What is the time complexity of Maximum Number of Visible Points?

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

What data structures are used to solve Maximum Number of Visible Points?

This problem uses Array, Math, Geometry, Sliding Window, Sorting. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Maximum Number of Visible Points?

Insight 1: Convert coordinates to angles relative to the viewing position. This allows us to work with angles instead of coordinates, simplifying the problem. Insight 2: The angular range wraps around 360 degrees, so we need to consider angles in a circular fashion to correctly handle points near 0 and 360 degrees.

Which companies ask LeetCode 1610. Maximum Number of Visible Points in interviews?

This problem is frequently asked at Anduril, Applied Intuition, Nuro, Nvidia.

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Maximum Number of Visible Points - LeetCode 1610 Solution

Maximum Number of Visible Points - Complete Solution Guide

Maximum Number of Visible Points is LeetCode problem 1610, a Hard 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 , an integer angle , and your location , where location = [pos x , pos y ] and points[i] = [x i , y i ] both denote integral coordinates on the X-Y plane. Initially, you are facing directly east from your position. You cannot move from your position, but you can rotate . In other words, pos x and pos y cannot be changed. Your field of view in degrees is represented by angle , determining how wide you can see from any given view direction. Let d be the amount in degr

Detailed Explanation

The problem asks us to find the maximum number of points visible from a fixed location, given an angle representing our field of view. We are given a list of points, our location coordinates, and the viewing angle. The points that lie within our field of view, which can be rotated, are considered visible. Points located at our position are always visible regardless of our rotation. The goal is to find the optimal rotation that maximizes the number of visible points, including those at our location.

Solution Approach

The solution first converts the given points to angles relative to the observer's location. Points at the same location as the observer are counted separately as they are always visible. The angles are then sorted. To handle the circular nature of the angles, the sorted angles list is duplicated, with each angle in the second copy increased by 360 degrees. A sliding window is used to find the maximum number of angles falling within the given view angle. Finally, the number of points at the observer's location is added to the maximum number of visible points found using the sliding window.

Step-by-Step Algorithm

Step 1: Iterate through the given list of points and count the number of points that are located at the same coordinates as the location.
Step 2: For each point not at the location, calculate the angle between the line connecting the location and the point and the positive x-axis. Use `atan2` function to obtain the angle and convert from radians to degrees.
Step 3: Store these angles in a list and sort them in ascending order.
Step 4: Create a new list `circular_angles` by appending a copy of the `angles` list to itself, adding 360 degrees to each angle in the appended list. This handles the wraparound case for angles.
Step 5: Use a sliding window technique on `circular_angles` to find the maximum number of points visible within the given angle. The window starts at index 0 and expands to the right.
Step 6: For each right endpoint of the window, shrink the window from the left as much as possible while ensuring that the difference between the right and left endpoints' angles does not exceed the given view angle.
Step 7: Update `max_points_in_fov` with the maximum window size (number of visible points) encountered so far.
Step 8: Finally, add the number of points at the location (calculated in step 1) to `max_points_in_fov` and return the result.

Key Insights

Insight 1: Convert coordinates to angles relative to the viewing position. This allows us to work with angles instead of coordinates, simplifying the problem.
Insight 2: The angular range wraps around 360 degrees, so we need to consider angles in a circular fashion to correctly handle points near 0 and 360 degrees.
Insight 3: Use a sliding window approach to efficiently find the maximum number of points within the specified angle after sorting. This helps avoid recomputing counts for similar viewing angles.

Complexity Analysis

Time Complexity: O(n log n)

Space Complexity: O(n)

Topics

This problem involves: Array, Math, Geometry, Sliding Window, Sorting.

Companies

Asked at: Anduril, Applied Intuition, Nuro, Nvidia.

Maximum Number of Visible Points | LeetCode 1610

Hard

Array Math Geometry Sliding Window Sorting

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

Solution Code

Available in:

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

class Solution {
    public int visiblePoints(List<List<Integer>> points, int angle, List<Integer> location) {
        int lx = location.get(0);
        int ly = location.get(1);

        List<Double> angles = new ArrayList<>();
        int points_at_location = 0;

        for (List<Integer> point : points) {
            int px = point.get(0);
            int py = point.get(1);

            if (px == lx && py == ly) {
                points_at_location++;
                continue;
            }

            angles.add(Math.toDegrees(Math.atan2(py - ly, px - lx)));
        }

        Collections.sort(angles);

        List<Double> circular_angles = new ArrayList<>(angles);
        for (Double a : angles) {
            circular_angles.add(a + 360);
        }

        int max_points_in_fov = 0;
        int left = 0;

        for (int right = 0; right < circular_angles.size(); right++) {
            while (circular_angles.get(right) - circular_angles.get(left) > angle) {
                left++;
            }

            max_points_in_fov = Math.max(max_points_in_fov, right - left + 1);
        }

        return max_points_in_fov + points_at_location;
    }
}

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Problem Info

Description

Problem 1610: Maximum Number of Visible Points - Detailed explanation and solution approaches.

Topics

Array Math Geometry Sliding Window Sorting

Companies

AndurilApplied IntuitionNuroNvidia

Stats

Difficulty:Hard

Languages:4

Companies:4

Problem #:1610

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 the given list of points and count the number of points that are located at the same coordinates as the location.
2
Step 2: For each point not at the location, calculate the angle between the line connecting the location and the point and the positive x-axis. Use `atan2` function to obtain the angle and convert from radians to degrees.
3
Step 3: Store these angles in a list and sort them in ascending order.
4
Step 4: Create a new list `circular_angles` by appending a copy of the `angles` list to itself, adding 360 degrees to each angle in the appended list. This handles the wraparound case for angles.
5
Step 5: Use a sliding window technique on `circular_angles` to find the maximum number of points visible within the given angle. The window starts at index 0 and expands to the right.
6
Step 6: For each right endpoint of the window, shrink the window from the left as much as possible while ensuring that the difference between the right and left endpoints' angles does not exceed the given view angle.
7
Step 7: Update `max_points_in_fov` with the maximum window size (number of visible points) encountered so far.
8
Step 8: Finally, add the number of points at the location (calculated in step 1) to `max_points_in_fov` and 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)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is dominated by sorting the angles, giving O(n log n). The space complexity is due to the `angles` and `circular_angles` lists, resulting in O(n) space.

📖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

Brute Force

Iterate through all possible rotation angles and calculate the number of visible points for each angle. This would involve discretizing the possible angles (e.g., in 1-degree increments).

Time: O(n * 360), where n is the number of points and 360 represents the possible angles.Space: O(1)

Trade-offs: This approach is simpler to understand but significantly less efficient than the sliding window approach, especially for large datasets. It's unlikely to pass the time limit for larger inputs.

✨

Key Insights

1Insight 1: Convert coordinates to angles relative to the viewing position. This allows us to work with angles instead of coordinates, simplifying the problem.
2Insight 2: The angular range wraps around 360 degrees, so we need to consider angles in a circular fashion to correctly handle points near 0 and 360 degrees.
3Insight 3: Use a sliding window approach to efficiently find the maximum number of points within the specified angle after sorting. This helps avoid recomputing counts for similar viewing angles.

📚

Key Terms Explained

For Students

Array

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

Sliding Window

A moving "window" over data to process subsets efficiently.

Sorting

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

Edge Cases

Edge case 1: Points at the same location as the observer. The solution correctly handles this by counting such points separately and adding them to the final result, as these points are always visible.
Edge case 2: Angles wrapping around 360 degrees. The solution addresses this by creating the `circular_angles` list, which effectively duplicates the angles and adds 360 degrees to the duplicates, allowing the sliding window to correctly consider points near the 0/360 degree boundary.
Edge case 3: Empty input point list. If the input `points` list is empty, the algorithm will correctly calculate zero visible points (plus any points at the location). The `angles` list will be empty, so the loops will not execute many times.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the angles. Using `atan` instead of `atan2` will only provide angles in the range [-90, 90] degrees, leading to incorrect results. `atan2` correctly handles all four quadrants.

✗Common mistake 2: Failing to account for the circular nature of angles. Forgetting to duplicate the angle list and add 360 degrees to the copies results in incorrect calculations when points are near the 0/360 degree boundary. The sliding window will not accurately capture the maximum visible points.

✗Common mistake 3: Integer overflow when calculating the angle. It's important to use double precision floating-point numbers to represent the angles and perform calculations. Using integers may result in precision loss or overflow, especially after multiplication and addition operations.

✅

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: Radar systems. Determining the number of targets within a specific radar beam's field of view.

🏦Real-world scenario 2: Surveillance systems. Optimizing camera placement and orientation to maximize the number of objects or areas under surveillance within a defined field of view.

🎮Real-world scenario 3: Astronomy. Calculating the number of stars visible from a specific location with a telescope within a given angular field of view.

💡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

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

👨‍💻 From Experience

I encountered a variation of this problem in a system design interview. The key was recognizing the underlying pattern.

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: November 18, 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 13

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

@AlgoMasterJul 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 do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

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 Number of Visible Points"?

For this hard-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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