Given the following sequence of numbers:

$224,57,45,13,18,7,4,6,2$

Build a min$-$heap

$($a$)$ What is the heap$-$property that must be held?

$($b$)$ Construct the complete tree. For every adding into the heap, show the intermediate steps of the percolate$/$up and

percolate$/$down in the complete tree and in the array

add $224$

iv$.$ add $13$

v$.$ add $18$

vii. add $-4$

ii$.$ add $57$

viii. add $6$

iii. add $45$

ix$.$ add $2$

vi$.$ add $7$

Draw the final result of built min heap

Based on the result of build$-$min heap from previous question, perform heap$-$sort. $($Hint: Build$-$min $+$ remove$-$min will lead

to the heap sort$)$

$($a$)$ Show the intermediate steps of the percolation $($up$/$down$)$ for every removeMin$()$

$($b$)$ Show the state of the array that keeps the results of the removeMin$()$ every step

Fill out the following array with the resulting array from previous question. Suppose the target $i=3$

$($a$)$ what is heap $[3]\text{'}$s parent? What is the formula to calculate its parent?

$($b$)$ what is heap $[3]\text{'}$s left child? What is the formula to calculate its left child?

$($c$)$ what is heap$[3]\text{'}$s right child? what is the formula to calculate its right child?