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1453. Maximum Number of Darts Inside of a Circular Dartboard

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.

 

Example 1:

Input: darts = [[-2,0],[2,0],[0,2],[0,-2]], r = 2
Output: 4
Explanation: Circle dartboard with center in (0,0) and radius = 2 contain all points.

Example 2:

Input: darts = [[-3,0],[3,0],[2,6],[5,4],[0,9],[7,8]], r = 5
Output: 5
Explanation: Circle dartboard with center in (0,4) and radius = 5 contain all points except the point (7,8).

 

Constraints:

  • 1 <= darts.length <= 100
  • darts[i].length == 2
  • -104 <= xi, yi <= 104
  • All the darts are unique
  • 1 <= r <= 5000

Solutions

Solution 1

Thinking

\(n\le 100\). A maximum covering disk of radius \(r\) can be assumed to pass through two points. For each pair at distance \(\le 2r\), compute the two candidate centers and count covered darts.

A single dart already gives \(1\). Compare distances against \(r\) with a small epsilon so boundary points are kept.

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class Solution:
    def numPoints(self, darts: list[list[int]], r: int) -> int:
        def countDarts(x, y):
            count = 0
            for x1, y1 in darts:
                if dist((x, y), (x1, y1)) <= r + 1e-7:
                    count += 1
            return count

        def possibleCenters(x1, y1, x2, y2):
            dx, dy = x2 - x1, y2 - y1
            d = sqrt(dx * dx + dy * dy)
            if d > 2 * r:
                return []
            mid_x, mid_y = (x1 + x2) / 2, (y1 + y2) / 2
            dist_to_center = sqrt(r * r - (d / 2) * (d / 2))
            offset_x = dist_to_center * dy / d
            offset_y = dist_to_center * -dx / d
            return [
                (mid_x + offset_x, mid_y + offset_y),
                (mid_x - offset_x, mid_y - offset_y),
            ]

        n = len(darts)
        max_darts = 1

        for i in range(n):
            for j in range(i + 1, n):
                centers = possibleCenters(
                    darts[i][0], darts[i][1], darts[j][0], darts[j][1]
                )
                for center in centers:
                    max_darts = max(max_darts, countDarts(center[0], center[1]))

        return max_darts

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class Solution {
    public int numPoints(int[][] darts, int r) {
        int n = darts.length;
        int maxDarts = 1;

        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                List<double[]> centers
                    = possibleCenters(darts[i][0], darts[i][1], darts[j][0], darts[j][1], r);
                for (double[] center : centers) {
                    maxDarts = Math.max(maxDarts, countDarts(center[0], center[1], darts, r));
                }
            }
        }
        return maxDarts;
    }

    private List<double[]> possibleCenters(int x1, int y1, int x2, int y2, int r) {
        List<double[]> centers = new ArrayList<>();
        double dx = x2 - x1;
        double dy = y2 - y1;
        double d = Math.sqrt(dx * dx + dy * dy);
        if (d > 2 * r) {
            return centers;
        }
        double midX = (x1 + x2) / 2.0;
        double midY = (y1 + y2) / 2.0;
        double distToCenter = Math.sqrt(r * r - (d / 2.0) * (d / 2.0));
        double offsetX = distToCenter * dy / d;
        double offsetY = distToCenter * -dx / d;

        centers.add(new double[] {midX + offsetX, midY + offsetY});
        centers.add(new double[] {midX - offsetX, midY - offsetY});
        return centers;
    }

    private int countDarts(double x, double y, int[][] darts, int r) {
        int count = 0;
        for (int[] dart : darts) {
            if (Math.sqrt(Math.pow(dart[0] - x, 2) + Math.pow(dart[1] - y, 2)) <= r + 1e-7) {
                count++;
            }
        }
        return count;
    }
}

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