1778. Shortest Path in a Hidden Grid - Detailed Explanation

Problem Statement

In this problem, you are placed in a hidden grid with unknown dimensions. You do not know the grid layout (where obstacles or open cells are), but you have access to an interface (called GridMaster) that lets you interact with the grid. The interface provides the following methods:

• canMove(direction):
  Returns true if you can move in the given direction ('U' for up, 'D' for down, 'L' for left, and 'R' for right) from your current cell; otherwise, it returns false.

• move(direction):
  Moves you one cell in the specified direction. You may assume that this operation will only be called if canMove(direction) is true.

• isTarget():
  Returns true if your current cell is the target cell; otherwise, it returns false.

You start at the cell you consider as coordinate (0, 0). Your goal is to determine the minimum number of steps required to reach the target cell. If the target is unreachable, return -1.

2. Example

Imagine the hidden grid is as follows (though you do not see this layout):

   1   1   1
   0   1   0
   T   1   1

• Here, 1 denotes accessible cells and 0 denotes obstacles.
• The target cell (T) is at position (2, 0) relative to the starting cell at (0, 0).
• One shortest path might be:
  (0,0) → (0,1) → (1,1) → (2,1) → (2,0), which takes 4 steps.
• The expected output would be 4.

If no path exists from (0, 0) to the target, the answer should be -1.

Hints for the Approach

• Hint 1:
  Since you cannot see the entire grid at once, explore the grid using Depth-First Search (DFS) with backtracking. As you explore, record each accessible cell (using coordinates relative to (0, 0)) in an internal map (or dictionary).

• Hint 2:
  During DFS, use the GridMaster methods to check moves, move into accessible cells, and then backtrack (i.e. move back to the previous cell) after exploring a branch. This ensures you explore every reachable cell.

• Hint 3:
  Once you have mapped all accessible cells and identified the target cell (if it exists), run a Breadth-First Search (BFS) on your discovered grid from the starting cell to compute the shortest path (i.e. the minimum number of steps) to the target.

Approaches

Approach Overview

1. DFS Exploration (Mapping the Grid):

   • Objective: Explore every accessible cell from the starting point (0, 0) using DFS.
   • Actions:
     • For each cell, record its relative coordinates and whether it is the target.
     • Try moving in each of the four directions. If a move is possible (i.e. canMove(direction) returns true), move into that cell, record it, and then recursively explore further.
     • Backtracking: After exploring a branch, move back (using the reverse direction) to ensure you continue exploring other paths.
   • Result: A map (or dictionary) of all accessible cells (with their coordinates) and the identification of the target cell’s coordinates (if found).

2. BFS for Shortest Path:

   • Objective: Use BFS on the discovered grid (from DFS) to compute the shortest path from (0, 0) to the target cell.
   • Actions:
     • Start BFS at (0, 0) and explore neighbors (up, down, left, right) that exist in your discovered grid.
     • Track the number of steps; once you reach the target cell, return the step count.
     • If BFS finishes without reaching the target, return -1.

Detailed Step-by-Step Walkthrough

Step 1: DFS to Discover the Grid

• Start at (0, 0):

  • Mark the starting cell as visited.
  • Check isTarget()—if true, record that (0, 0) is the target.

• Explore Neighbors:

  • For each direction ('U', 'D', 'L', 'R'), check if you can move (using canMove(direction)).
  • If you can move, update your coordinates accordingly (e.g., moving 'U' decreases the x-coordinate by 1).
  • Move into the new cell using move(direction) and call DFS recursively from the new cell.
  • After exploring from that cell, backtrack by moving in the opposite direction (for example, if you moved 'U', move 'D' to return).

• Record Data:

  • Use a dictionary (or map) to store each accessible cell’s coordinates (e.g., (x, y)) along with a flag indicating if it is the target.

Step 2: BFS to Find the Shortest Path

• Initialize BFS:
  • Start from (0, 0) (the starting position) with an initial step count of 0.
  • Use a queue to perform BFS and a set to mark visited cells (based on your discovered grid).
• BFS Process:
  • At each step, dequeue a cell, check if it is the target.
  • If not, enqueue all its neighboring cells (neighbors that exist in the discovered grid).
  • Increase the step count for each level of BFS.
• Finish:
  • If you reach the target cell, return the number of steps.
  • If BFS completes without reaching the target, return -1.

Pseudocode

DFS(x, y):
    mark (x, y) as visited in gridMap
    record gridMap[(x, y)] = gridMaster.isTarget()
    for each direction in {U, D, L, R}:
         if (x+dx, y+dy) not visited and gridMaster.canMove(direction):
             gridMaster.move(direction)
             DFS(x+dx, y+dy)
             gridMaster.move(opposite_direction)

BFS():
    Initialize queue with (0, 0, steps=0)
    while queue not empty:
         pop (x, y, steps)
         if (x, y) is target:
             return steps
         for each neighbor (x+dx, y+dy) in gridMap:
             if neighbor not visited in BFS:
                 add neighbor with steps+1
    return -1

Main:
    DFS(0, 0) using GridMaster interface
    if target not found in gridMap: return -1
    return BFS()

Complexity Analysis

• Time Complexity:

  • DFS Exploration: Each accessible cell is visited once. Let (N) be the number of accessible cells; time is (O(N)).

  • BFS Shortest Path: Again, each accessible cell is processed at most once, so (O(N)).

  • Overall: (O(N)) (plus constant factors for checking directions).

• Space Complexity:

  • The map storing accessible cells uses (O(N)) space.

  • The recursion stack in DFS may use (O(N)) in the worst case.

  • The BFS queue uses (O(N)) space.

  • Overall: (O(N)).

Code Implementation

Python Implementation