Exercise $3.$ The $\backslash ($ k $\backslash )-$ary heap.

$[20$ points$]$

A $\backslash ($ k $\backslash )-$ary heap is a generalization of the binary heaps we studied in class: it is a complete $\backslash ($ k $\backslash )-$ary tree $\backslash (\{\}^\{2\}\backslash )$ with the heap property, which means that for any node in the heap, its parent is of higher priority. When $\backslash ($ k$=2\backslash ),$ we have a familiar binary heap.

Though programmed almost identically to the binary heap no matter our choice of $\backslash ($ k $\backslash$geq $2\backslash ),$ the time complexities in a $\backslash ($ k $\backslash )-$ary heap depend on $\backslash ($ k $\backslash )$; deletemin takes roughly $\backslash ($ k $\backslash$log $\_\{$k$\}$ n $\backslash )$ operations, and insert takes roughly $\backslash (\backslash$log $\_\{$k$\}$ n $\backslash )$ operations.

In particular, we can implement build$-$max$-$k$-$heap, an algorithm that takes in a list of $\backslash ($ n $\backslash )$ distinct integers and an integer $\backslash (2\backslash$leq k