Efficiency
A measure of how well an algorithm uses resources, such as time and memory, to solve a problem.
We will focus on time efficiency, which is often the most critical factor when choosing an algorithm.
We look at the number of steps an algorithm performs in relation to input size $n$:
| Big O | Name | Example | Description |
|---|---|---|---|
| $O(1)$ | Constant | Accessing an array element | Time does not depend on input size. |
| $O(\log n)$ | Logarithmic | Binary search | Each step halves the problem size. |
| $O(n)$ | Linear | Summing a list | Work grows directly with input size. |
| $O(n \log n)$ | Log-linear | MergeSort, QuickSort | Efficient sorting algorithms. |
| $O(n^2)$ | Quadratic | Nested loops over $n$ | Work grows with the square of input size. |
| $O(2^n)$ | Exponential | Recursive subset generation | Work doubles with each extra input. |
| $O(n!)$ | Factorial | Traveling Salesman brute force | Extremely inefficient for large $n$. |
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