Problem Description
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. 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.
Key Insights
- The problem can be transformed into an angle calculation problem where we need to determine the angle of each point relative to the origin (location).
- The angles can be sorted and treated as a circular array to facilitate the sliding window approach.
- We can utilize a two-pointer technique to efficiently count how many points fall within the specified angle range.
Space and Time Complexity
Time Complexity: O(n log n) - due to sorting the angles. Space Complexity: O(n) - for storing the angles.
Solution
To solve this problem, we calculate the angle of each point relative to the positive x-axis (east direction) using the atan2 function. We then sort these angles. To handle the circular nature of angles, we can duplicate the angle list (adding 360 degrees to each angle) and use a sliding window approach to count the number of points that fall within the specified angle range. The result will be the maximum count of points visible at any rotation.