Best Position for a Service Centre — LeetCode 1515 Python Solution

The problem

A delivery company wants to build a new service center 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 center in a position such that the sum of the euclidean distances to all customers is minimum.

Example

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.

Python solution

class Solution:
    def getMinDistSum(self, positions: List[List[int]]) -> float:
        n = len(positions)
        x = y = 0
        for x1, y1 in positions:
            x += x1
            y += y1
        x, y = x / n, y / n
        decay = 0.999
        eps = 1e-6
        alpha = 0.5
        while 1:
            grad_x = grad_y = 0
            dist = 0
            for x1, y1 in positions:
                a = x - x1
                b = y - y1
                c = sqrt(a * a + b * b)
                grad_x += a / (c + 1e-8)
                grad_y += b / (c + 1e-8)
                dist += c
            dx = grad_x * alpha
            dy = grad_y * alpha
            x -= dx
            y -= dy
            alpha *= decay
            if abs(dx) <= eps and abs(dy) <= eps:
                return dist

Complexity

Pattern: Math and Number Theory

Find the closed form, the invariant, or the modular identity — and skip the loop entirely. LeetCode 1515. Best Position for a Service Centre is filed here on both counts: the reference solution below belongs to the algorithm family this hub collects, and LeetCode tags it Math and Geometry.

The math and number theory guide has the Python template for the pattern and the 485 LeetCode problems that use it.

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