom-right neighbor matrix[i+1][j+1].
4
Step 4: If the elements are not equal, immediately return `false` because the matrix is not Toeplitz.
5
Step 5: After the loops complete without finding any unequal neighbors, return `true` because the matrix is Toeplitz.

💡

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(m*n)

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is O(m*n) because we iterate through almost every element of the matrix once. The space complexity is O(1) because we only use a constant amount of extra space for loop counters and comparison operations, irrespective of the matrix 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

Diagonal-Based Iteration

Iterate through each diagonal starting from the first column of the first row and first row of the first column. For each diagonal, compare all elements to verify that they are same.

Time: O(m*n)Space: O(1)

Trade-offs: This approach is conceptually different, but it does not offer any performance advantages over the presented approach. Also, implementing this approach could be more complex than the provided solution.

✨

Key Insights

1Insight 1: The core concept is checking if matrix[i][j] == matrix[i+1][j+1] for every valid i and j.
2Insight 2: We can iterate through the matrix and compare adjacent diagonal elements efficiently.
3Insight 3: No additional data structures are necessary; we can perform the check in-place.

📚

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 0 rows or columns. The code handles it correctly, it will either return true (if the outer loop is skipped) or perform calculations on valid indices.
Edge case 2: A matrix with only one row or one column. The loops will not execute, and the function will return `true`, which is the correct behavior for a Toeplitz matrix.
Edge case 3: matrix with only one element. The loops will not execute, and the function returns true, because a single element is always a Toeplitz matrix.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Iterating beyond the matrix boundaries when comparing matrix[i][j] and matrix[i+1][j+1]. The provided code avoids this by iterating to rows-1 and cols-1.

✗Common mistake 2: Not handling empty matrix cases. The code correctly handles empty matrices or matrices with 0 dimensions.

✅

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: Identifying textures or patterns that exhibit Toeplitz-like properties. For example, certain structured noise patterns might resemble a Toeplitz structure.

🏦Real-world scenario 2: Signal processing: Analyzing signals that have a Toeplitz autocorrelation matrix. This often arises in stationary stochastic processes.

💡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

Use this problem to demonstrate clean code practices. Variable naming and code organization matter.

⚠️ 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 17, 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

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 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.

How should I approach easy-level algorithm problems?

Easy problems build foundational skills. Focus on: 1) Understanding the problem completely before coding, 2) Writing clean, readable code, 3) Handling all edge cases (empty input, single element, etc.), 4) Analyzing time and space complexity. Don't skip complexity analysis even for simple solutions.

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 "Toeplitz Matrix"?

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

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