Maximum Number of Darts Inside of a Circular Dartboard

Problem Description

Alice is throwing n darts on a very large wall. You are given an array darts where darts[i] = [xi, yi] is the position of the ith dart that Alice threw on the wall. Bob knows the positions of the n darts on the wall. He wants to place a dartboard of radius r on the wall so that the maximum number of darts that Alice throws lie on the dartboard. Given the integer r, return the maximum number of darts that can lie on the dartboard.

Key Insights

• The dartboard can be positioned anywhere on the wall, and the objective is to maximize the number of darts inside it.
• The distance from a dart to the center of the dartboard should be less than or equal to the radius r.
• It is essential to check various potential centers for the dartboard, particularly those that lie on the line segments between pairs of darts.

Space and Time Complexity

Time Complexity: O(n^2) - We check each pair of darts and calculate the potential centers. Space Complexity: O(1) - We use a constant amount of space for calculations.

Solution

To solve this problem, we will:

1. Iterate through each pair of darts to determine possible centers for the dartboard.
2. For each center, count how many darts fall within the radius r.
3. Use geometric properties to quickly compute if a dart lies within the dartboard using the distance formula.

The data structures used in this solution are simple arrays (for storing dart positions) and a few variables for counting darts and storing maximum counts. The algorithm primarily employs a brute-force approach enhanced by geometric insights.

Code Solutions