public class MyClass $\{$

$//$ Assume ints can get arbitrarily large but simple operations

$//$ on them still remain worst case O$(1).$

private int count$\_$i $=0$;

public void op$1()\{$

int temp $=$ count$\_$i;

for $($int i $=0$; i $<$ temp; i$++)\{$

count$\_$i $=$ count$\_$i $-1$;

$\}$

$//$ count$\_$i $==0$

$\}$

public void op$2()\{$

count$\_$i $+=7$;

$\}$

$\}$

In this problem, you will work through the steps that comprise an amortized analysis

on MyClass. You must use the potential method

$($a$)(10$ points$)$ Assuming D$0$ is the initial state of a MyClass, and Di is the state of

the MyClass object after the i$-$th operation, then define $($Di$).$

$($b$)(10$ points$)$ Prove that $($Di$)0.$

$($c$)(10$ points$)$ What is the amortized cost of op$1?$ Give both the non$-$asymptotic and

asymptotic cost. You may assume, for the sake of simplicity, that ci for op$1$ is the

number of times the for loop executes. Give the best possible big O and justify

your answer.

$($d$)(10$ points$)$ What is the amortized cost of op$2?$ Give both the non$-$asymptotic and

asymptotic cost. For your asymptotic cost, give the best possible big O$.$ Justify

your answer.