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.1
represents a tree in a cell that can be walked through, and this number is the tree's height.In one step, you can walk in any of the four directions: north, east, south, and west. If you are standing in a cell with a tree, you can choose whether to cut it off.
You must cut off the trees in order from shortest to tallest. When you cut off a tree, the value at its cell becomes 1
(an empty cell).
Starting from the point (0, 0)
, return the minimum steps you need to walk to cut off all the trees. If you cannot cut off all the trees, return -1
.
Note: The input is generated such that no two trees have the same height, and there is at least one tree needs to be cut off.
Example 1:
Input: forest = [[1,2,3],[0,0,4],[7,6,5]] Output: 6 Explanation: Following the path above allows you to cut off the trees from shortest to tallest in 6 steps.
Example 2:
Input: forest = [[1,2,3],[0,0,0],[7,6,5]] Output: -1 Explanation: The trees in the bottom row cannot be accessed as the middle row is blocked.
Example 3:
Input: forest = [[2,3,4],[0,0,5],[8,7,6]] Output: 6 Explanation: You can follow the same path as Example 1 to cut off all the trees. Note that you can cut off the first tree at (0, 0) before making any steps.
Constraints:
m == forest.length
n == forest[i].length
1 <= m, n <= 50
0 <= forest[i][j] <= 109
When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:
The brute force approach in this problem involves trying every single path to cut down all the trees. We'll explore all possible orders of visiting the trees and calculate the total distance for each order.
Here's how the algorithm would work step-by-step:
def cut_off_trees_for_golf_event_brute_force(forest):
number_of_rows = len(forest)
number_of_columns = len(forest[0])
trees_to_cut = []
for row_index in range(number_of_rows):
for column_index in range(number_of_columns):
tree_height = forest[row_index][column_index]
if tree_height > 1:
trees_to_cut.append((tree_height, row_index, column_index))
trees_to_cut.sort()
def calculate_manhattan_distance(start_row, start_column, end_row, end_column):
return abs(start_row - end_row) + abs(start_column - end_column)
import itertools
shortest_distance = float('inf')
# Try all permutations of tree cutting orders.
for permutation in itertools.permutations(trees_to_cut):
current_row = 0
current_column = 0
total_distance_for_permutation = 0
# Sum up distance between each tree.
for tree_height, next_row, next_column in permutation:
distance = calculate_manhattan_distance(current_row, current_column, next_row, next_column)
total_distance_for_permutation += distance
current_row = next_row
current_column = next_column
# Keep track of the smallest total distance seen
shortest_distance = min(shortest_distance, total_distance_for_permutation)
if shortest_distance == float('inf'):
return -1
else:
return shortest_distance
The key to efficiently cutting trees for the golf event is to visit them in increasing order of height, as smaller trees should always be cut before taller ones. We use a shortest path finding algorithm, like navigating a maze, to move from one tree to the next, always aiming for the closest uncut tree.
Here's how the algorithm would work step-by-step:
from collections import deque
def cut_off_trees(forest):
tree_locations = []
rows = len(forest)
cols = len(forest[0])
for row in range(rows):
for col in range(cols):
height = forest[row][col]
if height > 1:
tree_locations.append((height, row, col))
tree_locations.sort()
start_row, start_col = 0, 0
total_steps = 0
for height, target_row, target_col in tree_locations:
steps = find_shortest_path(forest, start_row, start_col, target_row, target_col)
if steps == -1:
return -1
total_steps += steps
start_row, start_col = target_row, target_col
forest[target_row][target_col] = 1
return total_steps
def find_shortest_path(forest, start_row, start_col, target_row, target_col):
rows = len(forest)
cols = len(forest[0])
queue = deque([(start_row, start_col, 0)])
visited = set()
visited.add((start_row, start_col))
# Define possible movement directions: up, down, left, right
directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]
while queue:
current_row, current_col, steps = queue.popleft()
if current_row == target_row and current_col == target_col:
return steps
for delta_row, delta_col in directions:
new_row = current_row + delta_row
new_col = current_col + delta_col
# Check boundary conditions and if cell is traversable
if 0 <= new_row < rows and 0 <= new_col < cols and \
forest[new_row][new_col] > 0 and (new_row, new_col) not in visited:
queue.append((new_row, new_col, steps + 1))
visited.add((new_row, new_col))
# If no path is found return -1
return -1
Case | How to Handle |
---|---|
Null or empty forest input | Return -1 immediately as no path can be calculated. |
Forest with only one cell | If the single cell has a tree (value > 1), return 0; otherwise, return -1 because all trees must be cut. |
No trees to cut (all values are 0 or 1) | If no tree exists (value > 1), return 0 if the starting position is already at a tree, else -1. |
Trees are not in strictly increasing order of height | Sort the trees by height before processing. |
No path exists between two trees | Return -1 if the BFS search for a specific tree returns no valid path. |
Large forest dimensions leading to potential memory issues with BFS | Implement BFS using an iterative approach with a queue to avoid stack overflow and optimize memory usage. |
Integer overflow when calculating distance or accumulated steps | Use long data type for distance calculations to prevent overflow. |
Forest containing a tree at the starting location (0, 0) | The algorithm should correctly handle starting at a tree and consider it cut. |