22C:21 Discrete Structures Problem 7.2. System Stacks on Recursion

Stacks – just to refresh your memory List FIFO – push/pop There is a built-in Stack ADT in java. However, I used my own implementation… … because I can … yes, it’s THAT easy … and a little personal touch is always better

The myStack class // Instance variables private static int[] stack; private static int top=0; private static int MAX=100; // Default constructor public myStack() { stack=new int[MAX]; top=-1; }

Push(element) public void push(int b) { if(top>=MAX) System.out.println("Overflow!"); return; } stack[++top]=b; this.print();

Pop() public int pop() { if(top<0) System.out.println("Underflow!"); return(-1); } int temp=top--; this.print(); return stack[temp];

Fibonacci Numbers F0 = 1; F1 = 1; Fn = Fn-1 + Fn-2 for n ≥ 2 So… F2 = 2, F3 = 3, F4 = 5, F5 = 8 …and so on

Modified Fibonacci Code public static BigInteger fibonacci(int n) { System.out.println("fibonacci("+n+") has been called..."); SysStack.push(n); if((n == 0) || (n == 1)) System.out.println("fibonacci("+n+") returning value..."); int garbage = SysStack.pop(); return new BigInteger ("1"); } else BigInteger number1 = fibonacci(n-1); BigInteger number2 = fibonacci(n-2); int moregrabage=SysStack.pop(); return number1.add(number2);

Output for n=3 fibonacci(3) has been called... Current Stack: -- 3 2

Output for n=3 contd. fibonacci(1) has been called... Current Stack: -- 1 2 3 fibonacci(1) returning value...

Output for n=3 contd. fibonacci(0) has been called... Current Stack: -- 2 3 fibonacci(0) returning value...

Output for n=3 contd. fibonacci(2) returning value... Current Stack: -- 3 fibonacci(1) has been called... 1

… should be empty at the end Output for n=3 contd. fibonacci(1) returning value... Current Stack: -- 3 fibonacci(3) returning value... … should be empty at the end

What happens for n=5? 5 Fibonacci(5) 4 3 5 System Stack Empty

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 3 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 2 Fibonacci(2) 3 1 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 2 1 2 Fibonacci(2) 3 4 Fibonacci(1) 1 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 2 Fibonacci(2) 3 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 2 1 2 Fibonacci(2) 3 4 Fibonacci(0) 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 2 Fibonacci(2) 3 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 3 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) 1 1 Fibonacci(1) 3 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 3 2 Fibonacci(3) System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(4) 4 3 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 1 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 1 Fibonacci(1) 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 2 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) Fibonacci(0) 2 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 2 Fibonacci(2) 2 System Stack

What happens for n=5? 5 Fibonacci(5) 4 3 Fibonacci(4) 4 5 System Stack

What happens for n=5? 5 Fibonacci(5) 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 1 2 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 1 2 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 2 2 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) Fibonacci(0) 2 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 2 1 Fibonacci(2) 2 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 1 3 5 System Stack

What happens for n=5? Fibonacci(5) 5 Fibonacci(3) 3 1 Fibonacci(1) 1 3 System Stack

What happens for n=5? 5 Fibonacci(5) Fibonacci(3) 3 3 5 System Stack

What happens for n=5? 5 Fibonacci(5) 5 System Stack

What happens for n=5? System Stack Empty

Note Actual system stack holds returned values Just the indices has been shown Compare with FastFibonacci()

Now you know!

Quiz 7 You are given a collection with 6 items. The priorities of these items are shown below: 9 30 25 22 7 50 1. Draw all possible max-heaps (as binary trees) for this collection of items. 2. Start with the max-heap (30, 50, 9, 25, 7, 22). Perform the following sequence of operations on this max-heap: Insert(45) Insert(29) Delete() Show the max-heap after each operation. 3. Answer in a sentence or two: Given a max-heap can you find the item with 2nd highest priority in O(1) time? Explain your answer. 4. Answer in a sentence or two: Given a max-heap can you find the item with smallest priority in O(1) time? Explain your answer.