1041. Robot Bounded In Circle - Detailed Explanation

Problem Statement

You are given a string of instructions consisting of the characters:

• G: move forward one unit
• L: turn left 90°
• R: turn right 90°

A robot starts at the coordinate (0, 0) facing North. The instructions are executed once and then repeated indefinitely. Determine if there exists a circle such that the robot never leaves the circle.

Example Input 1:

• Instructions: "GGLLGG"
  Output: true
  Explanation: The robot moves north 2 units, then turns left twice and moves south 2 units, returning to (0, 0). Even without returning to the origin, if the robot ends up not facing north, it will eventually loop around.

Example Input 2:

• Instructions: "GG"
  Output: false
  Explanation: The robot moves north 2 units and still faces north. Repeating these moves will make it go further away from the origin.

Example Input 3:

• Instructions: "GL"
  Output: true
  Explanation: The robot moves north 1 unit and then turns left. After a few cycles of these instructions, the robot will be confined within a circle.

Constraints:

• The instruction string contains only the characters 'G', 'L', and 'R'.
• The length of the instruction string is at most 100.

Hints Before the Solution

• Hint 1: Simulate one cycle (one complete pass of the instructions). After the cycle, check the robot’s position and direction.
• Hint 2: Think about what it means for the robot to be “bounded in a circle.” Notice that if the robot returns to the origin, it’s obviously bounded. Alternatively, if the robot doesn’t face North after one cycle, its subsequent moves will eventually form a loop.

Approach 1: Simulation (Optimal Approach)

Idea:

• Initialize the robot’s position at (0, 0) and its facing direction as North.
• Use a direction vector to represent the current orientation (for North, it is (0, 1)).
• Process each instruction:
  • G: Update the position using the current direction.
  • L/R: Update the direction accordingly.
• Key Insight:
  • If after one cycle the robot is at the origin, it will loop.
  • Or if the robot is not facing North (its initial direction), then after repeating the instructions at most 4 times the path will form a cycle.

Step-by-Step Outline:

• Start at (0, 0) with direction (0, 1).
• For each character in the instruction string:
  • If 'G', add the direction vector to the position.
  • If 'L', rotate the direction vector 90° left.
  • If 'R', rotate the direction vector 90° right.
• After processing the cycle, check:
  • If position == (0, 0): return true.
  • If direction != (0, 1): return true.
  • Otherwise, return false.

Visual Example for "GL":

1. Initial state: Position = (0, 0), Direction = North (0, 1).
2. Instruction 'G': Move forward → New Position = (0, 1), Direction = North.
3. Instruction 'L': Turn left → Direction changes from North (0, 1) to West (-1, 0).

• After one cycle, the position is (0, 1) and direction is West. Since the direction isn’t North, if you repeat the instructions, the robot’s path will eventually cycle, keeping it bounded.

Approach 2: Brute Force Simulation (Cycle Check)

Idea:

• Instead of checking after one cycle, simulate the instructions repeatedly (up to 4 cycles).
• The reasoning behind simulating 4 cycles is that after 4 repetitions the robot’s direction will cycle back to North.
• If the robot doesn’t return to the origin after 4 cycles, it will never be bounded.

Steps:

• Run the simulation for 4 cycles.
• If at any point the robot returns to (0, 0), then the robot is bounded.
• If after 4 cycles the position is not (0, 0), then the robot is unbounded.

Note:
This approach is less efficient than checking the conditions after one cycle but can be useful for understanding the cyclic behavior.

Code Implementation

Below are two implementations of the optimal simulation approach (Approach 1): one in Python and one in Java.

Python Code