What is the time complexity of Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts?

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

What is the space complexity of Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts?

The space complexity is O(n + m).

What data structures are used to solve Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts?

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

What are the key insights for solving Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts?

Insight 1: The maximum area is determined by the largest gap between horizontal cuts and the largest gap between vertical cuts. The positions of other cuts don't influence the largest possible piece. Insight 2: Sorting the cut locations allows us to easily find the largest gap between consecutive cuts by iterating through the sorted lists and computing the differences.

JavaScript is required for the full interactive experience. However, all content is accessible without JavaScript.

Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts - LeetCode 1465 Solution

Q: Which companies ask LeetCode 1465. Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts in interviews?

This problem is frequently asked at BNY Mellon, IXL.

Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts - Complete Solution Guide

Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts is LeetCode problem 1465, 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 a rectangular cake of size h x w and two arrays of integers horizontalCuts and verticalCuts where: horizontalCuts[i] is the distance from the top of the rectangular cake to the i th horizontal cut and similarly, and verticalCuts[j] is the distance from the left of the rectangular cake to the j th vertical cut. Return the maximum area of a piece of cake after you cut at each horizontal and vertical position provided in the arrays horizontalCuts and verticalCuts . Since the answer ca

Detailed Explanation

The problem asks us to find the maximum area of a piece of cake after making horizontal and vertical cuts. We are given the dimensions of the cake (h x w) and two arrays representing the locations of the horizontal and vertical cuts. The goal is to determine which combination of horizontal and vertical cut intervals yields the largest area, and return that area modulo 10^9 + 7.

Solution Approach

The solution involves first sorting the horizontal and vertical cuts. Then, we find the maximum difference between consecutive horizontal cuts, which represents the maximum possible height. Similarly, we find the maximum difference between consecutive vertical cuts, which represents the maximum possible width. Finally, we multiply the maximum height and maximum width and take the result modulo 10^9 + 7.

Step-by-Step Algorithm

Step 1: Sort the horizontalCuts and verticalCuts arrays in ascending order.
Step 2: Add 0 and h to the horizontalCuts array and 0 and w to the verticalCuts array. This ensures that we consider the edges of the cake.
Step 3: Iterate through the sorted horizontalCuts and calculate the maximum difference between adjacent cuts. This gives the maximum height of a piece.
Step 4: Iterate through the sorted verticalCuts and calculate the maximum difference between adjacent cuts. This gives the maximum width of a piece.
Step 5: Multiply the maximum height and maximum width, then take the modulo with (10^9 + 7).
Step 6: Return the result.

Key Insights

Insight 1: The maximum area is determined by the largest gap between horizontal cuts and the largest gap between vertical cuts. The positions of other cuts don't influence the largest possible piece.
Insight 2: Sorting the cut locations allows us to easily find the largest gap between consecutive cuts by iterating through the sorted lists and computing the differences.
Insight 3: Remember to include the cake edges (0 and h/w) as boundaries when calculating the maximum gaps, as the largest piece may be along the edge.

Complexity Analysis

Time Complexity: O(n log n + m log m)

Space Complexity: O(n + m)

Topics

This problem involves: Array, Greedy, Sorting.

Companies

Asked at: BNY Mellon, IXL.

Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts | LeetCode 1465

Medium

Array Greedy Sorting

Time: O(n log n + m log m)Space: O(n + m)

Solution Code

Available in:

import java.util.Arrays;

class Solution {
    public int maxArea(int h, int w, int[] horizontalCuts, int[] verticalCuts) {
        int MOD = 1000000007;

        int[] horizontalCutsExtended = Arrays.copyOf(horizontalCuts, horizontalCuts.length + 2);
        horizontalCutsExtended[horizontalCuts.length] = 0;
        horizontalCutsExtended[horizontalCuts.length + 1] = h;
        Arrays.sort(horizontalCutsExtended);

        int[] verticalCutsExtended = Arrays.copyOf(verticalCuts, verticalCuts.length + 2);
        verticalCutsExtended[verticalCuts.length] = 0;
        verticalCutsExtended[verticalCuts.length + 1] = w;
        Arrays.sort(verticalCutsExtended);

        long max_height = 0;
        for (int i = 1; i < horizontalCutsExtended.length; i++) {
            max_height = Math.max(max_height, horizontalCutsExtended[i] - horizontalCutsExtended[i-1]);
        }

        long max_width = 0;
        for (int i = 1; i < verticalCutsExtended.length; i++) {
            max_width = Math.max(max_width, verticalCutsExtended[i] - verticalCutsExtended[i-1]);
        }

        return (int)((max_height * max_width) % MOD);
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 1465: Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts - Detailed explanation and solution approaches.

Topics

Array Greedy Sorting

Companies

BNY MellonIXL

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:1465

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: Sort the horizontalCuts and verticalCuts arrays in ascending order.
2
Step 2: Add 0 and h to the horizontalCuts array and 0 and w to the verticalCuts array. This ensures that we consider the edges of the cake.
3
Step 3: Iterate through the sorted horizontalCuts and calculate the maximum difference between adjacent cuts. This gives the maximum height of a piece.
4
Step 4: Iterate through the sorted verticalCuts and calculate the maximum difference between adjacent cuts. This gives the maximum width of a piece.
5
Step 5: Multiply the maximum height and maximum width, then take the modulo with (10^9 + 7).
6
Step 6: Return the result.

💡

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 log n + m log m)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(n + m)

📝

Why?

Sorting the two arrays takes O(n log n) and O(m log m) time, respectively. The rest of the algorithm involves iterating through the sorted arrays, which takes O(n) and O(m) time. Therefore, the overall time complexity is dominated by the sorting steps. Space complexity depends on if the sort is in-place or not, plus the potential need for extra space to store extended arrays containing 0 and the height or width of the cake.

📖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

Using a Heap (Priority Queue)

Instead of sorting, we could insert all the cuts (including 0 and h/w for horizontal, and 0 and w for vertical) into a min-heap. Repeatedly remove the minimum element and calculate the difference with the next element. However, this is less efficient because building a heap is still O(n) time and calculating all diffs requires extracting n elements which gives an O(n log n) runtime.

Time: O(n log n + m log m)Space: O(n + m)

Trade-offs: Not really a beneficial alternative since it doesn't improve time complexity and requires more space.

✨

Key Insights

1Insight 1: The maximum area is determined by the largest gap between horizontal cuts and the largest gap between vertical cuts. The positions of other cuts don't influence the largest possible piece.
2Insight 2: Sorting the cut locations allows us to easily find the largest gap between consecutive cuts by iterating through the sorted lists and computing the differences.
3Insight 3: Remember to include the cake edges (0 and h/w) as boundaries when calculating the maximum gaps, as the largest piece may be along the edge.

📚

Key Terms Explained

For Students

Array

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

Greedy Algorithm

Making the best choice at each step without looking ahead.

Sorting

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

Edge Cases

Edge case 1: When either horizontalCuts or verticalCuts is empty. In this case, the maximum gap will simply be the height or width of the cake. The solution implicitly handles this by including 0 and h/w as cut locations.
Edge case 2: Large values of h and w that, when multiplied, could cause integer overflow. The solution handles this by using long long (or long in Java/C) data types for calculations and taking the modulo at the end.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to include the cake edges (0 and h/w) as potential cut locations. This can lead to an incorrect calculation of the maximum height or width.

✗Common mistake 2: Not taking the modulo at the end, which can lead to incorrect results for large values of h and w.

✗Common mistake 3: Using int instead of long (or long long) for calculations, which can lead to integer overflow errors.

✅

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: Resource allocation. Imagine dividing a piece of land into plots. Horizontal and vertical "cuts" represent the boundaries defined by laws or regulations. You want to determine the largest possible rectangular plot.

🏦Real-world scenario 2: Cutting materials in manufacturing (e.g., cloth, metal sheets, glass). Minimizing waste often involves finding the largest usable rectangular area after a series of cuts have been made.

💡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.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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: June 26, 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

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

@DevExpert92Jul 14

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

@AlgoMasterJul 7

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

@InterviewProJun 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.

What are common greedy problem patterns?

Activity selection (sort by end time), interval scheduling (sort by start/end), Huffman coding (always combine smallest), fractional knapsack (sort by value/weight ratio), and minimum spanning tree. Look for problems where local decisions don't affect future options.

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 "Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts"?

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

🤖 Need help with this problem?