Robot Bounded In Circle

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 degrees 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: instructions = "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: instructions = "GG"
Output: false
Explanation: The robot moves north indefinitely.

Example 3:

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

Constraints:

• 1 <= instructions.length <= 100
• instructions[i] is 'G', 'L' or, 'R'.

var isRobotBounded = function (instructions) {
    const origin = [0,0]
    let dir = 0;
    
    const moving = () => {
        for(let str of instructions){
            if(str === "L"){
                dir -= 1
                if(dir < 0) dir += 4;
            } else if (str === 'R') {
                dir += 1
                if(dir > 3) dir = dir % 4;
            } else {
                if(dir === 0){
                    origin[1] += 1
                } else if(dir === 1) {
                    origin[0] -= 1
                } else if(dir === 2) {
                    origin[1] -= 1
                } else {
                    origin[0] += 1
                }
            }
        }
    }
    
    let times = 0;
    while(times < 4){
        moving();
        if(origin[0] === 0 && origin[1] === 0) return true
        times += 1
    }
    return false
};

Approach 1: One Pass

Intuition

This solution is based on two facts about the limit cycle trajectory.

• After at most 4 cycles, the limit cycle trajectory returns to the initial point x = 0, y = 0. That is related to the fact that 4 directions (north, east, south, west) define the repeated cycles' plane symmetry.

Algorithm

• Let’s use numbers from 0 to 3 to mark the directions: north = 0, east = 1, south = 2, west = 3. In the array directions we could store corresponding coordinates changes, i.e. directions[0] is to go north, directions[1] is to go east, directions[2] is to go south, and directions[3] is to go west.
• The initial robot position is in the center x = y = 0, facing north idx = 0.
• Now everything is ready to iterate over the instructions.
• If the current instruction is R, i.e. to turn on the right, the next direction is idx = (idx + 1) % 4. Modulo here is needed to deal with the situation - facing west, idx = 3, turn to the right to face north, idx = 0.
• If the current instruction is L, i.e. to turn on the left, the next direction could written in a symmetric way idx = (idx - 1) % 4. That means we have to deal with negative indices. A more simple way is to notice that 1 turn to the left = 3 turns to the right: idx = (idx + 3) % 4.
• If the current instruction is to move, we simply update the coordinates: x += directions[idx][0], y += directions[idx][1].
• After one cycle we have everything to decide. It’s a limit cycle trajectory if the robot is back to the center: x = y = 0 or if the robot doesn't face north: idx != 0.

Complexity Analysis

• Time complexity: O(N), where N is a number of instructions to parse.
• Space complexity: O(1) because the array directions contains only 4 elements.