What is the time complexity of Maximum Area Rectangle With Point Constraints I?

The optimal solution has a time complexity of O(n^3).

What data structures are used to solve Maximum Area Rectangle With Point Constraints I?

This problem uses Array, Math, Binary Indexed Tree, Segment Tree, Geometry, Sorting, Enumeration. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum Area Rectangle With Point Constraints I?

Insight 1: The problem requires us to check all possible combinations of four points to form a rectangle. Insight 2: The constraint about no other points inside or on the border is crucial and needs careful checking.

Which companies ask LeetCode 3380. Maximum Area Rectangle With Point Constraints I in interviews?

This problem is frequently asked at UKG.

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Maximum Area Rectangle With Point Constraints I - LeetCode 3380 Solution

Maximum Area Rectangle With Point Constraints I - Complete Solution Guide

Maximum Area Rectangle With Point Constraints I is LeetCode problem 3380, 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 array points where points[i] = [x i , y i ] represents the coordinates of a point on an infinite plane. Your task is to find the maximum area of a rectangle that: Can be formed using four of these points as its corners. Does not contain any other point inside or on its border. Has its edges parallel to the axes. Return the maximum area that you can obtain or -1 if no such rectangle is possible. Example 1: Input: points = [[1,1],[1,3],[3,1],[3,3]] Output: 4 Explanation: We can ma

Detailed Explanation

The problem asks us to find the maximum area of a rectangle that can be formed from a given set of points. The rectangle must have its sides parallel to the axes, and crucially, it must not contain any other point from the input set, either strictly inside or on its border. The input is an array of (x, y) coordinate pairs. If no such rectangle can be formed, we return -1.

Solution Approach

The provided solution iterates through all possible pairs of points and checks if they can form the opposite corners of a rectangle. For each potential rectangle, it verifies that the other two corners also exist in the point set. Then, it checks if any other points from the input set lie inside or on the border of the potential rectangle. If both conditions are met, it calculates the area and updates the maximum area found so far.

Step-by-Step Algorithm

Step 1: Iterate through all possible pairs of points (i, j) using nested loops.
Step 2: Check if points i and j can form the diagonal of a rectangle (i.e., they have different x and y coordinates).
Step 3: Check if the other two potential corners of the rectangle (x1, y2) and (x2, y1) also exist in the input point set. Using a set data structure allows for O(1) lookup.
Step 4: If all four corners exist, calculate the potential rectangle's boundaries (min_x, max_x, min_y, max_y).
Step 5: Iterate through all other points in the input set to check if any of them fall inside or on the border of the rectangle. Exclude the four corner points from this check.
Step 6: If no points are found inside or on the border, calculate the area of the rectangle.
Step 7: Update the maximum area if the current rectangle's area is larger.
Step 8: After checking all possible rectangles, return the maximum area found. If no valid rectangle is found, return -1.

Key Insights

Insight 1: The problem requires us to check all possible combinations of four points to form a rectangle.
Insight 2: The constraint about no other points inside or on the border is crucial and needs careful checking.
Insight 3: Using a set data structure for point lookup greatly improves efficiency in checking if a potential rectangle corner exists in the provided input points.

Complexity Analysis

Time Complexity: O(n^3)

Space Complexity: O(n)

Topics

This problem involves: Array, Math, Binary Indexed Tree, Segment Tree, Geometry, Sorting, Enumeration.

Companies

Asked at: UKG.

Maximum Area Rectangle With Point Constraints I | LeetCode 3380

Medium

Array Math Binary Indexed Tree Segment Tree Geometry Sorting Enumeration

Time: O(n^3)Space: O(n)

Solution Code

Available in:

import java.util.HashSet;
import java.util.Set;

class Solution {
    public int maxRectangleArea(int[][] points) {
        int n = points.length;
        if (n < 4) {
            return -1;
        }

        Set<String> pointSet = new HashSet<>();
        for (int[] p : points) {
            pointSet.add(p[0] + "," + p[1]);
        }

        int maxArea = -1;

        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                int x1 = points[i][0];
                int y1 = points[i][1];
                int x2 = points[j][0];
                int y2 = points[j][1];

                if (x1 == x2 || y1 == y2) {
                    continue;
                }

                if (pointSet.contains(x1 + "," + y2) && pointSet.contains(x2 + "," + y1)) {
                    int minX = Math.min(x1, x2);
                    int maxX = Math.max(x1, x2);
                    int minY = Math.min(y1, y2);
                    int maxY = Math.max(y1, y2);

                    boolean isEmpty = true;
                    for (int k = 0; k < n; k++) {
                        int px = points[k][0];
                        int py = points[k][1];
                        
                        boolean isCorner = (px == x1 && py == y1) ||
                                            (px == x2 && py == y2) ||
                                            (px == x1 && py == y2) ||
                                            (px == x2 && py == y1);

                        if (!isCorner) {
                            if (minX <= px && px <= maxX && minY <= py && py <= maxY) {
                                isEmpty = false;
                                break;
                            }
                        }
                    }

                    if (isEmpty) {
                        int area = (maxX - minX) * (maxY - minY);
                        maxArea = Math.max(maxArea, area);
                    }
                }
            }
        }

        return maxArea;
    }
}

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Problem Info

Description

Problem 3380: Maximum Area Rectangle With Point Constraints I - Detailed explanation and solution approaches.

Topics

Array Math Binary Indexed Tree Segment Tree Geometry Sorting Enumeration

Companies

UKG

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:3380

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

8 steps

1
Step 1: Iterate through all possible pairs of points (i, j) using nested loops.
2
Step 2: Check if points i and j can form the diagonal of a rectangle (i.e., they have different x and y coordinates).
3
Step 3: Check if the other two potential corners of the rectangle (x1, y2) and (x2, y1) also exist in the input point set. Using a set data structure allows for O(1) lookup.
4
Step 4: If all four corners exist, calculate the potential rectangle's boundaries (min_x, max_x, min_y, max_y).
5
Step 5: Iterate through all other points in the input set to check if any of them fall inside or on the border of the rectangle. Exclude the four corner points from this check.
6
Step 6: If no points are found inside or on the border, calculate the area of the rectangle.
7
Step 7: Update the maximum area if the current rectangle's area is larger.
8
Step 8: After checking all possible rectangles, return the maximum area found. If no valid rectangle is found, return -1.

💡

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^3)

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The nested loops account for the O(n^2) factor. The inner loop which verifies the absence of points inside the rectangle accounts for the final factor of O(n).

📖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

Divide and Conquer

Divide the points into smaller subsets and recursively find the maximum area rectangle within each subset. Then, combine the results to find the overall maximum area. This would be complex and likely not provide significant performance benefits given the small constraint on the number of points (<= 10).

Time: Potentially O(n^2 log n), but complex to implement efficiently.Space: Potentially O(n log n).

Trade-offs: The complexity of implementation and potential overhead make this approach less suitable for small input sizes.

✨

Key Insights

1Insight 1: The problem requires us to check all possible combinations of four points to form a rectangle.
2Insight 2: The constraint about no other points inside or on the border is crucial and needs careful checking.
3Insight 3: Using a set data structure for point lookup greatly improves efficiency in checking if a potential rectangle corner exists in the provided input points.

📚

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: The input contains fewer than four points. The solution correctly returns -1.
Edge case 2: No valid rectangle can be formed. The solution correctly returns -1.
Edge case 3: There are duplicate points in the input. The problem statement specifies the input contains unique points so this case is avoided, but in other cases the solution can produce incorrect output.
Edge case 4: The points form a rectangle with area 0 (e.g., points are collinear). Since the coordinates are distinct, area 0 is not possible.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to check if the other two corners of the rectangle actually exist in the input set. This leads to calculating areas of non-existent rectangles.

✗Common mistake 2: Failing to properly check if other points lie *on* the border of the rectangle, and only checking points strictly inside.

✗Common mistake 3: Not considering all pairs of points as potential diagonal corners. This results in missing valid rectangles.

✗Common mistake 4: Integer overflow when calculating the area. Given the constraint that coordinates are up to 100, this is unlikely, but should be considered for larger coordinate ranges.

✅

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 - Finding rectangular regions of interest within an image that do not contain other features (e.g., object detection).

🏦Real-world scenario 2: Geographic Information Systems (GIS) - Identifying rectangular areas on a map that satisfy certain criteria (e.g., minimum size, no obstacles within the area).

🎮Real-world scenario 3: VLSI design - Determining the maximum size of a rectangular component that can be placed on a chip without overlapping existing components.

💡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

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ 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: October 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

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

❓

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 Area Rectangle With Point Constraints I"?

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.

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