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question:# Priority Queue Implementation and Optimization You are tasked to implement an efficient priority queue using different underlying data structures. The priority queue should support the following operations: * Insert an item with a given priority. * Remove and return the item with the smallest (or highest) priority. We'll start by implementing the priority queue using a simple linear array approach. Then, we'll move to an optimized version using a binary heap. Part 1: Linear Array-Based Priority Queue Implement a priority queue with the following methods: - `push(item, priority)`: Insert an item with the given priority. - `pop()`: Remove and return the item with the lowest priority. - `size()`: Return the number of items in the queue. Constraints: - The `priority` can be any integer (positive, negative, or zero). - The priority queue can have up to 10^4 items. Input and Output Formats - Input to `push`: An item (any data type) and an `priority` (integer). - Output of `pop`: The item with the lowest priority, or `None` if the queue is empty. - Output of `size`: Integer representing the number of items in the queue. ```python import itertools class PriorityQueueNode: def __init__(self, data, priority): self.data = data self.priority = priority def __repr__(self): return "{}: {}".format(self.data, self.priority) class PriorityQueue: def __init__(self, items=None, priorities=None): self.priority_queue_list = [] if items is None: return if priorities is None: priorities = itertools.repeat(None) for item, priority in zip(items, priorities): self.push(item, priority=priority) def size(self): return len(self.priority_queue_list) def push(self, item, priority=None): priority = item if priority is None else priority node = PriorityQueueNode(item, priority) for index, current in enumerate(self.priority_queue_list): if current.priority < node.priority: self.priority_queue_list.insert(index, node) return self.priority_queue_list.append(node) def pop(self): if self.size() == 0: return None return self.priority_queue_list.pop().data ``` # Example Usage: ```python pq = PriorityQueue() pq.push('task1', 3) pq.push('task2', 2) print(pq.pop()) # Outputs: 'task2' print(pq.size()) # Outputs: 1 ``` Part 2: Heap-Based Priority Queue (Advanced) To improve the efficiency of the priority queue, implement an optimized version using a binary heap. You may use Python's `heapq` module for this. Implement the following methods: - `push_heap(item, priority)`: Insert an item with the given priority using a heap. - `pop_heap()`: Remove and return the item with the lowest priority using a heap. - `size_heap()`: Return the number of items in the heap-based queue. Additional Constraints - Ensure the heap-based methods maintain the same previous constraints. - Expected time complexity for `push_heap` and `pop_heap` should be O(log n). **Note for Students**: Please provide both versions of your implementation. Compare and discuss the differences in performance observed during execution.
answer:class PriorityQueue: def __init__(self): self.queue = [] def push(self, item, priority): self.queue.append((priority, item)) self.queue.sort(key=lambda x: x[0]) def pop(self): if self.size() == 0: return None return self.queue.pop(0)[1] def size(self): return len(self.queue) import heapq class HeapPriorityQueue: def __init__(self): self.heap = [] def push_heap(self, item, priority): heapq.heappush(self.heap, (priority, item)) def pop_heap(self): if self.size_heap() == 0: return None return heapq.heappop(self.heap)[1] def size_heap(self): return len(self.heap)
question:# Question: Implement a Function to Find the Kth Largest Element Using Bitonic Sort You are given an array of integers whose length is guaranteed to be a power of two and a positive integer `k`. Your task is to find the `k`th largest element in the array using the bitonic sort algorithm. # Function Signature ```python def kth_largest_element(arr: List[int], k: int) -> int: pass ``` # Input * `arr` (List[int]): An array of integers with a length that is a power of two. * `k` (int): A positive integer representing the position (1-based) of the largest element to find in the sorted order. # Output * `int`: The `k`th largest element in the array. # Constraints * 1 ≤ len(arr) ≤ 2^10 * 1 ≤ arr[i] ≤ 10^6 * 1 ≤ k ≤ len(arr) # Example ```python arr = [3, 2, 1, 5, 6, 4] k = 2 # Result should be 5 because the sorted array is [1, 2, 3, 4, 5, 6] and the 2nd largest element is 5 assert kth_largest_element(arr, k) == 5 ``` # Requirements 1. Implement the `kth_largest_element` function. 2. The solution must use the `bitonic_sort` method as a core sorting mechanism. 3. Handle any edge cases, such as small arrays or out of bounds `k` values appropriately. # Note You can assume that the bitonic_sort function provided above is available and you do not need to implement this function. Focus on implementing the `kth_largest_element` function and use the `bitonic_sort` function within your implementation wherever necessary.
answer:from typing import List def bitonic_sort(arr: List[int], up: bool) -> List[int]: Helper function to sort an array using the bitonic sort. If `up` is True, the array is sorted in ascending order. If `up` is False, the array is sorted in descending order. def compare_and_swap(arr: List[int], i: int, j: int, direction: bool): if (direction == (arr[i] > arr[j])): arr[i], arr[j] = arr[j], arr[i] def bitonic_merge(arr: List[int], low: int, cnt: int, direction: bool): if cnt > 1: k = cnt // 2 for i in range(low, low + k): compare_and_swap(arr, i, i + k, direction) bitonic_merge(arr, low, k, direction) bitonic_merge(arr, low + k, k, direction) def bitonic_sort_recursive(arr: List[int], low: int, cnt: int, direction: bool): if cnt > 1: k = cnt // 2 bitonic_sort_recursive(arr, low, k, True) bitonic_sort_recursive(arr, low + k, k, False) bitonic_merge(arr, low, cnt, direction) n = len(arr) bitonic_sort_recursive(arr, 0, n, up) return arr def kth_largest_element(arr: List[int], k: int) -> int: n = len(arr) if n == 0 or k < 1 or k > n: raise ValueError("Invalid k value or empty array") sorted_arr = bitonic_sort(arr[:], up=False) return sorted_arr[k - 1]
question:# Scenario You are working on a powerful data analysis system that can handle various types of queries over a large set of numerical data. You need to implement a data structure that allows efficient range queries, such as calculating sums, finding the minimum or maximum values, and custom aggregations over defined ranges. # Task Implement and utilize a segment tree data structure for efficient range queries on an array. You will write a class that builds a segment tree from a given array and allows efficient range queries using a user-defined function. # Requirements 1. **Input**: The constructor will accept an array of integers and a function for segment merging. 2. **Output**: The `query` method will take two integers (L and R) representing the range [L, R] and return the result of the function applied over this range. # Constraints - The array will have at least one element and at most 10^5 elements. - The function provided will be a binary function that accepts two arguments of the same type as the array's elements and returns a value of that type. - The range query will always be within the bounds of the array. # Performance Requirements - The segment tree should be able to handle 10^5 elements, supporting O(log N) range queries. # Class Definition The class should be structured as follows: ```python class SegmentTree: def __init__(self, arr, function): # Implement the constructor to initialize the segment tree. def make_tree(self, i, l, r): # Build the tree by recursively dividing the array segments. def __query(self, i, L, R, l, r): # Perform a range query from L to R. def query(self, L, R): # Public method to start the query process. ``` # Example ```python # Initializing a Segment Tree with an array and a max function arr = [2, 4, 5, 3, 4] max_fn = max mytree = SegmentTree(arr, max_fn) # Making queries assert mytree.query(2, 4) == 5 assert mytree.query(0, 3) == 5 # Initializing a Segment Tree with an array and a sum function arr = [4, 5, 2, 3, 4, 43, 3] sum_fn = sum mytree = SegmentTree(arr, sum_fn) # Making queries assert mytree.query(1, 3) == 10 assert mytree.query(1, 6) == 60 ``` Write the implementation for the Segment Tree class based on the provided structure and requirements.
answer:class SegmentTree: def __init__(self, arr, function): Initialize the segment tree with the given array and function. self.n = len(arr) self.function = function self.tree = [None] * (4 * self.n) self.make_tree(arr, 0, 0, self.n - 1) def make_tree(self, arr, i, l, r): Recursively build the segment tree. if l == r: self.tree[i] = arr[l] else: mid = (l + r) // 2 left_child = 2 * i + 1 right_child = 2 * i + 2 self.make_tree(arr, left_child, l, mid) self.make_tree(arr, right_child, mid + 1, r) self.tree[i] = self.function(self.tree[left_child], self.tree[right_child]) def __query(self, i, L, R, l, r): Perform a range query from L to R within the segment [l, r]. if L > r or R < l: return None # return value outside of the range if L <= l and R >= r: return self.tree[i] mid = (l + r) // 2 left_result = self.__query(2 * i + 1, L, R, l, mid) right_result = self.__query(2 * i + 2, L, R, mid + 1, r) if left_result is None: return right_result if right_result is None: return left_result return self.function(left_result, right_result) def query(self, L, R): Public method to start the query process. return self.__query(0, L, R, 0, self.n - 1)
question:# Radix Sort Implementation: Large Dataset Sorting # Background You are tasked with implementing a highly efficient sorting algorithm to handle a vast dataset comprising non-negative integers. Your solution must leverage the Radix Sort algorithm to sort these integers. This sorting method prioritizes digit-wise sorting, offering exceptional performance for specific datasets. # Task Write a function `enhanced_radix_sort` that sorts a list of non-negative integers using the Radix Sort algorithm. The function should demonstrate an optimized implementation that handles various edge cases effectively. # Function Signature ```python def enhanced_radix_sort(arr: list[int]) -> list[int]: pass ``` # Input * `arr` (list of int): A list of non-negative integers that you need to sort. * Constraints: * `0 <= len(arr) <= 1_000_000` * `0 <= arr[i] <= 10^9` # Output * Return a list of integers sorted in non-decreasing order. # Constraints * Your implementation should not use built-in sort functions. * Optimize for large datasets with several edge cases (empty arrays, arrays with one element, etc.) * Handle digit extraction robustly to avoid mis-sorts. * Ensure efficient usage of memory. # Example ```python print(enhanced_radix_sort([170, 45, 75, 90, 802, 24, 2, 66])) # Output: [2, 24, 45, 66, 75, 90, 170, 802] print(enhanced_radix_sort([1, 4, 1, 2, 7, 5, 2])) # Output: [1, 1, 2, 2, 4, 5, 7] print(enhanced_radix_sort([])) # Output: [] print(enhanced_radix_sort([10**9, 10**9, 10**9])) # Output: [1000000000, 1000000000, 1000000000] ``` # Notes * The Radix Sort algorithm should illustrate sorting stability. * Consider edge cases like input arrays with large integers and repeated elements.
answer:def enhanced_radix_sort(arr: list[int]) -> list[int]: This function sorts a list of non-negative integers using Radix Sort. if not arr: return arr # Find the maximum number in the array to determine the number of digits max_num = max(arr) exp = 1 # Start with the least significant digit while max_num // exp > 0: arr = counting_sort(arr, exp) exp *= 10 return arr def counting_sort(arr, exp): A utility function to perform counting sort on arr[] according to the digit represented by exp. n = len(arr) output = [0] * n # Output array that will have sorted numbers count = [0] * 10 # Initialize count array as 0 for digits 0 to 9 # Store count of occurrences of digits for i in range(n): index = (arr[i] // exp) % 10 count[index] += 1 # Change count[i] so that count[i] now contains the actual position of this digit in output[] for i in range(1, 10): count[i] += count[i - 1] # Build the output array i = n - 1 while i >= 0: index = (arr[i] // exp) % 10 output[count[index] - 1] = arr[i] count[index] -= 1 i -= 1 # Copy the output array to arr[], so that arr[] now contains sorted numbers according to current digit for i in range(n): arr[i] = output[i] return arr