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 ith obstacle is at grid point obstacles[i] = (xi, yi). 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:
(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.
Example 1:
Input: commands = [4,-1,3], obstacles = []
Output: 25
Explanation:
The robot starts at (0, 0):
(0, 4).(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):
(0, 4).(2, 4), robot is at (1, 4).(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):
(0, 6).(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 <= 104commands[i] is either -2, -1, or an integer in the range [1, 9].0 <= obstacles.length <= 104-3 * 104 <= xi, yi <= 3 * 104231.Problem Overview: You control a robot on an infinite 2D grid starting at (0,0) facing north. The robot receives commands to move forward or rotate left/right. Some grid cells contain obstacles that block movement. The task is to simulate the robot’s path and return the maximum Euclidean distance squared from the origin reached during the simulation.
Approach 1: Naive Simulation with Obstacle Scan (O(n * k) time, O(1) space)
The straightforward solution simulates each command step-by-step. For every forward movement, the robot advances one unit at a time while checking if the next coordinate matches any obstacle. If obstacles are stored in a list, every step requires scanning the entire list to determine whether movement is blocked. With k obstacles and up to n movement steps, this results in O(n * k) time complexity. The logic is simple and demonstrates how robot movement works, but obstacle lookup becomes the bottleneck as the number of obstacles grows.
Approach 2: Simulate Robot Movement Using Direction Array + Hash Set (O(n) time, O(k) space)
The optimal solution still performs step-by-step simulation but removes the expensive obstacle scan. Store all obstacle coordinates in a hash set so membership checks run in O(1) average time. Maintain a direction array such as [(0,1),(1,0),(0,-1),(-1,0)] representing north, east, south, and west. Rotation commands simply update a direction index using modulo arithmetic, while forward commands iterate step-by-step and check the next coordinate in the set. If the coordinate exists in the set, movement stops for that command. Otherwise update the robot’s position and track the maximum distance squared from the origin.
This approach relies heavily on constant-time lookups from a Hash Table and simple coordinate updates using an Array of direction vectors. The problem becomes a clean Simulation task where each step performs only a few arithmetic operations and one hash lookup.
Recommended for interviews: Interviewers expect the hash set + direction array approach. The naive scan demonstrates understanding of the robot simulation, but the optimized version shows you recognize the need for constant-time obstacle checks. Using a direction array keeps the code concise and avoids repeated conditional logic for turns.
Simulate the movement of the robot by maintaining a variable for the current direction. Use an array to store the possible directions: north, east, south, and west. When a turn command is received, adjust the direction index. For movement commands, move the robot in the direction indexed by the current direction.
The solution defines the possible directions the robot can face using two arrays dx and dy. The direction is managed using a variable that cycles through the indices on executing turn commands. By maintaining a set of obstacles, it ensures efficient checks when the robot moves. The maximum squared distance is updated whenever the robot's position changes.
Python
C++
Java
JavaScript
Time Complexity: O(N + M), where N is the number of commands and M is the number of obstacles.
Space Complexity: O(M), where M is the number of obstacles.
We define a direction array dirs = [0, 1, 0, -1, 0] of length 5, where each pair of adjacent elements represents a direction. That is, (dirs[0], dirs[1]) represents north, (dirs[1], dirs[2]) represents east, and so on.
We use a hash table s to store the coordinates of all obstacles, so we can determine in O(1) time whether the next step will encounter an obstacle.
Additionally, we use two variables x and y to represent the robot's current coordinates, initially x = y = 0. The variable k represents the robot's current direction, and the answer variable ans represents the maximum squared Euclidean distance from the origin.
Next, we iterate over each element c in the array commands:
c = -2, the robot turns left 90 degrees, i.e., k = (k + 3) bmod 4;c = -1, the robot turns right 90 degrees, i.e., k = (k + 1) bmod 4;c units. We combine the robot's current direction k with the direction array dirs to obtain the increments along the x-axis and y-axis. We accumulate the increments step by step onto x and y, and check whether each new coordinate (nx, ny) is in the obstacle set. If not, we update the answer ans; otherwise, we stop the simulation and proceed to the next command.Finally, return the answer ans.
The time complexity is O(C times n + m) and the space complexity is O(m), where C is the maximum number of steps per move, and n and m are the lengths of the arrays commands and obstacles, respectively.
| Approach | Complexity |
|---|---|
| Simulate Robot Movement Using Direction Array | Time Complexity: O(N + M), where N is the number of commands and M is the number of obstacles. |
| Hash Table + Simulation | — |
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Naive Simulation with Obstacle Scan | O(n * k) | O(1) | When obstacle count is very small or during initial brute-force reasoning |
| Direction Array + Hash Set Simulation | O(n) | O(k) | General case. Fast obstacle lookup and clean movement simulation |
Walking Robot Simulation - Leetcode 874 - Python • NeetCodeIO • 12,313 views views
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