Toolpath simulation and verification in CNC Programming - Time & Space Complexity

Time Complexity: Toolpath simulation and verification

O(n)

Understanding Time Complexity

When simulating and verifying a CNC toolpath, we want to know how the time needed grows as the path gets longer or more detailed.

We ask: How does the simulation time change when the number of tool moves increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

FOR i = 1 TO N
  MOVE_TOOL_TO(X[i], Y[i], Z[i])
  CHECK_COLLISION()
  UPDATE_SIMULATION_DISPLAY()
NEXT i
    

This code simulates moving the tool through N points, checking for collisions and updating the display each time.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Loop over N tool positions.
• How many times: Exactly N times, once per tool move.

How Execution Grows With Input

Each new tool position adds one more set of operations: move, check, and update.

Input Size (n)	Approx. Operations
10	About 10 moves and checks
100	About 100 moves and checks
1000	About 1000 moves and checks

Pattern observation: The total work grows directly with the number of tool moves.

Final Time Complexity

Time Complexity: O(N)

This means the simulation time grows in a straight line as the number of tool moves increases.

Common Mistake

[X] Wrong: "The simulation time stays the same no matter how many moves there are."

[OK] Correct: Each move requires checking and updating, so more moves mean more work and more time.

Interview Connect

Understanding how simulation time grows helps you explain and improve CNC software performance in real projects.

Self-Check

"What if the collision check itself loops over M obstacles for each move? How would the time complexity change?"