# 1515. Best Position for a Service Centre

<https://leetcode.com/problems/best-position-for-a-service-centre>

## Description

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` 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.

In other words, you need to choose the position of the service centre `[xcentre, ycentre]` such that the following formula is minimized:

![](https://assets.leetcode.com/uploads/2020/06/25/q4_edited.jpg) Answers within `10^-5` of the actual value will be accepted.

**Example 1:**

![](https://assets.leetcode.com/uploads/2020/06/25/q4_e1.jpg)

```
**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.
```

**Example 2:**

![](https://assets.leetcode.com/uploads/2020/06/25/q4_e3.jpg)

```
**Input:** positions = [[1,1],[3,3]]
**Output:** 2.82843
**Explanation:** The minimum possible sum of distances = sqrt(2) + sqrt(2) = 2.82843
```

**Example 3:**

```
**Input:** positions = [[1,1]]
**Output:** 0.00000
```

**Example 4:**

```
**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!
```

**Example 5:**

```
**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.
```

**Constraints:**

* `1 <= positions.length <= 50`
* `positions[i].length == 2`
* `0 <= positions[i][0], positions[i][1] <= 100`

## ac

```java
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://jaywin.gitbook.io/leetcode/solutions/1515-best-position-for-a-service-centre.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.