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Cut Off Trees for Golf Event

Difficulty: Hard

Problem Description

You are asked to cut off all the trees in a forest for a golf event. The forest is represented as an m x n matrix. In this matrix, 0 means the cell cannot be walked through, 1 represents an empty cell that can be walked through, and a number greater than 1 represents a tree in a cell that can be walked through, with the number indicating the tree's height. You must cut off the trees in order from shortest to tallest, starting from the point (0, 0). Return the minimum steps needed to walk to cut off all the trees. If you cannot cut off all the trees, return -1.

Key Insights

  • Trees need to be cut in order of height, from shortest to tallest.
  • The movement is restricted to adjacent cells in four possible directions: north, east, south, and west.
  • The value of a cell changes to 1 after cutting a tree, allowing for subsequent navigation.
  • A breadth-first search (BFS) or Dijkstra's algorithm can be employed to find the shortest path to each tree.

Space and Time Complexity

Time Complexity: O(T * (m * n)), where T is the number of trees, as each BFS can potentially explore the entire grid. Space Complexity: O(m * n), for the queue used in BFS and the visited set.

Solution

To solve the problem, we can use a breadth-first search (BFS) algorithm to traverse the forest and find the shortest path to each tree. The algorithm involves the following steps:

  1. Extract all tree positions along with their heights from the matrix and sort them based on height.
  2. Initialize a BFS for each tree starting from the current position.
  3. Traverse the forest to find the shortest path to the next tree.
  4. Keep track of the total steps taken and update the current position after cutting each tree.
  5. If any tree cannot be reached, return -1.

This approach ensures that we visit each tree in the correct order and calculate the minimum steps required.

Code Solutions

from collections import deque

def cutOffTree(forest):
    def bfs(start, end):
        queue = deque([start])
        visited = set([start])
        steps = 0
        
        while queue:
            for _ in range(len(queue)):
                x, y = queue.popleft()
                if (x, y) == end:
                    return steps
                for dx, dy in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
                    nx, ny = x + dx, y + dy
                    if 0 <= nx < len(forest) and 0 <= ny < len(forest[0]) and (nx, ny) not in visited and forest[nx][ny] != 0:
                        visited.add((nx, ny))
                        queue.append((nx, ny))
            steps += 1
        return -1

    trees = [(i, j, forest[i][j]) for i in range(len(forest)) for j in range(len(forest[0])) if forest[i][j] > 1]
    trees.sort(key=lambda x: x[2])  # Sort trees by height

    total_steps = 0
    start = (0, 0)

    for x, y, _ in trees:
        steps = bfs(start, (x, y))
        if steps == -1:
            return -1
        total_steps += steps
        start = (x, y)  # Update start to the position of the cut tree
    
    return total_steps
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