EXERCISE $8\mathrm{.}5\mathrm{.}9$

$($a$)$ Let $A$ be a collection of objects. Describe an efficient method for converting A into a set. That is$,$ remove all duplicates from $A.$ What is the running time of this method?

EXERCISE $8\mathrm{.}5\mathrm{.}14$

$($a$)$ Suppose we are given a sequence $S$ of $n$ elements, on which a total order relation is defined. Describe an efficient method for determining whether there are two equal elements in $S.$ What is the running time of your method?

EXERCISE $8\mathrm{.}5\mathrm{.}19$

$($a$)$ Given a sequence of numbers, $(x_1,x_2,$dots,$x_n),$ the mode is the value that appears the most number of times in this sequence. Give an efficient algorithm to compute the mode for a sequence of $n$ numbers. What is the running time of your method?

EXERCISE $8\mathrm{.}5\mathrm{.}24$

$($a$)$ Bob has a set, $A,$ of $n$ nuts and a set, $B,$ of $n$ bolts, such that each nut has a unique matching bolt. Unfortunately, the nuts in A all look the same, and the bolts in $B$ all look the same as well. The only comparison that Bob can make is to take a nut$-$bolt pair $(a,b),$ such that ainA and binB, and test if the threads of a are larger, smaller, or a perfect match with the threads of $b.$ Describe an efficient algorithm for Bob to match up all of his nuts and bolts. What is the running time of this algorithm?