question:**Coding Problem: Prefix Trie (Autocomplete System)** # Problem Statement You are tasked with implementing a simplified version of an autocomplete system using a prefix trie (also known as a prefix tree). The system should allow you to efficiently insert words and find all words that start with a given prefix. You must implement two functions: - `insert(word)`: Inserts a word into the trie. - `find_words_with_prefix(prefix)`: Returns a list of all words in the trie that start with the given prefix. # Input - `word`: A string to be inserted into the trie. - `prefix`: A prefix string to search for in the trie. # Output - For the `find_words_with_prefix(prefix)` function, return a list of strings representing all words in the trie that start with the given prefix. # Constraints - The input words will consist only of lowercase English letters. - The maximum number of words that can be inserted into the trie is 10,000. - The length of each word will not exceed 100 characters. - The function `find_words_with_prefix(prefix)` should be efficient, with an average time complexity better than O(n) per prefix search, where n is the number of words in the trie. # Example ```python autocomplete = Trie() autocomplete.insert("hello") autocomplete.insert("hell") autocomplete.insert("helicopter") autocomplete.insert("help") autocomplete.insert("healthy") print(autocomplete.find_words_with_prefix("he")) # Output: ['hello', 'hell', 'helicopter', 'help', 'healthy'] print(autocomplete.find_words_with_prefix("hel")) # Output: ['hello', 'hell', 'helicopter', 'help'] print(autocomplete.find_words_with_prefix("hea")) # Output: ['healthy'] print(autocomplete.find_words_with_prefix("ho")) # Output: [] ``` # Implementation Details 1. Define a `TrieNode` class that contains a dictionary for child nodes and a boolean flag indicating the end of a word. 2. Create a `Trie` class that contains the root of the Trie and implements the `insert` and `find_words_with_prefix` methods. 3. Implement the `insert` method to add words to the Trie character by character. 4. Implement the `find_words_with_prefix` method to traverse the Trie from the root based on the given prefix and collect all the words that start with that prefix. # Template ```python class TrieNode: def __init__(self): self.children = {} self.is_end_of_word = False class Trie: def __init__(self): self.root = TrieNode() def insert(self, word: str): node = self.root for char in word: if char not in node.children: node.children[char] = TrieNode() node = node.children[char] node.is_end_of_word = True def find_words_with_prefix(self, prefix: str): def dfs(node, prefix, words): if node.is_end_of_word: words.append(prefix) for char, child in node.children.items(): dfs(child, prefix + char, words) node = self.root for char in prefix: if char not in node.children: return [] node = node.children[char] words = [] dfs(node, prefix, words) return words # Example usage autocomplete = Trie() autocomplete.insert("hello") autocomplete.insert("hell") autocomplete.insert("helicopter") autocomplete.insert("help") autocomplete.insert("healthy") print(autocomplete.find_words_with_prefix("he")) # Output: ['hello', 'hell', 'helicopter', 'help', 'healthy'] print(autocomplete.find_words_with_prefix("hel")) # Output: ['hello', 'hell', 'helicopter', 'help'] print(autocomplete.find_words_with_prefix("hea")) # Output: ['healthy'] print(autocomplete.find_words_with_prefix("ho")) # Output: [] ```

question:# Check for Duplicates in a Stream Write a function to determine if there are any duplicates in a stream of integers. The function will receive a list of integers that may contain duplicate values. Your goal is to identify if at least one number appears more than once in the list. Function Specification **Function**: `has_duplicates(nums: List[int]) -> bool` **Input**: - A list of integers, `nums`, where each integer can range from -10^7 to 10^7. - The length of the list, `len(nums)`, varies from 0 to 10^6. **Output**: - A single boolean value: `True` if any number appears more than once in the list, `False` otherwise. Examples ```python >>> has_duplicates([1, 2, 3, 4, 5]) False >>> has_duplicates([1, 2, 3, 4, 1]) True >>> has_duplicates([5, 5, 5, 5, 5]) True >>> has_duplicates([]) False ``` Constraints 1. The function should operate in linear time. 2. Use only a constant amount of extra space, excluding the input list. Hint - Consider using data structures that can efficiently track seen values to achieve the desired time complexity.

question:# Question # Evaluating Polish Notation Expression Write a Python function that evaluates an arithmetic expression given in Polish notation (prefix notation). In Polish notation, operators precede their operands. The function should handle the four basic arithmetic operators: addition, subtraction, multiplication, and division. Function Signature ```python def evaluate_polish_notation(expression: List[str]) -> float: Parameters: expression (List[str]): A list of strings representing the Polish notation expression. Returns: result (float): The result of evaluating the expression. ``` Input * A list `expression` containing a valid Polish notation expression. * Operators: `+`, `-`, `*`, `/` * Each operand is a real number (positive, negative, integer, or floating point). Output * A float representing the result of evaluating the given Polish notation expression. Example ```python >>> evaluate_polish_notation(["+", "4", "*", "3", "2"]) 10.0 >>> evaluate_polish_notation(["-", "/", "10", "5", "3"]) -1.0 >>> evaluate_polish_notation(["*", "+", "1", "2", "3"]) 9.0 ``` Constraints * The input list will always represent a valid Polish notation expression. * Each operation will have the correct number of operands. * Division by zero will not occur in the input. # Implementation Requirements 1. Use a stack-based approach to evaluate the expression. 2. Ensure the function is optimized for performance and can handle large input lists efficiently. 3. Write at least three test cases to validate the correctness of your implementation.

question:# Problem Statement: You are given an array of integers representing the heights of buildings along a street, and you need to determine the amount of water that will be trapped after a heavy rain. The water trapped between two buildings is determined by the shorter of the two heights and must account for any other buildings in between. Your task is to implement the function `calculate_trapped_water` which efficiently computes the total amount of trapped water. # Function Signature: ```python def calculate_trapped_water(heights: list[int]) -> int: pass ``` # Input: - `heights`: A list of integers `heights` representing the heights of buildings along the street (0 <= len(heights) <= 10^6), where each height is a non-negative integer (0 <= heights[i] <= 10^4). # Output: - An integer representing the total amount of trapped water. # Example: ```python heights = [0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 2, 1] print(calculate_trapped_water(heights)) # Output: 6 ``` # Constraints: - Time complexity should be O(n). - Space complexity should be O(n). # Additional Notes: - Your solution should handle edge cases like empty arrays, arrays with no space for trapping water (e.g., arrays that are strictly increasing or decreasing), and arrays with varying building heights. - Ensure the solution efficiently calculates the trapped water even for large input sizes. # Algorithm Guidance: - Consider using a two-pointer approach to solve the problem in linear time. - Optionally, you may use precomputed arrays for left and right max heights to aid in the calculation.