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question:Design a function `longest_consecutive_subarray` that returns the length of the longest subarray with consecutive integers in a given array. A subarray is a contiguous part of an array. For example, if the input array is `[1, 2, 3, 4]`, then `[1, 2, 3, 4]` is a subarray with consecutive integers because each element differs from the previous one by 1. If the input array is `[1, 9, 3, 10, 4, 20, 2]`, then the longest subarray with consecutive integers is `[1, 2, 3, 4]`. Write a function that takes an array of integers and returns the length of the longest subarray that contains consecutive integers. # Function Signature ```python def longest_consecutive_subarray(arr: List[int]) -> int: ``` # Input - An integer array `arr` with a size of up to 10^5 elements, where each element is in the range -10^9 to 10^9. # Output - Return the length of the longest subarray with consecutive integers. # Examples Example 1 ```python arr = [1, 94, 93, 1000, 92, 91, 90] assert longest_consecutive_subarray(arr) == 5 ``` - Explanation: The longest subarray with consecutive integers is `[90, 91, 92, 93, 94]`, which has a length of 5. Example 2 ```python arr = [1, 9, 3, 10, 4, 20, 2] assert longest_consecutive_subarray(arr) == 4 ``` - Explanation: The longest subarray with consecutive integers is `[1, 2, 3, 4]`, which has a length of 4. Example 3 ```python arr = [10, 12, 11, 13, 14, 15] assert longest_consecutive_subarray(arr) == 6 ``` - Explanation: The entire array `[10, 11, 12, 13, 14, 15]` is a subarray with consecutive integers, which has a length of 6. # Constraints - Elements in the subarray should be unique and consecutive integers. - Achieving an optimal solution in O(n) time complexity is ideal. You may assume that the provided input array will always be non-empty.
answer:from typing import List def longest_consecutive_subarray(arr: List[int]) -> int: if not arr: return 0 max_length = 0 num_set = set(arr) for num in arr: if num - 1 not in num_set: # Check if it's the start of a sequence current_num = num current_length = 1 while current_num + 1 in num_set: current_num += 1 current_length += 1 max_length = max(max_length, current_length) return max_length
question:You have N containers arranged in a row, each containing a certain number of marbles. Your task is to redistribute the marbles such that each container ends up with the same number of marbles, if possible. You can perform the following operation any number of times: pick one marble from a container and move it to an adjacent container (either the one immediately left or right). Write a program to determine if it is possible to redistribute the marbles equally among all containers, and if so, provide a sequence of moves to achieve it. ------ Input ------ The first line of the input contains a single integer N (1 leq N leq 1000), the number of containers. The second line contains N space-separated integers a_1, a_2, ..., a_N (0 le a_i le 10^6), where a_i represents the number of marbles in the i-th container. ------ Output ------ If it is not possible to redistribute the marbles equally, print "NO". Otherwise, print "YES" on the first line, followed by a sequence of moves. Each move should be of the form `i j`, indicating that a marble is moved from container i to container j (1 leq i, j leq N). If there are multiple valid sequences of moves, any will be accepted. ------ Constraints ------ 1 ≤ N ≤ 1000 0 ≤ a_i ≤ 10^6 ------ Sample Input 1 ------ 3 1 2 3 ------ Sample Output 1 ------ YES 1 2 2 3 ------ Explanation 1 ------ In this case, you can achieve equal distribution by moving one marble from container 1 to container 2, and then moving one marble from container 2 to container 3. ------ Sample Input 2 ------ 3 1 1 1 ------ Sample Output 2 ------ YES ------ Explanation 2 ------ All containers already have the same number of marbles, so no moves are necessary. ------ Sample Input 3 ------ 2 4 1 ------ Sample Output 3 ------ NO ------ Explanation 3 ------ It is not possible to redistribute the marbles equally because the total number of marbles (5) is not divisible by the number of containers (2).
answer:def redistribute_marbles(N, marbles): Determines if it is possible to redistribute marbles equally among containers and provides the moves to do so. :param N: Integer, the number of containers. :param marbles: List of integers, the number of marbles in each container. :return: Tuple, (String, List of moves (tuple of 2 integers)), or ("NO", []) if redistribution is not possible. total_marbles = sum(marbles) # Check if it is possible to redistribute marbles equally if total_marbles % N != 0: return "NO", [] target = total_marbles // N moves = [] for i in range(1, N): while marbles[i-1] > target: marbles[i-1] -= 1 marbles[i] += 1 moves.append((i, i + 1)) while marbles[i-1] < target: marbles[i-1] += 1 marbles[i] -= 1 moves.append((i + 1, i)) # Verify that all marbles are now equal to the target value for marble_count in marbles: if marble_count != target: return "NO", [] return "YES", moves
question:Problem King Arthur is organizing his yearly round table meeting with the N knights, each represented by a point in a 2D plane. The knights need to be divided into K distinct groups for discussion, and each group should have at least one knight. Arthur wants to position a round table at the optimal point so that the sum of the distances from each knight in a group to the table's center is minimized. Devise an algorithm that determines the positions of K tables such that the total sum of the distances from each knight to their assigned table is minimized. You must find the optimal positions of the round tables. Constraints: The input satisfies the following conditions. * 1 leq K leq N leq 10 * 0 leq x_i, y_i leq 10^4 Input N K x_1 y_1 x_2 y_2 ... x_N y_N The input is given in the following format. The number of knights N and the number of groups K are given as integers on the first line. From the second line onward, the coordinates x_i , y_i of each knight i are given as integers. Output Output the coordinates of the K tables as pairs of real numbers. The coordinates must not contain an absolute error greater than 10^{-3} . Examples Input 6 2 1 3 2 2 3 1 8 8 9 9 10 10 Output 2.000 2.000 9.000 9.000 Input 5 3 1 1 2 2 3 3 4 4 5 5 Output 1.500 1.500 3.500 3.500 5.000 5.000 Input 3 1 5 1 6 2 7 3 Output 6.000 2.000
answer:def k_means(knights, K): import numpy as np from scipy.spatial import distance def initialize_centroids(knights, K): np.random.seed(42) initial_indices = np.random.choice(len(knights), K, replace=False) return knights[initial_indices] def assign_knights_to_centroids(knights, centroids): clusters = {} for knight in knights: distances = [distance.euclidean(knight, centroid) for centroid in centroids] closest_centroid = np.argmin(distances) if closest_centroid in clusters: clusters[closest_centroid].append(knight) else: clusters[closest_centroid] = [knight] return clusters def update_centroids(clusters): new_centroids = [] for key in clusters.keys(): new_centroid = np.mean(clusters[key], axis=0) new_centroids.append(new_centroid) return new_centroids knights = np.array(knights) centroids = initialize_centroids(knights, K) for _ in range(100): # Run iterations (fixed number for convergence) clusters = assign_knights_to_centroids(knights, centroids) new_centroids = update_centroids(clusters) # Check for convergence by comparing old and new centroids if np.allclose(centroids, new_centroids, atol=1e-3): break centroids = new_centroids return centroids def find_optimal_tables(N, K, knight_coordinates): optimal_centroids = k_means(knight_coordinates, K) return optimal_centroids def main(): import sys input = sys.stdin.read data = input().split() N = int(data[0]) K = int(data[1]) knight_coordinates = [(int(data[i*2+2]), int(data[i*2+3])) for i in range(N)] tables = find_optimal_tables(N, K, knight_coordinates) for table in tables: print(f"{table[0]:.3f} {table[1]:.3f}") if __name__ == "__main__": main()
question:You are given a list of non-negative integers, with each integer representing the height of a vertical line on a graph. Your task is to write a program that finds two lines that, together with the x-axis, form a container such that the container holds the most amount of water. Return the maximum amount of water a container can store. Note that you may not slant the container. The container should be formed by two different lines chosen from the input list. Input A sequence of multiple datasets is given as input. The end of the input is indicated by a single zero. Each dataset contains the following format: n h1 h2 ... hn - The first line contains the integer n (2 ≤ n ≤ 1000), representing the number of vertical lines. - The second line contains n space-separated integers h1, h2, ..., hn (0 ≤ hi ≤ 10^4). The number of datasets does not exceed 50. Output For each dataset, output the maximum amount of water the container can store on one line. Examples Input 5 1 8 6 2 5 4 8 3 7 4 1 1 1 1 3 5 2 6 0 Output 49 3
answer:def max_area(heights): Return the maximum area of water that can be contained by two vertical lines. left, right = 0, len(heights) - 1 max_area = 0 while left < right: height = min(heights[left], heights[right]) width = right - left current_area = height * width max_area = max(max_area, current_area) if heights[left] < heights[right]: left += 1 else: right -= 1 return max_area def find_max_areas(datasets): For each dataset, find the max area of water the container can store each output in one line. results = [] for heights in datasets: results.append(max_area(heights)) return results def parse_input(input_data): lines = input_data.strip().split("n") datasets = [] i = 0 while i < len(lines): n = int(lines[i]) if n == 0: break heights = list(map(int, lines[i+1].split())) datasets.append(heights) i += 2 return datasets