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question:# Problem Description Given the provided implementation of an AVL Tree, your task is to complete the implementation to include the following functionalities: 1. **Delete Operation**: Implement a method to delete a key from the AVL tree while maintaining the tree's balance. 2. **Find Operation**: Implement a method to find a given key in the AVL tree. Return `True` if the key exists, otherwise return `False`. 3. **Pre-order Traversal**: Implement a method to perform a pre-order traversal of the AVL tree and return the elements in a list. 4. **Post-order Traversal**: Implement a method to perform a post-order traversal of the AVL tree and return the elements in a list. # Additional Details **Input and Output**: * The delete operation should take an `int` key as input and not return anything. * The find operation should take an `int` key as input and return a `bool` indicating the presence of the key. * The pre-order traversal operation should return a list of elements in pre-order. * The post-order traversal operation should return a list of elements in post-order. **Constraints**: * All keys are integers. * Input keys for insert/delete/find operations are within the range of `-10^6` to `10^6`. * Input keys are unique. **Performance Requirements**: * The implemented methods should maintain the AVL tree properties, ensuring O(log n) operations. # Implementation Complete the provided AVL tree class with the required methods. Here's the class header along with the imports and base structure: ```python Imports TreeNodes - assume the TreeNode class is already defined with necessary attributes. from tree.tree import TreeNode class AvlTree(object): An AVL tree. def __init__(self): self.node = None self.height = -1 self.balance = 0 def insert(self, key): # Implementation provided above. pass def delete(self, key): # To be implemented by the student. pass def find(self, key): # To be implemented by the student. pass def re_balance(self): # Implementation provided above. pass def update_heights(self, recursive=True): # Implementation provided above. pass def update_balances(self, recursive=True): # Implementation provided above. pass def rotate_right(self): # Implementation provided above. pass def rotate_left(self): # Implementation provided above. pass def in_order_traverse(self): # Implementation provided above. pass def pre_order_traverse(self): # To be implemented by the student. pass def post_order_traverse(self): # To be implemented by the student. pass ``` **Completion Requirements**: * Implement the `delete`, `find`, `pre_order_traverse`, and `post_order_traverse` methods. * Ensure the AVL tree remains balanced after each delete operation. * Conduct pre- and post-order tree traversals accurately.
answer:class TreeNode: A node in the AVL tree. def __init__(self, key): self.left = None self.right = None self.key = key self.height = 1 class AvlTree: An AVL tree. def __init__(self): self.root = None def insert(self, key): self.root = self._insert(self.root, key) def _insert(self, node, key): if not node: return TreeNode(key) if key < node.key: node.left = self._insert(node.left, key) else: node.right = self._insert(node.right, key) node.height = 1 + max(self._get_height(node.left), self._get_height(node.right)) balance = self._get_balance(node) # Left Left if balance > 1 and key < node.left.key: return self._rotate_right(node) # Right Right if balance < -1 and key > node.right.key: return self._rotate_left(node) # Left Right if balance > 1 and key > node.left.key: node.left = self._rotate_left(node.left) return self._rotate_right(node) # Right Left if balance < -1 and key < node.right.key: node.right = self._rotate_right(node.right) return self._rotate_left(node) return node def delete(self, key): self.root = self._delete(self.root, key) def _delete(self, node, key): if not node: return node if key < node.key: node.left = self._delete(node.left, key) elif key > node.key: node.right = self._delete(node.right, key) else: if not node.left: return node.right elif not node.right: return node.left temp_val = self._get_min_value_node(node.right) node.key = temp_val.key node.right = self._delete(node.right, temp_val.key) if not node: return node node.height = 1 + max(self._get_height(node.left), self._get_height(node.right)) balance = self._get_balance(node) # Left Left if balance > 1 and self._get_balance(node.left) >= 0: return self._rotate_right(node) # Left Right if balance > 1 and self._get_balance(node.left) < 0: node.left = self._rotate_left(node.left) return self._rotate_right(node) # Right Right if balance < -1 and self._get_balance(node.right) <= 0: return self._rotate_left(node) # Right Left if balance < -1 and self._get_balance(node.right) > 0: node.right = self._rotate_right(node.right) return self._rotate_left(node) return node def find(self, key): return self._find(self.root, key) def _find(self, node, key): if not node: return False if key == node.key: return True elif key < node.key: return self._find(node.left, key) else: return self._find(node.right, key) def pre_order_traverse(self): result = [] self._pre_order(self.root, result) return result def _pre_order(self, node, result): if not node: return result.append(node.key) self._pre_order(node.left, result) self._pre_order(node.right, result) def post_order_traverse(self): result = [] self._post_order(self.root, result) return result def _post_order(self, node, result): if not node: return self._post_order(node.left, result) self._post_order(node.right, result) result.append(node.key) def _rotate_right(self, y): x = y.left T2 = x.right x.right = y y.left = T2 y.height = 1 + max(self._get_height(y.left), self._get_height(y.right)) x.height = 1 + max(self._get_height(x.left), self._get_height(x.right)) return x def _rotate_left(self, x): y = x.right T2 = y.left y.left = x x.right = T2 x.height = 1 + max(self._get_height(x.left), self._get_height(x.right)) y.height = 1 + max(self._get_height(y.left), self._get_height(y.right)) return y def _get_height(self, node): if not node: return 0 return node.height def _get_balance(self, node): if not node: return 0 return self._get_height(node.left) - self._get_height(node.right) def _get_min_value_node(self, node): if node is None or node.left is None: return node return self._get_min_value_node(node.left)
question:Hailstone Sequence Analysis You are tasked with analyzing the Hailstone sequence (Collatz sequence) properties. To do so, you must write a function that not only generates the sequence for a given number `n`, but also determines the length of the sequence and the highest value encountered in the sequence. Function Signature ```python def hailstone_analysis(n: int) -> (int, int, list): ``` Input * An integer `n`, where `1 <= n <= 10^6`. Output * A tuple containing: 1. The length of the hailstone sequence. 2. The maximum value encountered in the sequence. 3. The hailstone sequence itself, as a list of integers. Constraints * Handle large inputs efficiently. * Ensure the function works for boundary value input cases. Performance Requirements * The function should be able to handle the upper limit efficiently (both time and space). Example ```python >>> hailstone_analysis(5) (6, 16, [5, 16, 8, 4, 2, 1]) >>> hailstone_analysis(1) (1, 1, [1]) >>> hailstone_analysis(10) (7, 16, [10, 5, 16, 8, 4, 2, 1]) ```
answer:def hailstone_analysis(n: int) -> (int, int, list): Generate the Hailstone sequence starting from n, and return the sequence length, the maximum value in the sequence, and the sequence itself. sequence = [] max_value = n while n != 1: sequence.append(n) if n % 2 == 0: n = n // 2 else: n = 3 * n + 1 max_value = max(max_value, n) sequence.append(1) # Add the final 1 to the sequence return len(sequence), max_value, sequence
question:# Array Filtering within Specified Bounds You are given a list of integers and two optional integer boundaries, `min_lim` and `max_lim`. Your task is to filter the list so that only the values within the range `[min_lim, max_lim]` are included. If a limit (`min_lim` or `max_lim`) is not provided (i.e., None), it should default to the smallest or largest value in the list, respectively. # Function Signature ```python def limit(arr: list, min_lim: int = None, max_lim: int = None) -> list: pass ``` # Input - `arr`: A list of integers. - `min_lim` (optional): An integer representing the inclusive lower bound of the range. If None, use the smallest value in `arr`. - `max_lim` (optional): An integer representing the inclusive upper bound of the range. If None, use the largest value in `arr`. # Output - Returns a new list containing only the integers in `arr` that lie within the range `[min_lim, max_lim]`. # Constraints - The length of `arr` will be between 0 and 10^6. - Elements of `arr` are within the range of -10^9 to 10^9. - `arr` contains only integers, but may be empty. # Performance Requirements - The function should run in O(n) time complexity where n is the length of the array. # Examples ```python # Example 1: # Input: arr = [1, 2, 3, 4, 5], min_lim = 2, max_lim = 4 # Output: [2, 3, 4] print(limit([1, 2, 3, 4, 5], 2, 4)) # Example 2: # Input: arr = [10, 20, 30, 40, 50], min_lim = 20, max_lim = None # Output: [20, 30, 40, 50] print(limit([10, 20, 30, 40, 50], 20, None)) # Example 3: # Input: arr = [0, -10, 5, 7], min_lim = None, max_lim = 5 # Output: [0, -10, 5] print(limit([0, -10, 5, 7], None, 5)) # Example 4: # Input: arr = [1, 2, 3], min_lim = 5, max_lim = 10 # Output: [] print(limit([1, 2, 3], 5, 10)) # Example 5: # Input: arr = [], min_lim = 1, max_lim = 3 # Output: [] print(limit([], 1, 3)) ``` Implement the `limit` function to achieve the described functionality.
answer:def limit(arr: list, min_lim: int = None, max_lim: int = None) -> list: Filters the array so that only values within the range [min_lim, max_lim] are included. If min_lim is None, it uses the smallest value in the array. If max_lim is None, it uses the largest value in the array. if not arr: return [] if min_lim is None: min_lim = min(arr) if max_lim is None: max_lim = max(arr) return [x for x in arr if min_lim <= x <= max_lim]
question:You are given an array of integers representing frequencies of elements. You need to efficiently perform two types of operations on this array: 1. Update the frequency of a specific element in the array. 2. Retrieve the prefix sum, which is the sum of frequencies from the beginning of the array up to a given index. To achieve this, implement a Fenwick Tree (Binary Indexed Tree). # Task Write a class `FenwickTree` with the following methods: - `__init__(self, freq: List[int])`: Initializes the Fenwick Tree with the given frequency array. - `update(self, index: int, value: int)`: Updates the frequency of the element at the specified index to the new value provided. - `prefix_sum(self, index: int) -> int`: Returns the sum of the frequencies from the start of the array up to the specified index. # Input - `freq`: A list of integers representing the initial frequencies. - `index`: An integer within the range of the list's length, indicating the position of the element to update. - `value`: An integer representing the new frequency value. - For `prefix_sum`, an integer `index` within the range of the list's length, indicating the end of the range for which prefix sum needs to be calculated. # Output - For `update`, no output is expected. - For `prefix_sum`, return the sum of the frequencies from the start of the array up to the specified index. # Constraints - The array `freq` can have up to `100,000` elements. - The frequency values and updated values can be any integer within the range of `[-1,000,000, 1,000,000]`. # Example ```python # Example usage: freq = [1, 7, 3, 0, 7, 8, 3] fenwick_tree = FenwickTree(freq) # Update the frequency at index 3 fenwick_tree.update(3, 5) # Query the prefix sum up to index 4 result = fenwick_tree.prefix_sum(4) # Expected output: 1 + 7 + 3 + 5 + 7 = 23 ``` # Performance Requirement - Both operations (update and prefix_sum) should be performed in O(log n) time complexity.
answer:class FenwickTree: def __init__(self, freq): Initializes the Fenwick Tree with the given frequency array. self.n = len(freq) self.tree = [0] * (self.n + 1) self.original = [0] * self.n for i in range(self.n): self.update(i, freq[i]) def update(self, index, value): Updates the frequency of the element at the specified index to the new value provided. delta = value - self.original[index] self.original[index] = value index += 1 while index <= self.n: self.tree[index] += delta index += index & -index def prefix_sum(self, index): Returns the sum of the frequencies from the start of the array up to the specified index. sum = 0 index += 1 while index > 0: sum += self.tree[index] index -= index & -index return sum