the last one in Weekly Contest 197.
Difficulty : Hard
Related Topics : Geometry
A delivery company wants to build a new service centre in a new city. The company knows the positions of all the customers in this city on a 2D-Map and wants to build the new centre in a position such that the sum of the euclidean distances to all customers is minimum.
Given an array
positions
wherepositions[i] = [xi, yi]
is the position of theith
customer on the map, return the minimum sum of the euclidean distances to all customers.In other words, you need to choose the position of the service centre
[xcentre, ycentre]
such that the following formula is minimized:Answers within
10^-5
of the actual value will be accepted.Input: positions = [[0,1],[1,0],[1,2],[2,1]] Output: 4.00000 Explanation: As shown, you can see that choosing [xcentre, ycentre] = [1, 1] will make the distance to each customer = 1, the sum of all distances is 4 which is the minimum possible we can achieve.
Input: positions = [[1,1],[3,3]] Output: 2.82843 Explanation: The minimum possible sum of distances = sqrt(2) + sqrt(2) = 2.82843
Input: positions = [[1,1]] Output: 0.00000
Input: positions = [[1,1],[0,0],[2,0]] Output: 2.73205 Explanation: At the first glance, you may think that locating the centre at [1, 0] will achieve the minimum sum, but locating it at [1, 0] will make the sum of distances = 3. Try to locate the centre at [1.0, 0.5773502711] you will see that the sum of distances is 2.73205. Be careful with the precision!
Input: positions = [[0,1],[3,2],[4,5],[7,6],[8,9],[11,1],[2,12]] Output: 32.94036 Explanation: You can use [4.3460852395, 4.9813795505] as the position of the centre.
1 <= positions.length <= 50
positions[i].length == 2
0 <= positions[i][0], positions[i][1] <= 100
- mine
- leetcode-cn Solution
Runtime: 9 ms, faster than 67.55%, Memory Usage: 37.5 MB, less than 100.00% of Java online submissions
private static int[][] dirs = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}}; public double getMinDistSum(int[][] positions) { double eps = 1e-7; double step = 1; double decay = 0.5; int n = positions.length; //get the center point double x = 0.0, y = 0.0; for (int[] pos : positions) { x += pos[0]; y += pos[1]; } x /= n; y /= n; while (step > eps) { boolean modified = false; //find the down direction for (int i = 0; i < 4; ++i) { double xNext = x + step * dirs[i][0]; double yNext = y + step * dirs[i][1]; if (getDist(xNext, yNext, positions) < getDist(x, y, positions)) { x = xNext; y = yNext; modified = true; break; } } // if not find the down direction, move less than before if (!modified) { step *= (1.0 - decay); } } return getDist(x, y, positions); } public double getDist(double xc, double yc, int[][] positions) { double ans = 0; for (int[] pos : positions) { ans += Math.sqrt((pos[0] - xc) * (pos[0] - xc) + (pos[1] - yc) * (pos[1] - yc)); } return ans; }