Asymptotic analysis evaluates an algorithm's performance as the input size $n$ approaches infinity ($n \rightarrow \infty$), allowing us to compare algorithms based on their theoretical scalability.
The visual comparison demonstrated that while constants affect performance for small inputs, the fundamental difference in growth rates ($O(n)$ vs $O(\log_2(n))$) ultimately determines efficiency for large systems. This formal simplification process is called asymptotic analysis.
When calculating the time complexity $O(f(n))$, we adhere to two fundamental simplification rules:
- Rule 1: Ignore Constant Factors
Any multiplicative constant within the running time function $f(n)$ is ignored. For example, an algorithm taking $f(n) = 50n$ steps is simplified to $O(n)$. The constant $50$ is considered implementation overhead that does not change the core growth pattern.
- Rule 2: Focus on the Dominating Term
When the running time is a sum of multiple terms, we keep only the term that grows fastest as $n$ increases—this is the dominating term. All lower-order terms are discarded.
For a complex time function:
$$f(n) = 2n^3 + 500n^2 + 1000$$The dominating term is $2n^3$. Ignoring constants and lower-order terms gives a final time complexity of $O(n^3)$.
Summary of Simplification Rules
| Rule | Description | Example |
|---|---|---|
| Ignore Constants | Drop multiplicative constant factors. | $50n \rightarrow O(n)$ |
| Dominating Term | Keep only the fastest-growing term. | $2n^3 + 500n^2 \rightarrow O(n^3)$ |