You are working with points in an n-dimensional Euclidean space, where each point is represented as a list (or array) of numeric coordinates. Assume: - Each point is represented as `list[float]`. - For any given operation, all points involved have the same dimensionality (same length of the list). - For the k-nearest-neighbors function, assume `1 <= k <= len(dataset)`. - If two points are at exactly the same distance from the query point, break ties by smaller index in the dataset. --- ### 1) Compute Euclidean distance between two n-dimensional points Write a function that computes the Euclidean distance between two points in n-dimensional space. The function should: - Accept two points as input, e.g. `p: list[float]` and `q: list[float]`. - Handle points of arbitrary dimensions (as long as both points have the same dimension). - Return the Euclidean distance as a `float`. The Euclidean distance between points `p` and `q` in n dimensions is: $$ \text{distance}(p, q) = \sqrt{\sum_{i=1}^{n} (p_i - q_i)^2} $$ **Examples:** - Distance between `[1, 2]` and `[4, 6]` in 2D space. - Distance between `[1, 2, 3]` and `[4, 5, 6]` in 3D space. You do not need to print anything; just implement the function that returns the distance. --- ### 2) Find k-nearest neighbors using Euclidean distance Using the Euclidean distance function from part (1) as the distance metric, write a function to find the k-nearest neighbors of a query point within a dataset. The function should: - Accept as input: - `query_vector: list[float]` — the query point. - `k: int` — the number of nearest neighbors to return. - `dataset: list[list[float]]` — a list of data points, where each data point is a list of floats of the same dimension as `query_vector`. - Compute the Euclidean distance from the `query_vector` to each point in `dataset`. - Return a `list[int]` of the indices of the **k nearest neighbors** in `dataset`, **sorted in order of increasing distance** to the query. - If two points have the same distance, return the point with the smaller index first. **Example dataset:** ```python dataset = [ [10, 11, 12], # index 0 [1, 2, 3], # index 1 [4, 5, 6], # index 2 [7, 8, 9], # index 3 [13, 14, 15] # index 4 ] ``` Given some `query_vector` and a value of `k`, your function should return the indices of the k closest points from this dataset, ordered from closest to farthest.