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question:# Coding Challenge: Improved Heap Sort with Simulation Heap Sort is a popular sorting algorithm that can be implemented using either a Max Heap or a Min Heap. The provided code snippets show two variations of Heap Sort: one using a Max Heap (`max_heap_sort`) and the other using a Min Heap (`min_heap_sort`). Your task is to **implement a more versatile and efficient version** of the Heap Sort algorithm. This improved version should handle both Ascending and Descending order based on function parameters and optionally simulate the sorting process by printing the array at each significant iteration. # Function Signature ```python def versatile_heap_sort(arr, ascending=True, simulation=False): Sorts an array using an optimized heap sort algorithm. Parameters: arr (List[int]): The array to be sorted. ascending (bool): If True, sort in ascending order, otherwise sort in descending order. simulation (bool): If True, print the array at each significant step for visualization. Returns: List[int]: The sorted array. ``` # Input * `arr` (List[int]): A list of integers to be sorted. * `ascending` (bool): If `True`, sort the array in ascending order (default). If `False`, sort the array in descending order. * `simulation` (bool): If `True`, print the array at each significant step. # Output * A sorted list of integers as per the sorting order specified. # Constraints * The array can have up to `10^5` elements. * Elements can be negative or positive integers. * Your solution should be efficient and handle edge cases appropriately. # Requirements 1. **Optimize the Heapify Process**: Improve the internal heapify process to reduce unnecessary iterations. 2. **Accommodate Both Heaps**: Depending on the `ascending` parameter, dynamically switch between Max Heap and Min Heap sorting logic. 3. **Simulation Mode**: If `simulation` is set to `True`, print the state of the array at each step where a significant change occurs (e.g., after each full heapify operation or each swap). # Example ```python # Example 1: Ascending Sort arr = [4, 10, 3, 5, 1] result = versatile_heap_sort(arr, ascending=True) # Expected Output: [1, 3, 4, 5, 10] # Example 2: Descending Sort with Simulation arr = [4, 10, 3, 5, 1] result = versatile_heap_sort(arr, ascending=False, simulation=True) # Expected Output (simulation steps printed and final result): # iteration 1: ... # iteration 2: ... # ... # Final Result: [10, 5, 4, 3, 1] ``` # Notes * Your implementation should account for edge cases, such as arrays with all identical elements or arrays that are already sorted. * Make sure your function is efficient and can handle large inputs within acceptable time limits.

answer:def versatile_heap_sort(arr, ascending=True, simulation=False): Sorts an array using an optimized heap sort algorithm. Parameters: arr (List[int]): The array to be sorted. ascending (bool): If True, sort in ascending order, otherwise sort in descending order. simulation (bool): If True, print the array at each significant step for visualization. Returns: List[int]: The sorted array. def heapify(arr, n, i, ascending): largest_or_smallest = i left = 2 * i + 1 right = 2 * i + 2 if ascending: # If left child exists and is greater than root if left < n and arr[left] > arr[largest_or_smallest]: largest_or_smallest = left # If right child exists and is greater than largest so far if right < n and arr[right] > arr[largest_or_smallest]: largest_or_smallest = right else: # If left child exists and is less than root if left < n and arr[left] < arr[largest_or_smallest]: largest_or_smallest = left # If right child exists and is less than largest so far if right < n and arr[right] < arr[largest_or_smallest]: largest_or_smallest = right # If largest or smallest is not root if largest_or_smallest != i: arr[i], arr[largest_or_smallest] = arr[largest_or_smallest], arr[i] # swap if simulation: print(f"heapify swap: {arr}") heapify(arr, n, largest_or_smallest, ascending) # Build a maxheap/minheap n = len(arr) for i in range(n//2 - 1, -1, -1): heapify(arr, n, i, ascending) if simulation: print(f"buildheap: {arr}") # One by one extract elements for i in range(n-1, 0, -1): arr[i], arr[0] = arr[0], arr[i] # swap if simulation: print(f"extract swap: {arr}") heapify(arr, i, 0, ascending) if simulation: print(f"post-extract heapify: {arr}") return arr

question:# Sliding Window Maximum Problem You are given an array of integers and a number `k`. Your task is to implement a function that finds the maximum elements of each sub-array (or sliding window) of length `k`. Function Signature ```python def max_sliding_window(arr: List[int], k: int) -> List[int]: ``` Input - `arr`: A list of integers representing the array. (1 ≤ len(arr) ≤ 10^5) - `k`: An integer representing the size of the sub-array (window). (1 ≤ k ≤ len(arr)) Output - Returns a list of integers representing the maximum value of each sub-array of length `k`. Constraints - The input array can contain both positive and negative integers. - Window size `k` will always be a valid size less than or equal to the length of the array. Scenario Imagine you are managing a dashboard that displays the maximum stock price in real-time over a rolling window of past `k` days. Implement this function so that data streaming into your system can be processed and queried for maximum values fast. Requirements 1. Your implementation must use a deque to ensure the sliding window operation is efficient. 2. Optimize for both time and space complexity as much as possible. 3. Handle edge cases where the array has fewer elements than the window size effectively. 4. Ensure accurate handling of array indices to avoid errors. Example ```python print(max_sliding_window([1,3,-1,-3,5,3,6,7], 3)) # Output: [3, 3, 5, 5, 6, 7] print(max_sliding_window([9,11,8,5,7,10], 2)) # Output: [11, 11, 8, 7, 10] print(max_sliding_window([1,-1], 1)) # Output: [1, -1] ``` Additional Notes - This is a classical problem suited for understanding and evaluating the efficient use of data structures to maintain real-time computations over a sliding window. Complete the function `max_sliding_window` based on the given signature and ensure it adheres to the provided constraints and requirements.

answer:from collections import deque from typing import List def max_sliding_window(arr: List[int], k: int) -> List[int]: Finds the maximum element in each sub-array (or sliding window) of size k. if not arr or k == 0: return [] deq = deque() result = [] for i in range(len(arr)): # Remove elements not part of the window if deq and deq[0] == i - k: deq.popleft() # Remove elements smaller than the current element from the deque while deq and arr[deq[-1]] < arr[i]: deq.pop() deq.append(i) # Append the current max to the result list if i >= k - 1: result.append(arr[deq[0]]) return result

question:# Merging k Sorted Linked Lists Imagine you work for a company that logs errors from multiple servers. Each server sends logs in a sorted order based on the time the errors occurred. Your task is to merge these logs into a single sorted list. Function Signature: ```python def merge_k_lists(lists: List[Optional[ListNode]]) -> Optional[ListNode]: ``` Input: * `lists` - A list of k linked lists, each linked list is sorted in ascending order. Output: * A single sorted linked list that merges all the k input linked lists. Constraints: 1. The total number of elements across all the linked lists will not exceed 10^6. 2. Each linked list will have at most 10^5 elements. 3. The linked lists will only contain integer values. Requirements: * Ensure your function handles edge cases gracefully (e.g., one or more lists being empty). * Aim for an efficient solution: aim for O(n log k) time complexity and O(k) space complexity. Example: ```python # Definition for singly-linked list. class ListNode: def __init__(self, x): self.val = x self.next = None # Example usage def print_list(node): while node: print(node.val, end=" -> ") node = node.next print("None") # Merging example lists a = ListNode(1) a.next = ListNode(4) a.next.next = ListNode(5) b = ListNode(1) b.next = ListNode(3) b.next.next = ListNode(4) c = ListNode(2) c.next = ListNode(6) lists = [a, b, c] result = merge_k_lists(lists) print_list(result) # Output should be 1 -> 1 -> 2 -> 3 -> 4 -> 4 -> 5 -> 6 -> None ``` Tips: * You may utilize Python's `heapq` module to manage the heap (priority queue) effectively. * Pay attention to memory usage, especially if some lists are much longer than others. Now, implement the function `merge_k_lists` based on the provided specification.

answer:from typing import List, Optional import heapq # Definition for singly-linked list. class ListNode: def __init__(self, x): self.val = x self.next = None def merge_k_lists(lists: List[Optional[ListNode]]) -> Optional[ListNode]: Merge k sorted linked lists and return it as one sorted list. :param lists: List[Optional[ListNode]] - A list of k sorted linked lists :return: Optional[ListNode] - A single merged sorted linked list if not lists or all(lst is None for lst in lists): return None min_heap = [] # Initialize the heap with the head nodes of each list for i, node in enumerate(lists): if node: heapq.heappush(min_heap, (node.val, i, node)) dummy = ListNode(0) current = dummy while min_heap: val, idx, node = heapq.heappop(min_heap) current.next = node current = current.next if node.next: heapq.heappush(min_heap, (node.next.val, idx, node.next)) return dummy.next

question:# Question You are given an array of integers and two optional boundaries, min_lim and max_lim. Your task is to implement a function that filters out values from the array which do not fall within these boundaries (inclusive). If min_lim or max_lim is not provided, treat them as unbounded in the respective direction. Function Signature ```python def limit_array(arr: List[int], min_lim: Optional[int] = None, max_lim: Optional[int] = None) -> List[int]: pass ``` Input - `arr`: A list of integers (1 <= len(arr) <= 10^6). - `min_lim`: Optional; an integer representing the lower boundary. - `max_lim`: Optional; an integer representing the upper boundary. Output - Returns a list of integers that fall within the specified boundaries. Constraints - If `min_lim` is None, treat it as no lower bound. - If `max_lim` is None, treat it as no upper bound. - The function should maintain O(n) time complexity and handle large input sizes efficiently. Examples ```python assert limit_array([1, 2, 3, 4, 5], 2, 4) == [2, 3, 4] assert limit_array([10, 20, 30, 40, 50], None, 30) == [10, 20, 30] assert limit_array([5, 15, 25, 35, 45], 20, None) == [25, 35, 45] assert limit_array([100, 200, 300, 400, 500], 250, 450) == [300, 400] assert limit_array([50, 40, 30, 20, 10], None, None) == [50, 40, 30, 20, 10] ``` Notes - You need to handle edge cases such as an empty array, and when min_lim is greater than max_lim. - Ensure the function is robust to handle large lists efficiently.

answer:from typing import List, Optional def limit_array(arr: List[int], min_lim: Optional[int] = None, max_lim: Optional[int] = None) -> List[int]: Returns a list of integers that fall within the specified boundaries. If min_lim or max_lim is None, it treats them as unbounded. if min_lim is None: min_lim = float('-inf') if max_lim is None: max_lim = float('inf') return [x for x in arr if min_lim <= x <= max_lim]