What is the time complexity of Maximum Sum of an Hourglass?

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

What data structures are used to solve Maximum Sum of an Hourglass?

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

What are the key insights for solving Maximum Sum of an Hourglass?

Insight 1: The core task is to iterate through all possible hourglass positions within the grid. The dimensions of the grid limit the possible positions. Insight 2: Since the size of the hourglass is fixed (3x3), we can directly calculate the sum of elements forming the hourglass for each position.

Which companies ask LeetCode 2428. Maximum Sum of an Hourglass in interviews?

This problem is frequently asked at Nutanix, Zoho.

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Maximum Sum of an Hourglass - LeetCode 2428 Solution

Maximum Sum of an Hourglass - Complete Solution Guide

Maximum Sum of an Hourglass is LeetCode problem 2428, 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 m x n integer matrix grid . We define an hourglass as a part of the matrix with the following form: Return the maximum sum of the elements of an hourglass . Note that an hourglass cannot be rotated and must be entirely contained within the matrix. Example 1: Input: grid = [[6,2,1,3],[4,2,1,5],[9,2,8,7],[4,1,2,9]] Output: 30 Explanation: The cells shown above represent the hourglass with the maximum sum: 6 + 2 + 1 + 2 + 9 + 2 + 8 = 30. Example 2: Input: grid = [[1,2,3],[4,5,6],[7

Detailed Explanation

The problem requires finding the maximum sum of an hourglass shape within a given m x n integer matrix (grid). An hourglass is defined as a specific 3x3 pattern where the middle element of the second row is included, but the first and last elements of the second row are not. The hourglass must be fully contained within the bounds of the matrix, and we need to return the largest sum among all possible hourglasses in the grid.

Solution Approach

The provided solution uses a brute-force approach. It iterates through all possible starting positions for an hourglass within the grid. For each position, it calculates the sum of the elements that form the hourglass shape. It then compares this sum with the current maximum sum and updates the maximum sum if the current hourglass sum is greater. This process continues until all possible hourglass positions have been considered.

Step-by-Step Algorithm

Step 1: Determine the dimensions of the grid (m rows and n columns).
Step 2: Initialize a variable `max_sum` to store the maximum hourglass sum encountered so far (initialized to 0).
Step 3: Iterate through the grid using nested loops. The outer loop iterates from `i = 0` to `m - 2`, and the inner loop iterates from `j = 0` to `n - 2`. These loops define the top-left corner of the hourglass.
Step 4: Inside the inner loop, calculate the sum of the elements forming the hourglass centered at `grid[i+1][j+1]`. This involves summing `grid[i][j] + grid[i][j+1] + grid[i][j+2] + grid[i+1][j+1] + grid[i+2][j] + grid[i+2][j+1] + grid[i+2][j+2]`.
Step 5: Compare the calculated hourglass sum (`current_sum`) with the current `max_sum`. If `current_sum` is greater than `max_sum`, update `max_sum` to `current_sum`.
Step 6: After iterating through all possible positions, return the `max_sum`.

Key Insights

Insight 1: The core task is to iterate through all possible hourglass positions within the grid. The dimensions of the grid limit the possible positions.
Insight 2: Since the size of the hourglass is fixed (3x3), we can directly calculate the sum of elements forming the hourglass for each position.
Insight 3: The problem can be solved without using extra space. We only need to keep track of the maximum hourglass sum encountered so far.

Complexity Analysis

Time Complexity: O(m*n)

Space Complexity: O(1)

Topics

This problem involves: Array, Matrix, Prefix Sum.

Companies

Asked at: Nutanix, Zoho.

Maximum Sum of an Hourglass | LeetCode 2428

Medium

Array Matrix Prefix Sum

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

Solution Code

Available in:

class Solution {
    public int maxSum(int[][] grid) {
        int m = grid.length;
        int n = grid[0].length;
        int max_sum = 0;
        
        for (int i = 0; i <= m - 3; i++) {
            for (int j = 0; j <= n - 3; j++) {
                int current_sum = (
                    grid[i][j] + grid[i][j+1] + grid[i][j+2] +
                    grid[i+1][j+1] +
                    grid[i+2][j] + grid[i+2][j+1] + grid[i+2][j+2]
                );
                if (current_sum > max_sum) {
                    max_sum = current_sum;
                }
            }
        }
        
        return max_sum;
    }
}

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Problem Info

Description

Problem 2428: Maximum Sum of an Hourglass - Detailed explanation and solution approaches.

Topics

Array Matrix Prefix Sum

Companies

NutanixZoho

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:2428

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

6 steps

1
Step 1: Determine the dimensions of the grid (m rows and n columns).
2
Step 2: Initialize a variable `max_sum` to store the maximum hourglass sum encountered so far (initialized to 0).
3
Step 3: Iterate through the grid using nested loops. The outer loop iterates from `i = 0` to `m - 2`, and the inner loop iterates from `j = 0` to `n - 2`. These loops define the top-left corner of the hourglass.
4
Step 4: Inside the inner loop, calculate the sum of the elements forming the hourglass centered at `grid[i+1][j+1]`. This involves summing `grid[i][j] + grid[i][j+1] + grid[i][j+2] + grid[i+1][j+1] + grid[i+2][j] + grid[i+2][j+1] + grid[i+2][j+2]`.
5
Step 5: Compare the calculated hourglass sum (`current_sum`) with the current `max_sum`. If `current_sum` is greater than `max_sum`, update `max_sum` to `current_sum`.
6
Step 6: After iterating through all possible positions, return the `max_sum`.

💡

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 the solution iterates through all possible positions of the hourglass in the grid. The nested loops iterate (m-2) * (n-2) times, which is proportional to m*n. The space complexity is O(1) because the solution uses a constant amount of extra space, regardless of the size of the input grid. We only use a few integer variables to store dimensions and the maximum sum.

📖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

Prefix Sum Approach

A prefix sum array could be pre-computed to allow for constant time calculation of sums of sub-matrices. This could potentially improve the constant factors of the calculation.

Time: O(m*n) for preprocessing, O(m*n) for hourglass calculation. Total O(m*n)Space: O(m*n) for the prefix sum array

Trade-offs: The prefix sum approach provides the same overall time complexity but uses more memory. It might be faster in practice if the hourglass sum calculation is significantly more expensive than array accesses, but in this case, the brute force approach is simple and sufficiently efficient.

✨

Key Insights

1Insight 1: The core task is to iterate through all possible hourglass positions within the grid. The dimensions of the grid limit the possible positions.
2Insight 2: Since the size of the hourglass is fixed (3x3), we can directly calculate the sum of elements forming the hourglass for each position.
3Insight 3: The problem can be solved without using extra space. We only need to keep track of the maximum hourglass sum encountered so far.

📚

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: When the grid dimensions are exactly 3x3, there is only one possible hourglass. The algorithm correctly calculates and returns its sum.
Edge case 2: The problem statement specifies m,n >=3. The algorithm will work correctly on the boundary. It is guaranteed to have at least one valid hourglass configuration.
Edge case 3: The grid contains very large positive numbers. This is handled correctly since we're just summing integers. Overflow is not a consideration given constraints.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the indices of the elements forming the hourglass. It's easy to make off-by-one errors.

✗Common mistake 2: Not considering the constraints of the grid and trying to access elements outside of the grid's boundaries.

✗Common mistake 3: Forgetting to initialize the `max_sum` variable. If not initialized, it may contain garbage value, leading to incorrect results.

✅

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 - detecting patterns in images (though hourglass is an unusual pattern).

🏦Real-world scenario 2: Financial data analysis - identifying patterns in a matrix of financial data to predict market trends or detect anomalies.

🎮Real-world scenario 3: Game development - searching for specific configurations in a game board.

💡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

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ 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 28, 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 "Maximum Sum of an Hourglass"?

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

🤖 Need help with this problem?