Walking Robot Simulation

Medium

Jane Street

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Topics:

Arrays

A robot on an infinite XY-plane starts at point (0, 0) facing north. The robot receives an array of integers commands, which represents a sequence of moves that it needs to execute. There are only three possible types of instructions the robot can receive:

-2: Turn left 90 degrees.
-1: Turn right 90 degrees.
1 <= k <= 9: Move forward k units, one unit at a time.

Some of the grid squares are obstacles. The i^th obstacle is at grid point obstacles[i] = (x_i, y_i). If the robot runs into an obstacle, it will stay in its current location (on the block adjacent to the obstacle) and move onto the next command.

Return the maximum squared Euclidean distance that the robot reaches at any point in its path (i.e. if the distance is 5, return 25).

Note:

There can be an obstacle at (0, 0). If this happens, the robot will ignore the obstacle until it has moved off the origin. However, it will be unable to return to (0, 0) due to the obstacle.
North means +Y direction.
East means +X direction.
South means -Y direction.
West means -X direction.

Example 1:

Input: commands = [4,-1,3], obstacles = []

Output: 25

Explanation:

The robot starts at (0, 0):

Move north 4 units to (0, 4).
Turn right.
Move east 3 units to (3, 4).

The furthest point the robot ever gets from the origin is (3, 4), which squared is 3² + 4²= 25 units away.

Example 2:

Input: commands = [4,-1,4,-2,4], obstacles = [[2,4]]

Output: 65

Explanation:

The robot starts at (0, 0):

Move north 4 units to (0, 4).
Turn right.
Move east 1 unit and get blocked by the obstacle at (2, 4), robot is at (1, 4).
Turn left.
Move north 4 units to (1, 8).

The furthest point the robot ever gets from the origin is (1, 8), which squared is 1² + 8² = 65 units away.

Example 3:

Input: commands = [6,-1,-1,6], obstacles = [[0,0]]

Output: 36

Explanation:

The robot starts at (0, 0):

Move north 6 units to (0, 6).
Turn right.
Turn right.
Move south 5 units and get blocked by the obstacle at (0,0), robot is at (0, 1).

The furthest point the robot ever gets from the origin is (0, 6), which squared is 6² = 36 units away.

Constraints:

1 <= commands.length <= 10⁴
commands[i] is either -2, -1, or an integer in the range [1, 9].
0 <= obstacles.length <= 10⁴
-3 * 10⁴ <= x_i, y_i <= 3 * 10⁴
The answer is guaranteed to be less than 2³¹.

Try coding it in LeetCode

Walking Robot Simulation

Medium

Jane Street

5 views

Topics:

Arrays

A robot on an infinite XY-plane starts at point (0, 0) facing north. The robot receives an array of integers commands, which represents a sequence of moves that it needs to execute. There are only three possible types of instructions the robot can receive:

-2: Turn left 90 degrees.
-1: Turn right 90 degrees.
1 <= k <= 9: Move forward k units, one unit at a time.

Some of the grid squares are obstacles. The i^th obstacle is at grid point obstacles[i] = (x_i, y_i). If the robot runs into an obstacle, it will stay in its current location (on the block adjacent to the obstacle) and move onto the next command.

Return the maximum squared Euclidean distance that the robot reaches at any point in its path (i.e. if the distance is 5, return 25).

Note:

There can be an obstacle at (0, 0). If this happens, the robot will ignore the obstacle until it has moved off the origin. However, it will be unable to return to (0, 0) due to the obstacle.
North means +Y direction.
East means +X direction.
South means -Y direction.
West means -X direction.

Example 1:

Input: commands = [4,-1,3], obstacles = []

Output: 25

Explanation:

The robot starts at (0, 0):

Move north 4 units to (0, 4).
Turn right.
Move east 3 units to (3, 4).

The furthest point the robot ever gets from the origin is (3, 4), which squared is 3² + 4²= 25 units away.

Example 2:

Input: commands = [4,-1,4,-2,4], obstacles = [[2,4]]

Output: 65

Explanation:

The robot starts at (0, 0):

Move north 4 units to (0, 4).
Turn right.
Move east 1 unit and get blocked by the obstacle at (2, 4), robot is at (1, 4).
Turn left.
Move north 4 units to (1, 8).

The furthest point the robot ever gets from the origin is (1, 8), which squared is 1² + 8² = 65 units away.

Example 3:

Input: commands = [6,-1,-1,6], obstacles = [[0,0]]

Output: 36

Explanation:

The robot starts at (0, 0):

Move north 6 units to (0, 6).
Turn right.
Turn right.
Move south 5 units and get blocked by the obstacle at (0,0), robot is at (0, 1).

The furthest point the robot ever gets from the origin is (0, 6), which squared is 6² = 36 units away.

Constraints:

1 <= commands.length <= 10⁴
commands[i] is either -2, -1, or an integer in the range [1, 9].
0 <= obstacles.length <= 10⁴
-3 * 10⁴ <= x_i, y_i <= 3 * 10⁴
The answer is guaranteed to be less than 2³¹.

Try coding it in LeetCode

Solution

Clarifying Questions

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:

What are the constraints on the magnitude and type of the movement commands (e.g., integers, positive/negative, range)?
How are obstacles represented, and what are the constraints on their coordinates (e.g., integer coordinates, range)?
What should the robot do if a movement command would cause it to collide with an obstacle?
What is the initial position and direction of the robot? If not specified, can I assume a default?
What is the expected output format? For example, should I return the final coordinates as a tuple/array/string?

Brute Force Solution

Approach

The robot starts at a point and tries to move according to a series of commands. A brute force approach means we directly simulate every single movement, step-by-step, following the commands and checking for obstacles at each position.

Here's how the algorithm would work step-by-step:

Start the robot at its initial position, facing a specific direction.
For each command, consider what it tells the robot to do: either turn or move forward.
If the command is to turn, simply change the robot's facing direction.
If the command is to move forward a certain number of steps, repeatedly move the robot one step at a time in its current facing direction.
Before each step, check if the intended next position is blocked by an obstacle.
If there's an obstacle, don't move forward and proceed to the next command.
If there's no obstacle, move the robot to the new position.
After completing all commands, calculate the distance from the robot's final position to the starting point.

Code Implementation

def walking_robot_simulation_brute_force(commands, obstacles):
    x_coordinate = 0
    y_coordinate = 0
    facing_direction = 0  # 0: North, 1: East, 2: South, 3: West
    directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]

    for command in commands:
        if command == -1:
            # Turn right
            facing_direction = (facing_direction + 1) % 4

        elif command == -2:
            # Turn left
            facing_direction = (facing_direction - 1) % 4

        else:
            # Move forward
            for _ in range(command):
                next_x = x_coordinate + directions[facing_direction][0]
                next_y = y_coordinate + directions[facing_direction][1]

                # Check if the next position is an obstacle
                if (next_x, next_y) in set(map(tuple, obstacles)):
                    break

                # Update the position
                x_coordinate = next_x
                y_coordinate = next_y

    # Calculate the squared Euclidean distance from the origin.
    distance = x_coordinate**2 + y_coordinate**2
    return distance

Big(O) Analysis

Time Complexity

O(n*k) – Let n be the number of commands in the input array and k be the maximum value of any forward movement command. For each of the n commands, if the command is a turn, it takes O(1) time. However, if the command is a move forward, the robot moves at most k steps, and in each step, it checks for obstacles. Therefore, in the worst case, it iterates through all commands and potentially moves the maximum number of steps k for each move command. Hence the time complexity is O(n*k).

Space Complexity

O(O) – The algorithm's space complexity is primarily determined by the need to store the obstacle locations to efficiently check for collisions. The plain English explanation necessitates checking if the robot's intended next position is blocked by an obstacle. This likely implies storing the obstacles' coordinates in a data structure such as a set or hash map for quick lookup. If we assume there are O obstacles, the space complexity is O(O), where O is the number of obstacles.

Optimal Solution

Approach

The robot walks around a grid, following commands and avoiding obstacles. We need to keep track of its position and direction as it moves, while also efficiently checking if a potential move would lead to a collision with an obstacle.

Here's how the algorithm would work step-by-step:

Imagine the robot starts at a specific spot, facing a certain direction (like North, South, East, or West).
Read each command one by one. The commands can either tell the robot to turn or to move forward.
If the command is to turn, simply update the direction the robot is facing.
If the command is to move forward, check each step of the move to see if there's an obstacle in the way.
Before moving one step, check if the new location contains an obstacle. If it does, stop moving and continue with the next command.
If there's no obstacle, move the robot one step forward and repeat the obstacle check until the robot has moved the specified number of steps or encounters an obstacle.
After processing all the commands, the robot's final location is the answer.

Code Implementation

def walking_robot_simulation(commands, obstacles):
    x_coordinate = 0
    y_coordinate = 0
    direction = 0  # 0: North, 1: East, 2: South, 3: West
    max_distance_squared = 0

    # Store obstacles as a set of tuples for efficient lookup
    obstacle_set = set((x, y) for x, y in obstacles)

    # Define movement offsets for each direction
    direction_offsets = [(0, 1), (1, 0), (0, -1), (-1, 0)]

    for command in commands:
        if command == -2:
            direction = (direction - 1) % 4
        elif command == -1:
            direction = (direction + 1) % 4
        else:
            # Move forward, checking for obstacles at each step
            for _ in range(command):
                new_x = x_coordinate + direction_offsets[direction][0]
                new_y = y_coordinate + direction_offsets[direction][1]

                # If next step has an obstacle, stop moving
                if (new_x, new_y) in obstacle_set:
                    break

                x_coordinate = new_x
                y_coordinate = new_y

                # Update the maximum squared distance
                max_distance_squared = max(max_distance_squared, x_coordinate * x_coordinate + y_coordinate * y_coordinate)

    return max_distance_squared

Big(O) Analysis

Time Complexity

O(m + k*l) – Let m be the number of commands, k be the maximum move distance in a command, and l be the number of obstacles. The outer loop iterates through each of the m commands. For each move command, the robot attempts to move up to k steps. In each step, it checks for collision against the set of obstacles. Therefore, the worst-case time complexity for move commands is k*l. The turning commands take constant time, O(1). Thus the overall time complexity is O(m + k*l).

Space Complexity

O(K) – The space complexity is determined by the storage of obstacles. The plain English explanation mentions avoiding obstacles, implying that the obstacle locations are stored in some data structure. If we assume there are K obstacles provided as input, the space required to store their locations (e.g., in a set or hash table) would be proportional to K. Other variables, like the robot's current position and direction, take up constant space. Therefore, the auxiliary space complexity is O(K), where K is the number of obstacles.

Edge Cases

Case	How to Handle
Empty commands array.	Initialize robot's position to (0,0) and return the list containing only (0,0).
Empty obstacles array.	Robot moves according to the commands without any collision detection.
Obstacles at origin (0,0).	Robot cannot start at an obstacle, and any move towards it from adjacent squares should be prevented.
Commands array contains only -1 or -2 (turns).	The robot rotates but does not move forward, so it stays at (0,0) and the returned path contains only (0,0).
Commands array contains large positive numbers, potentially leading to large coordinates and integer overflow.	Use long type to store coordinates, or use modulo arithmetic if the problem allows coordinates within a certain range.
Obstacles are very close to each other.	Ensure the robot doesn't pass between them in a single forward move command.
Maximum number of commands and/or obstacles leading to potential memory constraints.	Confirm coordinate ranges are within reasonable memory limits.
Robot encounters the same obstacle multiple times from different directions.	The obstacle set allows for efficient collision detection regardless of the approach angle.