You’re correct that if the number of characters (
numChars) is treated as a constant, the complexity simplifies accordingly.
Simplified Complexity
When considering numChars as a constant (e.g., 26 for the English alphabet), the time complexity can be analyzed differently:
- Constant Factor:
- If
numCharsis constant, we can denote it as CCC. Thus, the complexity of the recursive calls would be O(Cn)O(C^n)O(Cn).
- Final Complexity:
- When CCC is constant, CnC^nCn behaves like an exponential growth function in terms of
n. Therefore, in this context, the complexity can be expressed as O(1)⋅O(nn)O(1) \cdot O(n^n)O(1)⋅O(nn) or simply O(nn)O(n^n)O(nn) for very largen.
Overall Time Complexity
Therefore, when considering numChars as constant, the complexity simplifies to:
O(nn)O(n^n)O(nn)
This indicates that the time grows polynomially with respect to the length of the strings being generated.
Conclusion
In summary, when numChars is constant, we can reduce the complexity to a more manageable form, and it becomes clear that the recursive nature still causes an exponential increase in calls with respect to n. This emphasizes the efficiency concerns when n grows large.
For more details about recursive algorithms and time complexity, you might explore these resources: