Maximum Number of Visible Points

This problem can be solved using geometry, sorting, and the two-pointer technique.

First, we calculate the angles formed by the points and the east direction from our location.
Then, we sort the angles and apply a sliding window to find the maximum number of points that can be seen within the given angle.

Here’s the code:

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from math import atan2, degrees

class Solution:
    def visiblePoints(self, points: List[List[int]], angle: int, location: List[int]) -> int:
        # Function to calculate angle
        def calc_angle(x, y):
            return degrees(atan2(y - location[1], x - location[0]))

        # List to store angles
        angles = []
        same_location = 0

        for x, y in points:
            if x == location[0] and y == location[1]:
                same_location += 1
            else:
                angles.append(calc_angle(x, y))

        # Sort the angles
        angles.sort()

        # Duplicate the angles to handle the circular case
        angles += [x + 360 for x in angles]

        # Apply sliding window technique to find max points within angle
        max_points = l = 0
        for r in range(len(angles)):
            while angles[r] - angles[l] > angle:
                l += 1
            max_points = max(max_points, r - l + 1)

        return max_points + same_location

Explanation:

We calculate the angles formed by the points and the east direction from our location using the atan2 function.
If a point is at our location, we increase the count of same_location.
We sort the angles and duplicate them to handle the circular case.
We use a sliding window technique to find the maximum number of points that can be seen within the given angle, and add the count of points at our location.

The time complexity of this solution is (O(n \log n)) due to the sorting, and the space complexity is (O(n)), where (n) is the length of the points array.

To tackle the problem “Maximum Number of Visible Points”, you should have good understanding of geometry, trigonometry, sorting, and sliding window concepts.

10 Prerequisite LeetCode Problems

Here are 10 problems to prepare for this:

“Two Sum II - Input array is sorted” (LeetCode Problem #167): Understanding the basic two-pointer technique is crucial before approaching problems with similar patterns.
“3Sum” (LeetCode Problem #15): This problem helps to practice sorting and using the two-pointer technique to find sets of numbers that meet certain conditions.
“Sort Colors” (LeetCode Problem #75): This problem helps to reinforce the idea of sorting in problems.
“Minimum Window Substring” (LeetCode Problem #76): It’s an essential problem for understanding the sliding window concept.
“Sliding Window Maximum” (LeetCode Problem #239): This problem requires a good grasp of the sliding window concept and using data structures like Deque.
“Find the Duplicate Number” (LeetCode Problem #287): This problem exposes you to concepts such as Floyd’s Tortoise and Hare which are important for understanding cyclic data.
“K Closest Points to Origin” (LeetCode Problem #973): It’s a good problem to understand geometry related problems and usage of sorting in them.
“Meeting Rooms II” (LeetCode Problem #253): This problem uses a sliding window concept in the context of meeting scheduling.
“Subarray Product Less Than K” (LeetCode Problem #713): This problem gives practice on applying the sliding window concept to subarray problems.
“Angle Between Hands of a Clock” (LeetCode Problem #1344): This problem helps you to understand how to deal with angle calculation in problems.

These cover geometry, trigonometry, sorting, and sliding window concepts, which will be useful for “Maximum Number of Visible Points”.

Problem Classification

Problem Statement: You are given an array points, an integer angle, and your location, where location = [posx, posy] and points[i] = [xi, yi] 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, posx and posy 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 degrees that you rotate counterclockwise. Then, your field of view is the inclusive range of angles [d - angle/2, d + angle/2]. You can see some set of points if, for each point, the angle formed by the point, your position, and the immediate east direction from your position is in your field of view.

There can be multiple points at one coordinate. There may be points at your location, and you can always see these points regardless of your rotation. Points do not obstruct your vision to other points.

Return the maximum number of points you can see.

Input: points = [[2,1],[2,2],[3,3]], angle = 90, location = [1,1] Output: 3 Explanation: The shaded region represents your field of view. All points can be made visible in your field of view, including [3,3] even though [2,2] is in front and in the same line of sight.

Input: points = [[2,1],[2,2],[3,4],[1,1]], angle = 90, location = [1,1] Output: 4 Explanation: All points can be made visible in your field of view, including the one at your location.

Input: points = [[1,0],[2,1]], angle = 13, location = [1,1] Output: 1 Explanation: You can only see one of the two points, as shown above.

Constraints:

1 <= points.length <= 105 points[i].length == 2 location.length == 2 0 <= angle < 360 0 <= posx, posy, xi, yi <= 100

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Alternatively, if you’re working on a specific problem, you might ask something like:

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.