$1.$ Prove that finding the minimum $($or maximum$)$ in a sequence of n

numbers x$1,$ x$2,...,$ xn requires at least n $1$ comparisons.

$2.$ Finding the smallest and the second smallest elements can also use

heap, instead of tournament. Design an efficient algorithm for finding

the smallest and the second smallest elements using the heap data

structure. Compare heap method and tournament method, which one

is more efficient.

$3.$In the algorithm for finding the k$-$th smallest element of n elements

shown in Figure $5\mathrm{.}8,$ we first partition the n elements into $5-$element

subsets. If we divide the n elements into subsets of $3$ elements, is the

time complexity still linear? How about subsets of $7$ elements? Prove that finding the minimum $($or maximum$)$ in a sequence of $n$

numbers $x_1,x_2,$dots,$x_n$ requires at least $n-1$ comparisons. Finding the smallest and the second smallest elements can also use

heap, instead of tournament. Design an efficient algorithm for finding

the smallest and the second smallest elements using the heap data

structure. Compare heap method and tournament method, which one

is more efficient. In the algorithm for finding the $k-$th smallest element of $n$ elements

shown in Figure $5\mathrm{.}8,$ we first partition the $n$ elements into $5-$element

subsets. If we divide the $n$ elements into subsets of $3$ elements, is the

time complexity still linear? How about subsets of $7$ elements?