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question:Valeria is working on a project that requires her to follow a particular sequence of tasks. She has a plan consisting of n tasks, numbered from 1 to n. The tasks must be executed sequentially from task 1 to task n, but there are dependencies between some tasks. A dependency means that one task must be completed before another task can begin. To manage her workflow, Valeria needs to know the order in which she should execute the tasks. If there are multiple valid orders, Valeria will choose any of them. Write a program to determine the order in which Valeria should execute the tasks. If it’s not possible to execute all tasks due to a circular dependency, your program should output "Impossible". The first line of the input contains two integers n and m (1 ≤ n ≤ 100, 0 ≤ m ≤ 10000) — the number of tasks and the number of dependencies. The next m lines contain two integers a and b (1 ≤ a, b ≤ n), representing a dependency that task a must be completed before task b. Output the numbers of the tasks in the order they should be executed. If there are multiple correct orders, output any one of them. If it is not possible to execute all tasks, output "Impossible". Example: Input: 4 4 1 2 2 3 3 4 4 1 Output: Impossible Input: 5 4 1 2 2 3 3 4 4 5 Output: 1 2 3 4 5
answer:from collections import defaultdict, deque def find_task_order(n, m, dependencies): Determines the task execution order based on given dependencies. Args: n (int): Number of tasks. m (int): Number of dependencies. dependencies (list of tuple): List of (a, b) representing task a must be completed before task b. Returns: list or str: List of tasks in the order they should be executed or "Impossible" if circular dependency exists. # Graph to store the dependencies graph = defaultdict(list) indegree = [0] * (n + 1) # Indegree array to keep track of incoming edges # Building the graph and indegree array for a, b in dependencies: graph[a].append(b) indegree[b] += 1 # Queue to maintain the order of tasks with no dependencies queue = deque([i for i in range(1, n + 1) if indegree[i] == 0]) order = [] while queue: current = queue.popleft() order.append(current) for neighbor in graph[current]: indegree[neighbor] -= 1 if indegree[neighbor] == 0: queue.append(neighbor) if len(order) == n: return order else: return "Impossible" # Example usage: # Input 1: 4 tasks with circular dependencies # print(find_task_order(4, 4, [(1, 2), (2, 3), (3, 4), (4, 1)])) # Output: "Impossible" # Input 2: 5 tasks with valid dependencies # print(find_task_order(5, 4, [(1, 2), (2, 3), (3, 4), (4, 5)])) # Output: [1, 2, 3, 4, 5]
question:You are given an array a of n integers. You need to select exactly two distinct elements from the array and remove them. After that, you will replace these two elements with their sum. This operation can be repeated as many times as desired until only one element remains in the array. Your task is to determine the maximum possible value of the final remaining element in the array after performing the operations optimally. The first line contains a single integer t (1 le t le 1000) — the number of test cases. The first line of each test case contains an integer n (2 le n le 100) — the length of the array a. The second line of each test case contains n integers a_1, a_2, ldots, a_n (1 le a_i le 100) — the elements of array a. For each test case, print a single integer: the maximum possible value of the final remaining element in the array after performing the described operations. # Example Input: ``` 3 4 1 2 2 3 3 5 5 5 5 1 1 1 1 1 ``` Output: ``` 8 15 5 ``` # Explanation In the first test case, one of the optimal sequences of operations is: - Remove 3 and 2, resulting in the array [1, 2, 5] - Remove 5 and 2, resulting in the array [1, 7] - Remove 7 and 1, resulting in the array [8] In the second test case, we can sum the 5s directly, resulting in 15. In the third test case, since all 1s can be summed to eventually become 5.
answer:def max_final_element(t, test_cases): results = [] for i in range(t): n, arr = test_cases[i] results.append(sum(arr)) return results # Example usage for the function t = 3 test_cases = [ (4, [1, 2, 2, 3]), (3, [5, 5, 5]), (5, [1, 1, 1, 1, 1]) ] print(max_final_element(t, test_cases)) # Expected output: [8, 15, 5]
question:Given a grid of size n x m, you need to find the number of unique paths from the top-left corner to the bottom-right corner, where you can only move either down or right at any point in time. The first line of input contains two integers n and m (1 ≤ n, m ≤ 100) representing the number of rows and columns in the grid. Print the number of unique paths from the top-left corner to the bottom-right corner. Example: Input: 3 4 Output: 10 In the example, there are 10 unique paths from (1,1) to (3,4).
answer:def unique_paths(n, m): Returns the number of unique paths in a n x m grid from top-left to bottom-right. # Create a 2D list to store the number of paths to each cell dp = [[1] * m for _ in range(n)] # Iterate over the grid starting from cell (1, 1) for i in range(1, n): for j in range(1, m): dp[i][j] = dp[i - 1][j] + dp[i][j - 1] # The bottom-right corner will have the number of unique paths return dp[-1][-1]
question:You are given an array of integers representing the height of trees in a forest. You need to collect a certain amount of wood by cutting down the tops of trees. The problem is to determine the maximum height at which you should cut the trees so you can collect at least the required amount of wood. Formally, given an array of integers `heights` of length `n` where `heights[i]` is the height of the ith tree, and an integer `m` representing the minimum amount of wood you need to collect, your task is to find the maximum integer `h` such that when all trees taller than `h` are cut to `h`, the amount of wood collected is at least `m`. The amount of wood collected from a tree is the difference between its height and `h` if the tree's height is greater than `h`; otherwise, no wood is collected from that tree. # Input - The first line contains two integers `n` and `m` (1 leq n leq 10^5, 1 leq m leq 10^{18}). - The second line contains `n` integers representing the heights of the trees: `heights_1, heights_2, ..., heights_n` (1 leq heights_i leq 10^9). # Output - Print a single integer `h` representing the maximum height at which you should cut the trees to collect at least `m` wood. # Example Input ``` 4 7 20 15 10 17 ``` Output ``` 15 ``` # Explanation If you cut the trees at height h = 15: - Tree of height 20: 20 - 15 = 5 wood - Tree of height 15: 15 - 15 = 0 wood - Tree of height 10: 10 - 15 = 0 wood (because height < h) - Tree of height 17: 17 - 15 = 2 wood Total wood collected = 5 + 0 + 0 + 2 = 7 wood. No larger height h can achieve at least 7 wood.
answer:def max_height_to_cut_trees(heights, m): Given a list of tree heights and a required amount of wood, this function returns the maximum height at which trees should be cut to collect at least the required amount of wood. :param heights: List[int] - list of tree heights :param m: int - required amount of wood :return: int - maximum height at which to cut trees def wood_collected(cut_height): return sum(max(height - cut_height, 0) for height in heights) low, high = 0, max(heights) answer = 0 while low <= high: mid = (low + high) // 2 if wood_collected(mid) >= m: answer = mid low = mid + 1 else: high = mid - 1 return answer