A storekeeper is a game in which the player pushes boxes around in a warehouse trying to get them to target locations.
The game is represented by an m x n grid of characters grid where each element is a wall, floor, or box.
Your task is to move the box 'B' to the target position 'T' under the following rules:
'S' represents the player. The player can move up, down, left, right in grid if it is a floor (empty cell).'.' represents the floor which means a free cell to walk.'#' represents the wall which means an obstacle (impossible to walk there).'B' and one target cell 'T' in the grid.Return the minimum number of pushes to move the box to the target. If there is no way to reach the target, return -1.
Example 1:
Input: grid = [["#","#","#","#","#","#"],
["#","T","#","#","#","#"],
["#",".",".","B",".","#"],
["#",".","#","#",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: 3
Explanation: We return only the number of times the box is pushed.
Example 2:
Input: grid = [["#","#","#","#","#","#"],
["#","T","#","#","#","#"],
["#",".",".","B",".","#"],
["#","#","#","#",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: -1
Example 3:
Input: grid = [["#","#","#","#","#","#"],
["#","T",".",".","#","#"],
["#",".","#","B",".","#"],
["#",".",".",".",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: 5
Explanation: push the box down, left, left, up and up.
Constraints:
m == grid.lengthn == grid[i].length1 <= m, n <= 20grid contains only characters '.', '#', 'S', 'T', or 'B'.'S', 'B', and 'T' in the grid.This approach uses a BFS strategy, commonly used for exploring shortest paths, which also tracks states consisting of the player's and the box’s positions along with the number of pushes made. Each state can be represented as a tuple consisting of these values. The BFS runs until the box reaches its target position or all possible states are exhausted. To ensure efficiency, already visited states are stored and checked before exploration to prevent cyclic paths.
This Python solution uses a BFS approach to explore all possible ways the player can push the box to the target. It tracks the player's and the box's current positions and checks all possible moves. The BFS ensures the shortest path is found by exploring in layers. The main components include state validation, tracking visited states, and maintaining a queue for the BFS exploration.
Java
Time Complexity: O(m * n) due to traversing each cell, where m and n are the grid dimensions.
Space Complexity: O(m * n) for the space required to store states in the queue and visited set.
This approach optimizes the BFS approach even further by using the A* search algorithm, which is a popular choice for pathfinding and graph traversal. In A*, each state is evaluated with a heuristic to estimate the minimum cost to reach the target. The heuristic can be the Manhattan distance from the box to the target which guides the search more directly towards optimal solutions more efficiently.
This C++ solution uses the A* algorithm which is advantageous over BFS by utilizing a heuristic approach. The heuristic guides the exploration to significantly reduce unnecessary steps by taking into consideration the Manhattan distance to the target. The state of the game is abstracted into a struct and processed through a priority queue, ensuring that the paths with the least cost are expanded first.
JavaScript
Time Complexity: O((m * n)^2) in the worst case because of priority queue operations and heuristic calculations.
Space Complexity: O(m * n) for the storage of states.
| Approach | Complexity |
|---|---|
| Breadth-First Search (BFS) with State Tracking | Time Complexity: O(m * n) due to traversing each cell, where m and n are the grid dimensions. |
| A* Search for Optimal Pushes | Time Complexity: O((m * n)^2) in the worst case because of priority queue operations and heuristic calculations. |
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