\begin{align*}
<N!>& <N! > < N!> < N! > < N ! > <N> <N > < N> < N > <!> - \\
<P!>& <P! > < P!> < P! > < P ! > <P> <P > < P> < P > <!> - \\
<Q!>& <Q! > < Q!> < Q! > < Q ! > <Q> <Q > < Q> < Q > <!> - \\
<I!>& <I! > < I!> < I! > < I ! > <I> <I > < I> < I > <!> - \\
<U!>& <U! > < U!> < U! > < U ! > <U> <U > < U> < U > <!> - \\
<B!>& <B! > < B!> < B! > < B ! > <B> <B > < B> < B > <!>
\end{align*}Big-O notation is a mathematical notation used to describe the time or space complexity of an algorithm. It is used to analyze the efficiency of an algorithm and how it will perform as the input size grows. It is commonly used in computer science to compare and evaluate different algorithms.
Complexity is calculated by looking at the worst-case scenario of an algorithm, which is represented by the term with the highest power. The coefficients and lower-order terms are ignored as the input size grows larger, making Big-O notation a simplified way to express the performance of an algorithm.
O(1) complexity means that the algorithm's performance does not change as the input size increases, whereas O(n) complexity means that the algorithm's performance will increase linearly as the input size grows. In other words, an algorithm with O(1) complexity will always have a constant runtime, while an algorithm with O(n) complexity will take longer to run as the input size increases.
The best-case scenario for an algorithm's complexity is O(1), meaning that the algorithm will always have a constant runtime regardless of the input size. This is the most efficient and fastest scenario for an algorithm.
Yes, it is possible for two algorithms to have the same Big-O notation but different runtimes. This is because Big-O notation only looks at the growth rate of an algorithm as the input size increases, and does not take into account other factors such as the actual runtime or the size of the input. Therefore, two algorithms with the same Big-O notation may have different runtimes due to differences in their implementation or other factors.