On an infinite plane, a robot initially stands at (0, 0) and faces north.  The robot can receive one of three instructions:

• "G": go straight 1 unit;
• "L": turn 90 degrees to the left;
• "R": turn 90 degress to the right.

The robot performs the instructions given in order, and repeats them forever.

Return true if and only if there exists a circle in the plane such that the robot never leaves the circle.

 

Example 1:

Input: "GGLLGG"
Output: true
Explanation: 
The robot moves from (0,0) to (0,2), turns 180 degrees, and then returns to (0,0).
When repeating these instructions, the robot remains in the circle of radius 2 centered at the origin.

Example 2:

Input: "GG"
Output: false
Explanation: 
The robot moves north indefinitely.

Example 3:

Input: "GL"
Output: true
Explanation: 
The robot moves from (0, 0) -> (0, 1) -> (-1, 1) -> (-1, 0) -> (0, 0) -> ...

 

Note:

1. 1 <= instructions.length <= 100
2. instructions[i] is in {'G', 'L', 'R'}

 

Approach #1: Math. [Java]

class Solution {
    public boolean isRobotBounded(String instructions) {
        int x = 0, y = 0, t = 0;
        int[][] dirs = {{0, 1}, {1, 0}, {0, -1}, {-1, 0}};
        for(int i = 0; i < instructions.length(); ++i) {
            if (instructions.charAt(i) == 'R') {
                t = (t + 1) % 4;
            } else if (instructions.charAt(i) == 'L') {
                t = (t + 3) % 4;
            } else {
                x += dirs[t][0];
                y += dirs[t][1];   
            }
        }
        
        return x == 0 && y == 0 || t > 0;
    }
}

　　

Analysis:

In order for the robot to stay within a circle, you need to move in a cycle. The only way you move in a cycle is if you end where you start (the origin at (0, 0)).

The minimum number of instructions you need to repeat is 4 in order to figure out if you are in a cycle.

For example, if each instruction only rotates 90 degrees, you need to repeat the instructions 4 times to possibly end where you start.

If each instruction rotates 180 degrees, you need to repeat the instructions 2 times to possibly end where you start.

If each instruction rotates 270 degrees, you need to repeat the instructions 4 times to possibly end where you start.

 

Reference:

https://leetcode.com/problems/robot-bounded-in-circle/discuss/290915/Python-Concise-%2B-Explanation