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question:# Merge and Sort Array Merge Sort is a classic algorithm used for sorting an array with the efficiency of O(n log(n)) time complexity. This algorithm works by recursively dividing the array into two halves, sorting those halves recursively, and finally merging the sorted halves together. Given this context, you will be implementing the merge function that merges two sorted arrays into one sorted array. **Objective:** Write a function `merge_two_sorted_arrays` that merges two sorted lists (`left` and `right`) into a single sorted list. You need to ensure the resulting list is sorted in ascending order. **Function Signature:** ```python def merge_two_sorted_arrays(left: List[int], right: List[int]) -> List[int]: ``` **Input:** * `left`: A list of integers sorted in non-decreasing order. * `right`: Another list of integers sorted in non-decreasing order. **Output:** * A list of integers sorted in non-decreasing order that contains all the elements from both `left` and `right`. **Constraints:** * Both `left` and `right` can be empty. * The elements of both `left` and `right` are within the range of [-10^9, 10^9]. * The lengths of `left` and `right` are between 0 and 10^6. **Performance Requirements:** * The function must run within an efficient time complexity, ideally O(n), where n is the total number of elements from both lists combined. **Example:** ```python >>> merge_two_sorted_arrays([1, 3, 5], [2, 4, 6]) [1, 2, 3, 4, 5, 6] >>> merge_two_sorted_arrays([-5, 3, 8], [-6, -4, 0, 10]) [-6, -5, -4, 0, 3, 8, 10] >>> merge_two_sorted_arrays([], [2, 3, 4]) [2, 3, 4] >>> merge_two_sorted_arrays([7, 8, 9], []) [7, 8, 9] ```
answer:from typing import List def merge_two_sorted_arrays(left: List[int], right: List[int]) -> List[int]: Merges two sorted arrays into one sorted array. :param left: List[int] - a sorted list of integers. :param right: List[int] - another sorted list of integers. :return: List[int] - a merged sorted list containing all elements from left and right. # Initialize pointers for both lists i, j = 0, 0 merged = [] # Traverse both lists and append the smaller element to the merged list while i < len(left) and j < len(right): if left[i] < right[j]: merged.append(left[i]) i += 1 else: merged.append(right[j]) j += 1 # Append any remaining elements from the left list while i < len(left): merged.append(left[i]) i += 1 # Append any remaining elements from the right list while j < len(right): merged.append(right[j]) j += 1 return merged
question:# Context As part of your final assessment in understanding sorting algorithms, you are asked to implement an efficient sorting method. Shell Sort, a generalization of Insertion Sort, is found to improve the performance significantly on larger datasets. # Task Implement a function that performs Shell Sort on a given list of integers. # Function Signature ```python def shell_sort(arr: List[int]) -> List[int]: pass ``` # Parameters - `arr` (List[int]): A list of integers to be sorted. # Returns - `List[int]`: A new list of integers sorted in non-decreasing order. # Constraints - 1 ≤ len(arr) ≤ 10^5 - -10^9 ≤ arr[i] ≤ 10^9 # Performance Requirements - Aim for a time complexity better than O(n^2) in typical cases. # Example ```python assert shell_sort([12, 34, 54, 2, 3]) == [2, 3, 12, 34, 54] assert shell_sort([5, 5, 5, 5, 5]) == [5, 5, 5, 5, 5] assert shell_sort([4, 3, 2, 1]) == [1, 2, 3, 4] assert shell_sort([]) == [] assert shell_sort([1]) == [1] ``` # Edge Cases - Already sorted array. - Array with all elements the same. - Very large array up to the constraint limits. Use these details to test robustness and correctness of the implementation.
answer:def shell_sort(arr): Perform Shell Sort on the input list and return the sorted list. n = len(arr) gap = n // 2 while gap > 0: for i in range(gap, n): temp = arr[i] j = i while j >= gap and arr[j - gap] > temp: arr[j] = arr[j - gap] j -= gap arr[j] = temp gap //= 2 return arr
question:Nearest Neighbor Optimization You are working on a classification problem where you need to identify the class of a query vector based on the nearest neighbor in the given training dataset. The current implementation uses a simple Euclidean distance to find the nearest neighbor which can become inefficient with larger datasets. Your task is to enhance the existing implementation by reducing its bottlenecks and handling possible edge cases. Task Given the following functions: - `distance(x, y)`: Calculates the Euclidean distance between two vectors `x` and `y`. - `nearest_neighbor(x, tSet)`: Finds the nearest neighbor of vector `x` from the dataset `tSet`, which is a dictionary where keys are vectors and values are their corresponding class labels. Optimize the nearest neighbor implementation to handle larger datasets more efficiently. Additional Constraints: 1. The input vectors and training set shall abide by their respective types (tuples and dictionary with tuple keys). 2. Take special care of edge cases, including empty training sets and multiple equidistant points. Input - A tuple `x` representing the query vector. - A dictionary `tSet` where keys are tuples in the format (float, float, ..., float) and values are their respective classes. Output - Return the class of the nearest neighbor from the training set. Performance Requirement - Handle inputs efficiently for larger datasets, potentially using space-time trade-off implementations or different data structures. # Example: ```python x = (1.0, 2.0) tSet = { (2.0, 3.0): 'ClassA', (4.0, 5.0): 'ClassB', (1.5, 1.8): 'ClassA' } nearest_neighbor(x, tSet) # Expected output: 'ClassA' ``` Ensure your implementation handles edge cases and optimizes the nearest neighbor search effectively.
answer:import numpy as np from sklearn.neighbors import KDTree def distance(x, y): Calculates the Euclidean distance between two vectors x and y. return np.linalg.norm(np.array(x) - np.array(y)) def nearest_neighbor(x, tSet): Finds the nearest neighbor of vector x from the dataset tSet. tSet is a dictionary where keys are vectors and values are their corresponding class labels. if not tSet: raise ValueError("Training set is empty") # Extract point vectors and their corresponding classes from the dictionary points = np.array(list(tSet.keys())) classes = list(tSet.values()) # Build KDTree for efficient nearest neighbor search tree = KDTree(points) # Find the nearest neighbor dist, ind = tree.query([x], k=1) # Return the class of the nearest neighbor return classes[ind[0][0]]
question:You are required to implement a function `enhanced_hailstone(n: int) -> list[int]` that generates the hailstone sequence for a given integer `n`. The function should also handle and return appropriate results for edge cases such as invalid input types or values. # Function Signature ```python def enhanced_hailstone(n: int) -> list[int]: pass ``` # Input * An integer `n` where `n > 0`. # Output * A list of integers representing the hailstone sequence starting from `n` and ending at `1`. # Constraints * The input integer `n` will always be positive and greater than zero. * The sequence must be computed in a reasonable time for numbers up to `10^6`. # Example ```python assert enhanced_hailstone(1) == [1] assert enhanced_hailstone(5) == [5, 16, 8, 4, 2, 1] assert enhanced_hailstone(7) == [7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1] ``` # Notes * Your solution should correctly handle and return edge cases such as `n = 1`. * While explicit handling of invalid input is not required due to constraints, consider how you might structure checks or handle unexpected input types in practice.
answer:def enhanced_hailstone(n: int) -> list[int]: Generate the hailstone sequence for a given integer n. Parameters: n (int): A positive integer Returns: list[int]: The hailstone sequence starting from n and ending at 1 if not isinstance(n, int) or n <= 0: raise ValueError("Input must be a positive integer") sequence = [] while n != 1: sequence.append(n) if n % 2 == 0: n = n // 2 else: n = 3 * n + 1 sequence.append(1) # Ending the sequence by adding 1 return sequence