Exercises

$2\mathrm{.}2-1$

Express the function $\frac{n^3}{1000}-100n^2-100n+3$ in terms of $-$notation.

$2\mathrm{.}2-2$

Consider sorting $n$ numbers stored in array $A$ by first finding the smallest element

of A and exchanging it with the element in $A[1].$ Then find the second smallest

element of $A,$ and exchange it with $A[2].$ Continue in this manner for the first $n-1$

elements of $A.$ Write pseudocode for this algorithm, which is known as selection

sort. What loop invariant does this algorithm maintain? Why does it need to run

for only the first $n-1$ elements, rather than for all $n$ elements? Give the best$-$case

and worst$-$case running times of selection sort in $-$notation.

$2\mathrm{.}2-3$

Consider linear search again $($see Exercise $2\mathrm{.}1-3).$ How many elements of the in$-$

put sequence need to be checked on the average, assuming that the element being

searched for is equally likely to be any element in the array? How about in the

worst case? What are the average$-$case and worst$-$case running times of linear

search in $-$notation? Justify your answers.

$2\mathrm{.}2-4$

How can we modify almost any algorithm to have a good best$-$case running time?