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Best Position for a Service Centre

Difficulty: Hard

Problem Description

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. Given an array positions where positions[i] = [xi, yi] is the position of the ith customer on the map, return the minimum sum of the euclidean distances to all customers.

Key Insights

  • The problem can be approached as a geometric optimization problem where the goal is to find a point in 2D space that minimizes the sum of distances to a set of given points.
  • The function to minimize is continuous and differentiable, which allows for the use of optimization techniques.
  • An iterative approach, such as gradient descent or using libraries for numerical optimization, can be effective for finding the optimal location.

Space and Time Complexity

Time Complexity: O(N * log(1/epsilon)), where N is the number of customer positions and epsilon is the precision desired for the result.

Space Complexity: O(1), as we are using a constant amount of extra space regardless of the input size.

Solution

To solve this problem, we can use a numerical optimization technique to find the point (x_centre, y_centre) that minimizes the sum of distances to all customer locations. One effective algorithm to apply here is the gradient descent method or a variation of it, which iteratively refines the position of the service center based on the gradient of the distance function.

  1. Define a function that calculates the total distance from a given point to all customer points.
  2. Start with an initial guess for the service center's coordinates, typically the centroid of the customer positions.
  3. Iteratively adjust the coordinates by evaluating the gradient of the distance function and moving in the direction that reduces the total distance.
  4. Continue iterating until the change in distance is less than a specified threshold (epsilon).

Code Solutions

import numpy as np

def getMinDistSum(positions):
    # Convert positions to a numpy array for easier manipulation
    positions = np.array(positions)
    
    # Initialize the center position as the mean of the customer positions
    center = np.mean(positions, axis=0)
    
    # Function to calculate the total distance
    def total_distance(center):
        return np.sum(np.sqrt(np.sum((positions - center) ** 2, axis=1)))
    
    # Gradient descent parameters
    step_size = 0.1
    epsilon = 1e-6
    while True:
        # Compute the gradient
        distances = np.sqrt(np.sum((positions - center) ** 2, axis=1))
        gradient = np.zeros(2)
        for i in range(len(positions)):
            if distances[i] > 0:  # Avoid division by zero
                gradient += (center - positions[i]) / distances[i]
        
        # Update the center position
        new_center = center - step_size * gradient
        
        # Check for convergence
        if np.linalg.norm(new_center - center) < epsilon:
            break
        center = new_center
    
    return total_distance(center)

# Example usage
positions = [[0,1],[1,0],[1,2],[2,1]]
print(getMinDistSum(positions))
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