What is the time complexity of Sort the Matrix Diagonally?

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

What data structures are used to solve Sort the Matrix Diagonally?

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

What are the key insights for solving Sort the Matrix Diagonally?

Insight 1: The row - column value (r - c) is constant for all elements lying on the same diagonal. Insight 2: Using a hash map (or dictionary) with the diagonal identifier (r - c) as the key allows efficient grouping of elements belonging to the same diagonal.

Which companies ask LeetCode 1329. Sort the Matrix Diagonally in interviews?

This problem is frequently asked at Quora, Yandex.

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Sort the Matrix Diagonally - LeetCode 1329 Solution

Q: What is the space complexity of Sort the Matrix Diagonally?

The space complexity is O(m*n).

Sort the Matrix Diagonally - Complete Solution Guide

Sort the Matrix Diagonally is LeetCode problem 1329, 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

A matrix diagonal is a diagonal line of cells starting from some cell in either the topmost row or leftmost column and going in the bottom-right direction until reaching the matrix's end. For example, the matrix diagonal starting from mat[2][0] , where mat is a 6 x 3 matrix, includes cells mat[2][0] , mat[3][1] , and mat[4][2] . Given an m x n matrix mat of integers, sort each matrix diagonal in ascending order and return the resulting matrix . Example 1: Input: mat = [[3,3,1,1],[2,2,1,2],[1,1,1

Detailed Explanation

The problem requires sorting each diagonal of a given m x n matrix in ascending order. A matrix diagonal is defined as a line of cells starting from any cell in the topmost row or leftmost column and going in the bottom-right direction. The task is to return the modified matrix with its diagonals sorted.

Solution Approach

The solution uses a hash map (or dictionary) to group the elements of each diagonal. The key of the hash map is the difference between the row and column indices (r - c), which uniquely identifies each diagonal. After grouping, each diagonal is sorted in ascending order. Finally, the sorted elements are placed back into their original positions in the matrix, thereby creating the sorted matrix which will be returned.

Step-by-Step Algorithm

Step 1: Initialize a hash map (or dictionary) to store the elements of each diagonal. The key will be r - c, where r is the row index and c is the column index.
Step 2: Iterate through the matrix and for each element mat[r][c], append it to the list associated with the key r - c in the hash map.
Step 3: Iterate through the hash map and sort each list of elements (i.e., each diagonal) in ascending order.
Step 4: Iterate through the matrix again. For each element mat[r][c], retrieve the sorted list associated with the key r - c from the hash map and replace mat[r][c] with the next element from the sorted list.
Step 5: Return the modified matrix.

Key Insights

Insight 1: The row - column value (r - c) is constant for all elements lying on the same diagonal.
Insight 2: Using a hash map (or dictionary) with the diagonal identifier (r - c) as the key allows efficient grouping of elements belonging to the same diagonal.
Insight 3: Sorting each diagonal separately and then placing the sorted elements back into their respective positions will achieve the desired outcome.

Complexity Analysis

Time Complexity: O(m*n*log(min(m,n)))

Space Complexity: O(m*n)

Topics

This problem involves: Array, Sorting, Matrix.

Companies

Asked at: Quora, Yandex.

Sort the Matrix Diagonally | LeetCode 1329

Medium

Array Sorting Matrix

Time: O(m*n*log(min(m,n)))Space: O(m*n)

Solution Code

Available in:

import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Collections;
import java.util.Map;

class Solution {
    public int[][] diagonalSort(int[][] mat) {
        int m = mat.length;
        int n = mat[0].length;
        Map<Integer, List<Integer>> diagonals = new HashMap<>();

        for (int r = 0; r < m; r++) {
            for (int c = 0; c < n; c++) {
                if (!diagonals.containsKey(r - c)) {
                    diagonals.put(r - c, new ArrayList<>());
                }
                diagonals.get(r - c).add(mat[r][c]);
            }
        }

        for (int key : diagonals.keySet()) {
            Collections.sort(diagonals.get(key), Collections.reverseOrder());
        }

        for (int r = 0; r < m; r++) {
            for (int c = 0; c < n; c++) {
                mat[r][c] = diagonals.get(r - c).remove(diagonals.get(r - c).size() - 1);
            }
        }

        return mat;
    }
}

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Problem Info

Description

Problem 1329: Sort the Matrix Diagonally - Detailed explanation and solution approaches.

Topics

Array Sorting Matrix

Companies

QuoraYandex

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:1329

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: Initialize a hash map (or dictionary) to store the elements of each diagonal. The key will be r - c, where r is the row index and c is the column index.
2
Step 2: Iterate through the matrix and for each element mat[r][c], append it to the list associated with the key r - c in the hash map.
3
Step 3: Iterate through the hash map and sort each list of elements (i.e., each diagonal) in ascending order.
4
Step 4: Iterate through the matrix again. For each element mat[r][c], retrieve the sorted list associated with the key r - c from the hash map and replace mat[r][c] with the next element from the sorted list.
5
Step 5: Return the modified matrix.

💡

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*log(min(m,n)))

💾

Space Complexity

O(m*n)

📝

Why?

The time complexity is O(m*n*log(min(m,n))) because we iterate through the entire matrix (m*n) and for each diagonal, we sort the elements which takes O(k*log(k)) where k is the length of the diagonal. The maximum length of a diagonal can be min(m,n). Therefore, the overall time complexity becomes O(m*n*log(min(m,n))). The space complexity is O(m*n) because, in the worst case, we might need to store all the elements of the matrix in the hash map.

📖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

In-place sorting without hash map

Instead of using a hash map, we can iterate through each possible diagonal (defined by the starting row and column) and sort the diagonal elements directly in the matrix. This avoids the extra space of the hash map.

Time: O(m*n*log(min(m,n)))Space: O(1)

Trade-offs: This approach uses less space (O(1)) compared to the hash map approach (O(m*n)), but the code is more complex and harder to read. The hash map approach is often preferred for its clarity, unless memory constraints are very tight.

✨

Key Insights

1Insight 1: The row - column value (r - c) is constant for all elements lying on the same diagonal.
2Insight 2: Using a hash map (or dictionary) with the diagonal identifier (r - c) as the key allows efficient grouping of elements belonging to the same diagonal.
3Insight 3: Sorting each diagonal separately and then placing the sorted elements back into their respective positions will achieve the desired outcome.

📚

Key Terms Explained

For Students

Array

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

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

Edge case 1: Empty matrix (m=0 or n=0). The provided solutions inherently handle this edge case by returning an empty or unchanged matrix, since the loops won't execute.
Edge case 2: Matrix with only one row or one column. The diagonal identification (r - c) and sorting still work correctly in these cases.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the diagonal identifier. The diagonal identifier is r - c, not r + c. Using r + c would group elements along anti-diagonals (top-left to bottom-right).

✗Common mistake 2: Not sorting the diagonals in ascending order as specified in the problem statement. The sorting step is crucial for the correctness of the solution.

✗Common mistake 3: Not handling negative indices when storing diagonals in the hash map. The diagonal identifier (r - c) can be negative, so the hash map implementation or language-specific features like `defaultdict` must handle negative keys correctly.

✅

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. Diagonal sorting can be used as a part of image processing algorithms for noise reduction or feature extraction along diagonal lines of pixels.

🏦Real-world scenario 2: Data analysis. Analyzing data trends along diagonals in a spreadsheet or a data matrix can reveal patterns or correlations in datasets such as sales data or market trends. The algorithm can be adapted to identify and sort these trends.

💡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: August 9, 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

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

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

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

@AlgoMasterJun 30

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Sort the Matrix Diagonally"?

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 sorting techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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