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question:You are a treasure hunter on a grid-based island map. The island is represented by a 2D grid of size n x m (1 ≤ n, m ≤ 1000) where each cell is either land or water. The grid is given as a list of strings, where each character is either '.' representing land, or '#' representing water. You start at the top-left corner of the grid (0, 0) and your goal is to reach the bottom-right corner of the grid (n-1, m-1) while only being able to move on land ('.') and without passing through water ('#'). You can move up, down, left, or right from a cell to an adjacent cell. Determine if it is possible to reach the bottom-right corner from the top-left corner. Input Format: - The first line contains two integers, n and m, the dimensions of the grid. - The next n lines contain m characters each, representing the grid. Output Format: - Print "YES" (without the quotes) if it's possible to reach the bottom-right corner from the top-left corner, otherwise print "NO" (without the quotes). Example Input: ``` 5 5 ..... .#. .#. .#. ..... ``` Example Output: ``` YES ```

answer:def is_path_possible(n, m, grid): Determines if there is a path from the top-left corner to the bottom-right corner of the grid, considering only land cells ('.') and not crossing water cells ('#'). if grid[0][0] == '#' or grid[n-1][m-1] == '#': return "NO" from collections import deque directions = [(-1, 0), (1, 0), (0, -1), (0, 1)] # Up, Down, Left, Right queue = deque([(0, 0)]) visited = [[False] * m for _ in range(n)] visited[0][0] = True while queue: x, y = queue.popleft() if x == n - 1 and y == m - 1: return "YES" for dx, dy in directions: nx, ny = x + dx, y + dy if 0 <= nx < n and 0 <= ny < m and not visited[nx][ny] and grid[nx][ny] == '.': visited[nx][ny] = True queue.append((nx, ny)) return "NO"

question:There is a unique tree called the "Magical Tree", which has the property that for any given integer n, you can perform a certain number of transformations to obtain "Magic Leaves". A transformation consists of selecting a node U in the tree and converting it into its immediate children nodes. Each node U has a label that is a positive integer, and the value of the node is defined as the sum of the labels of all the nodes in the subtree rooted at U. The Magic Leaves are those nodes that cannot be further transformed, i.e., leaf nodes. Given the structure of the Magical Tree and its node labels, your task is to determine the minimum number of transformations required to maximize the sum of the values of all Magic Leaves. # Input - The first line contains an integer n (1 <= n <= 10^5) — the number of nodes in the tree. - The second line contains n space-separated integers where the ith integer is the label of the ith node. - The next n-1 lines each contain two integers u and v (1 <= u, v <= n) representing an edge between nodes u and v in the tree. # Output Output a single integer — the minimum number of transformations required to maximize the sum of the values of all Magic Leaves. # Example Input ``` 5 1 5 3 7 4 1 2 1 3 3 4 3 5 ``` Output ``` 2 ``` # Explanation In this example: - Initially, nodes 1, 4, and 5 are leaves. - By performing one transformation on node 3, nodes 4 and 5 become leaves with their subtree values remaining the same. - By performing a second transformation on node 3, the maximum sum of Magic Leaves is increased.

answer:def min_transformations(n, labels, edges): from collections import defaultdict, deque if n == 1: return 0 adj = defaultdict(list) for u, v in edges: adj[u].append(v) adj[v].append(u) degrees = [0] * (n + 1) for i in range(1, n + 1): degrees[i] = len(adj[i]) leaves = deque() for i in range(1, n + 1): if degrees[i] == 1: leaves.append(i) transformations = 0 while len(leaves) > 1: new_leaves = deque() while leaves: leaf = leaves.popleft() for neighbor in adj[leaf]: degrees[neighbor] -= 1 if degrees[neighbor] == 1: new_leaves.append(neighbor) leaves = new_leaves transformations += 1 return transformations # Example usage n = 5 labels = [1, 5, 3, 7, 4] edges = [(1, 2), (1, 3), (3, 4), (3, 5)] print(min_transformations(n, labels, edges)) # Output: 2

question:You are given a binary tree where each node contains an integer value, and your task is to find the subtree with the maximum sum. A subtree of a tree consists of a node and all its descendants. The sum of a subtree is the sum of all the node values in that subtree. Write a function that takes the root of the binary tree as input and returns the sum of the subtree with the maximum sum. The function signature is: ```python def max_subtree_sum(root: Optional[TreeNode]) -> int: ``` # Example Consider the binary tree depicted below: ``` 1 / 2 3 / / 4 5 6 7 ``` For this tree, the sum of each subtree is as follows: - Subtree rooted at 4: 4 - Subtree rooted at 5: 5 - Subtree rooted at 6: 6 - Subtree rooted at 7: 7 - Subtree rooted at 2 (including 4 and 5): 11 - Subtree rooted at 3 (including 6 and 7): 16 - Subtree rooted at 1 (including entire tree): 28 In this example, the subtree with the maximum sum is the entire tree itself, with a sum of 28. Therefore, the function will return 28. # Constraints - The number of nodes in the tree is between 1 and 10^5. - The value of each node is between -10^4 and 10^4. # Notes - You may define your own `TreeNode` class if needed. - Your solution should be efficient and handle large inputs within a reasonable time frame. - You may assume the input tree is well-formed and does not contain cycles.

answer:from typing import Optional class TreeNode: def __init__(self, val: int = 0, left: Optional['TreeNode'] = None, right: Optional['TreeNode'] = None): self.val = val self.left = left self.right = right def max_subtree_sum(root: Optional[TreeNode]) -> int: Finds the subtree with the maximum sum in a binary tree. max_sum = float('-inf') def subtree_sum(node: Optional[TreeNode]) -> int: nonlocal max_sum if not node: return 0 left_sum = subtree_sum(node.left) right_sum = subtree_sum(node.right) total_sum = node.val + left_sum + right_sum max_sum = max(max_sum, total_sum) return total_sum subtree_sum(root) return max_sum

question:You are given an array of integers, representing the heights of buildings on a street. The array is of length n and the i-th element represents the height of the i-th building. You are allowed to increase the height of any building by 1 unit in one operation. The goal is to make all buildings have the same height with the minimum number of operations. The first line contains one integer n (1 le n le 10^5) — the number of buildings. The second line contains n integers h_1, h_2, dots, h_n (1 le h_i le 10^9) — the heights of the buildings. Output the minimum number of operations needed to make all buildings have the same height. In the first example, we have the building heights [1, 2, 3] and can: 1. Increase the height of the first building to 3 (2 operations). 2. Increase the height of the second building to 3 (1 operation). In the second example, we can: 1. Increase the height of the first building to 5 (2 operations). 2. Increase the height of the second building to 5 (1 operation). 3. Increase the height of the third building to 5 (1 operation). Note that the heights cannot be reduced, only increased.

answer:def min_operations_to_equal_heights(n, heights): Returns the minimum number of operations needed to make all buildings have the same height. Parameters: n (int): Number of buildings. heights (list): List of integers representing the heights of the buildings. Returns: int: Minimum number of operations. max_height = max(heights) return sum(max_height - height for height in heights)