Problem Description

This problem presents a scenario where you are in control of a robot placed in an unknown position within a room. The room is set up as a grid with "1" representing an empty space that the robot can move through, and "0" representing a wall that the robot cannot pass. Your task is to create an algorithm that instructs the robot to clean the entire room, covering every "1" cell in the grid.

The robot is equipped with a set of APIs that allow it to:

1. Move forward into an adjacent cell (move method), unless there is a wall which will cause the robot to stay put.
2. Turn left or right by 90 degrees without moving from its current cell (turnLeft and turnRight methods).
3. Clean the cell it is currently on (clean method).

One crucial detail to note is that the initial orientation of the robot is facing "up," and all four edges of the grid are walled off. Furthermore, you must write this algorithm without any knowledge of the room's layout or the robot's starting position.

Flowchart Walkthrough

First, let’s analyze the problem using the Flowchart. Here's a step-by-step walkthrough:

1. Is it a graph?

   • Yes: The problem can be modeled as a graph where each cell in the room is a node and the possible movements to adjacent cells represent the edges.

2. Need to solve for kth smallest/largest?

   • No: The task is not about finding the kth smallest or largest element.

3. Involves Linked Lists?

   • No: The problem does not involve manipulating or navigating through linked lists.

4. Does the problem have small constraints?

   • Yes: Although the size of the room can be large, the complexity of the problem and the nature of the robot's movement stipulate small effective constraints for the immediate choices of movement at any given step.

5. Brute force / Backtracking?

   • Yes: The nature of the problem, needing to explore potentially all cells and cleaning them without prior knowledge of which cells are clean or blocked, points to the suitability of a brute-force approach with backtracking. We need to recursively explore each direction, backtrack when a path is blocked, and ensure all possible cells are visited.

Conclusion: The flowchart suggests using a backtracking approach to solve the problem, exploring all directions and managing dead ends effectively.

Intuition

The solution to this problem involves a depth-first search (DFS) strategy. The idea behind DFS is to exhaustively explore a path until hitting an obstacle (in this case, a wall), backtrack, and then try another path. This principle is well-suited for the problem since we need to ensure every navigable cell is visited and cleaned.

Since the robot's starting point is unknown, and the room's layout is also unknown, we can imagine the robot's initial position as the origin of a coordinate system, (0, 0).

For the DFS to work, we need to:

• Keep track of cells that have already been visited to avoid redundant cleaning and to prevent the robot from going in cycles.
• Identify a way to represent the robot's orientation since it can face four different directions (up, right, down, and left). We use a direction vector (dirs) for this purpose.

The dfs function in the solution recursively explores the grid by trying to move in the current facing direction, cleaning the cell, and then attempting to move forward in the next direction. If a move is successful (indicating an empty cell), the robot moves forward, and the search continues from the new cell position. After exploring one direction, the robot turns 90 degrees to the right to check the next direction.

Once the robot hits a wall and cannot move forward:

• It backtracks to the previous cell by performing a 180-degree turn (turnRight twice), moving forward one cell, and then realigning to the original orientation by another 180-degree turn.
• It then continues the DFS from the previous cell in the next direction.

This process continues until all reachable cells have been visited and cleaned.

Learn more about Backtracking patterns.

Solution Approach

The solution uses depth-first search (DFS) to navigate and clean the room. DFS is a popular algorithm for exploring all nodes in a graph or all vertices in a grid by moving as far as possible until you can no longer proceed, then backtracking.

Here's how the algorithm in the provided solution works:

1. Data Structure for Visited Cells: A set named vis is used to keep track of visited coordinates. This prevents the robot from cleaning the same cell multiple times and ensures that the algorithm terminates.

2. Directional Handling: The dirs tuple provides the relative coordinates when moving up, right, down, and left respectively, assuming an orientation of facing up initially. These directions correspond to the changes in the robot's coordinates given its current direction.

3. Recursive DFS Function: The heart of the solution is the dfs(i, j, d) function, which takes the robot's current x and y coordinates (i and j), and its current direction (d).

   • It starts by cleaning the current cell and adding the cell to the vis (visited set).
   • The robot then attempts to move in its current direction and three more directions by turning right sequentially.
   • After attempting to move in each direction:
     • If the move is successful, the robot has discovered a new cell, and dfs is called recursively from that new position.
     • After the recursive call (which represents backtracking to the source cell), the robot performs two 90-degree turns to face the opposite direction (robot.turnRight called twice), moves forward to return to the original cell, and then realigns to the initial direction by turning another 180 degrees in total.
   • It then turns right to change to the next direction and continue the DFS.

4. Triggering DFS: The DFS begins by calling dfs(0, 0, 0) from the robot's initial position, which is treated as the origin (0, 0) and its initial direction (up).

Throughout this process, the robot continues to move, turn, and clean until it has reached and cleaned all accessible cells. The vis set ensures that the robot never revisits a cell, efficiently covering the entire grid.

By using these approaches, the robot is able to clean the entire room—represented by the 1 cells in the grid—without any prior knowledge of the room's layout or its own initial position.

Example Walkthrough

Imagine a room represented by the following 3x3 grid and the robot starts at the cell marked R (which we consider (0, 0) for our coordinate system), facing upwards. 1 indicates an empty space, and 0 indicates a wall:

Room:
1 1 1
0 1 0
R 1 1

Step-by-step using the DFS approach:

1. Initialization: The robot starts at (0, 0). The cell is automatically cleaned and is added to the visited set vis.

2. Exploration:

   • The robot tries to move forward (up) into (0, 1). Since it's free, the cell is cleaned, and this position is added to vis. The function dfs(0, 1, 0) is called.
   • From (0,1), the robot continues and moves up to (0,2), which is also free. The robot cleans it and then attempts to go right, but there is a wall, so it turns right and tries to go down but hits another wall. It turns right again, and now it moves left, cleans (0,1), and returns to (0,2) and faces down (original direction).
   • The sequence repeats until it faces upward again, and since all directions are visited, it returns to (0,1). The robot tries to go right, and it's blocked, so it tries to go down and arrives at (0,0), which is already cleaned and in vis.
   • Now the robot tries to move right from (0,0). It can't because there is a wall, so it rotates right and moves down to (1,0). The robot cleans the cell, then tries to go left (hit wall), up (cleaned and in vis), right, and finally down. Since down is the only direction it hasn't come from, it moves to (2,0) and cleans it.
   • At (2,0), the robot repeats the process: trying each direction, cleaning new cells, and moving accordingly. It soon finds it can move to (2,1) and cleans it.
   • At (2,1), the robot cannot move right or down because of walls; it moves up and finds it's already visited (1,1). After rotating right, it can move left to (2,0) but since it's been visited, the robot rotates right again and finally faces the last unexplored direction at (2,1) which is the cell on its left (2,2).
   • At (2,2), which is the last cell, the robot can't move up or right due to walls, and the left and down directions lead to already cleaned cells. It has now finished cleaning.

By the end of this process, the robot has covered the entire accessible area, and each 1 has been visited and turned into a 2 (indicating the robot has cleaned it).

Resulting visited cells:

2 2 2
0 2 0
2 2 2

This example shows how the robot effectively uses the DFS method to clean the entire room. Each step is guided by whether the space has been visited or if there's an obstacle, and the visited set vis ensures that no space gets cleaned twice. The right turning mechanism helps the robot to systematically explore all four directions in its current location.

Solution Implementation

Time and Space Complexity

The time complexity of the above code is O(4^(N-M)), where N is the total number of cells in the room and M is the number of obstacles. This is because the algorithm has to visit each non-obstacle cell once and at each cell, it makes up to 4 decisions – move in 4 possible directions. The recursion may go up to 4 branches at each level but would not revisit cells that are already visited, summarized by visited set vis.

The space complexity of the DFS is O(N) for the recursive call stack as well as the space to hold the set of visited cells (in the worst case where there are no obstacles and we can move to every cell). However, in a densely packed room with obstructions, the number of visited states will be less than N.

Learn more about how to find time and space complexity quickly using problem constraints.