Roughing strategies (adaptive, pocket) in CNC Programming - Time & Space Complexity

Time Complexity: Roughing strategies (adaptive, pocket)

O(n^2)

Understanding Time Complexity

When using roughing strategies in CNC programming, it's important to understand how the time to complete the toolpath changes as the size of the area to be machined grows.

We want to know how the number of tool movements increases when the workpiece area gets bigger.

Scenario Under Consideration

Analyze the time complexity of the following roughing toolpath generation snippet.

// Roughing pocket with adaptive strategy
for x from 0 to width step step_over {
  for y from 0 to height step step_over {
    if inside_pocket(x, y) {
      move_tool_to(x, y);
      cut_material();
    }
  }
}
    

This code moves the tool in a grid pattern over the pocket area, cutting material where needed.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Nested loops scanning the pocket area in steps.
• How many times: The outer loop runs about width/step_over times, and the inner loop runs about height/step_over times for each outer loop.

How Execution Grows With Input

As the pocket size grows, the number of tool moves grows roughly with the area size.

Input Size (n)	Approx. Operations
10 x 10	100 moves
100 x 100	10,000 moves
1000 x 1000	1,000,000 moves

Pattern observation: Doubling the width and height quadruples the number of moves, showing growth with the area.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to complete the roughing grows with the square of the input size, as the tool covers a two-dimensional area.

Common Mistake

[X] Wrong: "The time grows linearly because the tool just moves along one path."

[OK] Correct: The tool moves in two directions (width and height), so the total moves multiply, not just add up.

Interview Connect

Understanding how toolpath complexity grows helps you explain machining efficiency and resource planning in real CNC projects.

Self-Check

"What if the step_over distance doubled? How would the time complexity change?"