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question:You have a collection of N books, each with a unique width Wi. You need to arrange these books on a bookshelf of width W so that the total width of the books on the shelf does not exceed W. However, there's a special condition: You must place as many books as possible on the shelf without exceeding the width limit W. If there are multiple ways to place the maximum number of books, choose the arrangement where the total unused width is minimized. Write a function that, given the widths of the books and the width of the bookshelf, determines the maximum number of books that can be placed on the shelf and the minimum unused width. Input Format: - The first line of input contains two integers N and W, the number of books and the width of the bookshelf respectively. - The second line contains N space-separated integers representing the widths of the books. Output Format: - Print two integers: the maximum number of books that can be placed on the shelf and the minimum unused width on the shelf. Constraints: - 1 ≤ N ≤ 1000 - 1 ≤ W ≤ 10000 - 1 ≤ Wi ≤ 1000 Example Input: ``` 5 10 2 3 5 6 1 ``` Example Output: ``` 3 0 ``` Explanation: - The optimal arrangement is to place books with widths 2, 3, and 5, which sum exactly to 10. Thus, 3 books can be placed with no unused width.
answer:def max_books_on_shelf(n, W, widths): Given the number of books, the width of the bookshelf, and the widths of each book, determines the maximum number of books that can fit on the shelf and the minimum unused width. Parameters: n (int): The number of books. W (int): The width of the bookshelf. widths (list of int): The widths of the books. Returns: tuple: A tuple containing two integers: - The maximum number of books that can be placed on the shelf. - The minimum unused width with the given configuration. from itertools import combinations max_books = 0 min_unused_width = W for r in range(1, n+1): for combo in combinations(widths, r): total_width = sum(combo) if total_width <= W: unused_width = W - total_width if r > max_books or (r == max_books and unused_width < min_unused_width): max_books = r min_unused_width = unused_width return max_books, min_unused_width # Input prompt for example implementation if __name__ == "__main__": n, W = map(int, input().split()) widths = list(map(int, input().split())) result = max_books_on_shelf(n, W, widths) print(result[0], result[1])
question:A tree is a connected graph without cycles. A subtree is any connected subgraph of a tree. Given a tree with N nodes, each node has a certain value assigned to it. Define the score of a subtree as the product of values of all its nodes. You are required to find the maximum score among all possible subtrees of the given tree. Input The first line contains an integer N (1 ≤ N ≤ 50), the number of nodes in the tree. The second line contains N integers v1, v2, ..., vN (1 ≤ vi ≤ 1000), where vi is the value assigned to the i-th node. Each of the next N-1 lines contains two integers u and v (1 ≤ u, v ≤ N), indicating that there is an edge between node u and node v. Output Print the maximum score among all possible subtrees of the given tree. Since the numbers can be large, output the result modulo 1000000007. Examples Input 3 2 3 4 1 2 1 3 Output 12 Input 4 3 5 2 6 1 2 1 3 3 4 Output 30 Note In the first example, the subtree consisting of nodes [1, 2, 3] has the maximum score, which is 2 * 3 * 4 = 24. Since 24 modulo 1000000007 is still 24, the output is 24. In the second example, the subtree consisting of nodes [1, 2, 3, 4] has the maximum score, which is 3 * 5 * 2 * 6 = 180. Since 180 modulo 1000000007 is still 180, the output is 180.
answer:MOD = 1000000007 def max_tree_score(N, values, edges): from collections import defaultdict import sys sys.setrecursionlimit(10000) # Convert the edges list into an adjacency list tree = defaultdict(list) for u, v in edges: tree[u].append(v) tree[v].append(u) def dfs(node, parent): product = values[node - 1] for child in tree[node]: if child != parent: product = product * dfs(child, node) % MOD return product # We need to find the maximum product subtree starting from any node max_score = max(dfs(i, -1) for i in range(1, N + 1)) % MOD return max_score
question:# Task You are given a string that represents a sequence of parentheses. Your task is to determine if the sequence is valid. A sequence of parentheses is considered valid if and only if it is balanced: every opening parenthesis has a corresponding closing parenthesis, and the pairs of parentheses are properly nested. # Input/Output - `[input]` string `s` A string representing the sequence of parentheses. `1 <= s.length <= 100` The string consists only of the characters `(` and `)`. - `[output]` a boolean value `true` if the sequence of parentheses is valid, `false` otherwise.
answer:def is_valid_parentheses(s): Checks if the parentheses in the sequence are valid (i.e., balanced and properly nested). Args: s (str): A string representing the sequence of parentheses. Returns: bool: True if the sequence is valid, otherwise False. stack = [] for char in s: if char == '(': stack.append(char) elif char == ')': if not stack: return False stack.pop() return not stack
question:The owner of a small shopping mall wants to keep track of customer spending. Each customer has a record of the prices of items they purchased. The owner needs to find out how many customers have spent more than a given amount in total during their shopping visits. -----Input----- - The first line of input contains an integer T, the number of test cases. - The first line of each test case contains two integers, N and K, where N is the number of customers, and K is the threshold amount. - The next N lines contain a list of integers separated by spaces, representing the prices of items each customer purchased. -----Output----- - For each test case, output the number of customers whose total spending exceeds the threshold amount K. -----Constraints----- - 1 ≤ T ≤ 100 - 1 ≤ N ≤ 100 - 1 ≤ K ≤ 10^6 - 1 ≤ number of prices per customer ≤ 100 - 1 ≤ each price ≤ 10^4 -----Example----- Input: 2 3 500 100 200 250 400 150 100 100 100 200 2 1000 1000 900 200 100 Output: 2 1 -----Explanation----- Example case 1: The total spending of the three customers are 550 (100 + 200 + 250), 550 (400 + 150), and 500 (100 + 100 + 100 + 200). Two customers have spent more than 500 in total. Example case 2: The total spending of the two customers are 1000 and 1200 (900 + 200 + 100). One customer has spent more than 1000 in total.
answer:def count_customers_exceeding_threshold(T, test_cases): results = [] for case in test_cases: N, K, customers = case count = 0 for customer in customers: if sum(customer) > K: count += 1 results.append(count) return results # Helper function to process input data def process_input(input_data): lines = input_data.strip().split("n") T = int(lines[0]) index = 1 test_cases = [] for _ in range(T): N, K = map(int, lines[index].split()) customers = [] for i in range(1, N+1): customer = list(map(int, lines[index + i].split())) customers.append(customer) test_cases.append((N, K, customers)) index += N + 1 return T, test_cases