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This approach involves using a BFS algorithm to find the shortest path between the starting point and each tree in increasing order of their heights. We first list all trees with their positions and heights, sort them, and for each sorted tree, compute the shortest path using BFS. If any tree is unreachable, we return -1.
The time complexity of this approach is O(T * m * n) where T is the number of trees. This is because we perform BFS from the starting position to each tree. The space complexity is O(m * n) due to the storage of the queue and visited matrix in BFS.
1from typing import List
2from collections import deque
3
4class Solution:
5 def cutOffTree(self, forest: List[List[int]]) -> int:
6 def bfs(sr, sc, tr, tc):
7 if sr == tr and sc == tc:
8 return 0
9 queue = deque([(sr, sc, 0)])
10 visited = set((sr, sc))
11 while queue:
12 r, c, steps = queue.popleft()
13 for dr, dc in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
14 nr, nc = r + dr, c + dc
15 if 0 <= nr < len(forest) and 0 <= nc < len(forest[0]) and (nr, nc) not in visited and forest[nr][nc] > 0:
16 if nr == tr and nc == tc:
17 return steps + 1
18 queue.append((nr, nc, steps + 1))
19 visited.add((nr, nc))
20 return -1
21
22 trees = sorted((height, r, c)
23 for r, row in enumerate(forest)
24 for c, height in enumerate(row)
25 if height > 1)
26
27 steps = 0
28 sr = sc = 0
29 for _, tr, tc in trees:
30 s = bfs(sr, sc, tr, tc)
31 if s == -1:
32 return -1
33 steps += s
34 sr, sc = tr, tc
35 return stepsThe Python solution uses BFS to explore paths in the forest grid to reach trees by height order. It lists all trees with their respective heights and sorts them before computing the shortest path between consecutively taller trees using BFS. If a tree is unreachable, it returns -1.
Using Bidirectional BFS can help in optimizing the pathfinding step by searching from both the start and the goal concurrently. The aim is to decrease the search space substantially as the two searches meet in the middle. This method is an extension to the regular BFS approach, potentially optimizing the traversal cost considerably under specific configurations.
The potential benefits of bidirectional search in space complexity can be realized when effectively reducing the number of explored nodes, as time complexity ideally shortens from O(b^d) to roughly O(b^(d/2)), where b is the branching factor and d the solution depth.
1from typing import List, Tuple
2
Python realizes the bidirectional search using dual BFS queues for start and target, merging states reached from both ends. It prioritizes advancing the search queue with fewer elements, thereby optimizing space and efficiency. The express serialization of nodes helps manage state transitions seamlessly.