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question:# Advanced Sorting Problem: Optimized Bucket Sort Given an array of `n` integers, write an optimized implementation of Bucket Sort that minimizes the worst-case time complexity and space usage. Instead of using a fixed bucket count, adaptively determine the number of buckets and ensure efficient sorting within each bucket. Use a more efficient sorting mechanism, such as Timsort, for sorting the contents of the buckets. Function Signature ```python def optimized_bucket_sort(arr: List[int]) -> List[int]: pass ``` Input - An array of integers, `arr`, with `1 ≤ len(arr) ≤ 10^6`. - Values in `arr` could range from `-10^9` to `10^9`. Output - A sorted array in ascending order. Constraints - Minimize the worst-case time complexity of your implementation. - Ensure that your solution works within a reasonable space complexity. Example ```python assert optimized_bucket_sort([4, 1, 3, 9, 7]) == [1, 3, 4, 7, 9] assert optimized_bucket_sort([10, -1, 2, 5, 0, 6]) == [-1, 0, 2, 5, 6, 10] assert optimized_bucket_sort([4, 4, 4, 4, 4]) == [4, 4, 4, 4, 4] ``` Ensure that your code handles large input sizes efficiently and consider performance optimizations and edge cases discussed in the algorithm analysis.
answer:from typing import List def optimized_bucket_sort(arr: List[int]) -> List[int]: if len(arr) <= 1: return arr num_buckets = len(arr) min_val, max_val = min(arr), max(arr) if min_val == max_val: return arr bucket_range = (max_val - min_val) / num_buckets # Create buckets buckets = [[] for _ in range(num_buckets)] # Distribute input array values into buckets for num in arr: idx = int((num - min_val) / bucket_range) if idx == num_buckets: idx -= 1 # Handle the boundary case where num == max_val buckets[idx].append(num) # Sort each bucket and gather results sorted_arr = [] for bucket in buckets: sorted_arr.extend(sorted(bucket)) return sorted_arr
question:# Maximum Subarray Sum Scenario: You are working on a financial analysis program and need to find the period with the highest profit in a given year. The data for each month's profit/loss is given as a list of integers. You need to write a function that will determine the maximum profit that could be obtained by considering any contiguous subarray within this list. Function Signature: ```python def max_subarray(array: List[int]) -> int: ``` Input: - `array` (List[int]): A list of integers representing the profit or loss for each month. `1 <= len(array) <= 10^5`, `-10^6 <= array[i] <= 10^6`. Output: - `int`: The maximum sum of any contiguous subarray within the provided list. Example: ```python # Example 1: array = [1, 2, -3, 4, 5, -7, 23] print(max_subarray(array)) # Output: 25 (subarray: [4, 5, -7, 23]) # Example 2: array = [-1, -2, -3, -4] print(max_subarray(array)) # Output: -1 (single element subarray: [-1]) ``` Constraints: - The array must contain at least one element. - The array can contain both positive and negative integers. - The function should have a time complexity of O(n) and a space complexity of O(1). Explanation: Implement the "Kadane's Algorithm" to solve this problem efficiently. Iterate through the array while maintaining the `max_now` and `max_so_far` values as described in the analysis. Consider any potential edge cases like arrays with all negative numbers and make sure to handle them correctly.
answer:def max_subarray(array): Returns the maximum sum of any contiguous subarray. max_now = max_so_far = array[0] for num in array[1:]: max_now = max(num, max_now + num) max_so_far = max(max_so_far, max_now) return max_so_far
question:You are tasked with implementing a more optimized version of a prime checking function. This function should be able to handle very large numbers efficiently by incorporating one or more optimization techniques or probabilistic approaches for determining primality, such as the Miller-Rabin primality test. # Function Signature ```python def is_prime_optimized(n: int) -> bool: Determine whether a given number `n` is prime. Parameters: n (int): An integer to be checked for primality. Returns: bool: True if `n` is a prime number, False otherwise. ``` # Inputs and Outputs - **Input**: - `n` is an integer where `0 <= n <= 10^18`. - **Output**: - Return `True` if `n` is a prime number, otherwise `False`. # Constraints - The function should work efficiently for the given range of `n`. # Performance Requirements - Time complexity should aim to be better than O(√n) for large numbers, leveraging probabilistic methods if necessary. - Space complexity should remain O(1) or only slightly more if storage of intermediate computational states is necessary. # Example ```python assert is_prime_optimized(1) == False assert is_prime_optimized(2) == True assert is_prime_optimized(3) == True assert is_prime_optimized(4) == False assert is_prime_optimized(29) == True assert is_prime_optimized(1_000_000_007) == True # Example of a large prime number ``` Your task is to implement the `is_prime_optimized` function, ensuring it meets the performance and accuracy requirements stated above.
answer:def is_prime_optimized(n: int) -> bool: Determine whether a given number `n` is prime using the Miller-Rabin primality test. Parameters: n (int): An integer to be checked for primality. Returns: bool: True if `n` is a prime number, False otherwise. if n <= 1: return False if n <= 3: return True if n % 2 == 0 or n % 3 == 0: return False # Miller-Rabin test with a predetermined set of bases def miller_rabin_test(d, n): import random a = random.randint(2, n - 2) x = pow(a, d, n) if x == 1 or x == n - 1: return True while d != n - 1: x = (x * x) % n d *= 2 if x == 1: return False if x == n - 1: return True return False # Find d such that n-1 = d * 2^r d = n - 1 while d % 2 == 0: d //= 2 # Perform the Miller-Rabin test multiple times k = 5 # Number of iterations for _ in range(k): if not miller_rabin_test(d, n): return False return True
question:# Question: You are required to implement an enhanced version of the `OrderedStack` class that maintains a stack where elements are always sorted in ascending order from bottom to top. This stack should have the following functionalities: 1. **Push**: Insert an element into the stack while maintaining the order. 2. **Pop**: Remove and return the top element from the stack. 3. **Peek**: Return the top element without removing it from the stack. 4. **Is_empty**: Check whether the stack is empty or not. 5. **Size**: Return the number of elements in the stack. Implement the class `OrderedStack` with the following method signatures: ```python class OrderedStack: def __init__(self): # Initialize an empty list to store stack elements pass def is_empty(self) -> bool: # Return True if the stack is empty, False otherwise pass def push(self, item: int) -> None: # Insert 'item' into the stack, maintaining sorted order pass def pop(self) -> int: # Remove and return the top element of the stack pass def peek(self) -> int: # Return the top element without removing it from the stack pass def size(self) -> int: # Return the number of elements in the stack pass ``` # Input and Output: * The `push` method takes an integer as input and returns nothing. * The `pop` method returns the top element of the stack. * The `peek` method returns the top element of the stack without removing it. * The `is_empty` method returns a boolean indicating whether the stack is empty. * The `size` method returns an integer indicating the number of elements in the stack. # Constraints: * Elements in the stack are integers. * Assume all input operations are valid (i.e., no `pop` or `peek` will be called on an empty stack). # Example: ```python stack = OrderedStack() stack.push(5) stack.push(2) stack.push(8) stack.push(1) print(stack.peek()) # Output: 8 print(stack.pop()) # Output: 8 print(stack.size()) # Output: 3 print(stack.peek()) # Output: 5 print(stack.is_empty()) # Output: False ``` Ensure that your implementation efficiently manages the ordering of stack elements during the push operation.
answer:class OrderedStack: def __init__(self): # Initialize an empty list to store stack elements self.stack = [] def is_empty(self) -> bool: # Return True if the stack is empty, False otherwise return len(self.stack) == 0 def push(self, item: int) -> None: # Insert 'item' into the stack, maintaining sorted order if self.is_empty(): self.stack.append(item) else: insert_pos = len(self.stack) for i in range(len(self.stack)): if item <= self.stack[i]: insert_pos = i break self.stack.insert(insert_pos, item) def pop(self) -> int: # Remove and return the top element of the stack if not self.is_empty(): return self.stack.pop() else: raise IndexError("pop from empty stack") def peek(self) -> int: # Return the top element without removing it from the stack if not self.is_empty(): return self.stack[-1] else: raise IndexError("peek from empty stack") def size(self) -> int: # Return the number of elements in the stack return len(self.stack)