Simulate robot path and detect boundedness
Robot Path Simulation and Boundedness Detection
A robot begins its journey at coordinates (0, 0) on an infinite 2D Cartesian plane, initially facing North. It processes a sequence of commands provided as a string s. The command set includes:
- 'G': Move forward one unit in the current direction.
- 'L': Turn left 90 degrees.
- 'R': Turn right 90 degrees.
After executing the command string s once, the robot repeats the entire string s a total of k times. The value of k can be extremely large, up to 10^9.
Your task is to implement an algorithm that:
- Returns the robot's final position (x, y) and its facing direction after a single complete execution of the command string
s. - Determines whether the robot's trajectory remains within a bounded region if the command string
swere to be repeated indefinitely.
You must provide the time and space complexity of your solution. Crucially, your algorithm must avoid simulating all k repetitions to achieve the final state or determine boundedness.
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Core dilemma
The central engineering conflict is how to predict the long-term behavior (boundedness) of a system that repeats a fixed sequence of operations, without explicitly simulating an astronomically large number of repetitions. This requires leveraging the cyclical nature of robot movement and direction changes to determine the net effect of one or more command string executions.
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Simulate robot path and detect boundedness
Robot Path Simulation and Boundedness Detection
A robot begins its journey at coordinates (0, 0) on an infinite 2D Cartesian plane, initially facing North. It processes a sequence of commands provided as a string s. The command set includes:
- 'G': Move forward one unit in the current direction.
- 'L': Turn left 90 degrees.
- 'R': Turn right 90 degrees.
After executing the command string s once, the robot repeats the entire string s a total of k times. The value of k can be extremely large, up to 10^9.
Your task is to implement an algorithm that:
- Returns the robot's final position (x, y) and its facing direction after a single complete execution of the command string
s. - Determines whether the robot's trajectory remains within a bounded region if the command string
swere to be repeated indefinitely.
You must provide the time and space complexity of your solution. Crucially, your algorithm must avoid simulating all k repetitions to achieve the final state or determine boundedness.