Exercise XIII
Exercise For each of the $T(n)$ running time listed below, figure out the asymptotic runtime and express it in Big-Oh notation. Then rank the functions in growth rate order, starting with the slowest growth rate (i.e those resulting in a fast runtime), and ending with the fastest growth rate (worst runtime). If two functions have the same asymptotic runtime, then rank them based on the original expressions (including constants).
| $T(n)$ | Big-Oh | Rank |
|---|---|---|
| $\left ( \lg \frac{n}{4} \right )^3$ | ||
| $(n^2-4)/(n+2)$ | ||
| $\left ( 3n + \lg n \right )^2$ | ||
| $2 \lg n^2$ | ||
| $\left ( 2^n \right )^2 + \lg n$ | ||
| $n^2 \lg 16$ | ||
| $2n\lg\lg n$ | ||
| $4n^2+n(10 + \lg n)$ |
Solution
| $T(n)$ | Big-Oh | Rank |
|---|---|---|
| $\left ( \lg \frac{n}{4} \right )^3$ | $\left ( \lg n - \lg 4 \right )^3 = \left ( \lg n - 2 \right )^3 \in O(\lg^3 n)$ | 2 |
| $(n^2-4)/(n+2)$ | $(n-2)(n+2)/(n+2) \in O(n)$ | 3 |
| $\left ( 3n + \lg n \right )^2$ | $9n^2+\lg^2 n + 6n\lg n \in O(n^2)$ | 7 |
| $2 \lg n^2$ | $4 \lg n \in O(\lg n)$ | 1 |
| $\left ( 2^n \right )^2 + \lg n$ | $4^n+\lg n \in O(4^n)$ | 8 |
| $n^2 \lg 16$ | $4n^2 \in O(n^2)$ | 5 |
| $2n\lg\lg n$ | $O(n\lg\lg n)$ | 4 |
| $4n^2+n(10 + \lg n)$ | $4n^2+10n+n\lg n \in O(n^2)$ | 6 |
Resources
Logarithmic identities are essential for solving the above exercise.