1. More on Recursion

2. What’s the answer?
What value is returned by the following recursive function Mystery for an input value of n?
Mystery (integer n)
If n = 1 then
return 1
else
return Mystery(n-1) + 1
end if
end function Mystery

3. Answer
Mystery(n) = n

4. What does this iterative method do?
public static void mystery (String s) {
for (int j = s.length() – 1; j >= 0; j --)
System.out.print (s.charAt(j)); }
// How would we write it recursively?

5. Recursive version

6. What is the output? public static void mystery (int n) { if (n < 1)
System.out.print (n); }
else
mystery(n/2);
System.out.print (”, “ + n);
} // end mystery
When call is
mystery(1) mystery(2) mystery(3) mystery(4) mystery(7) mystery(30)
RUN CODE ON NEXT SLIDE TO SEE ANSWERS.

7. public class Nov53 {
public static void main (String [] args)
mystery(1);
System.out.println ();
mystery(2);
mystery(3);
mystery(4);
mystery(7);
mystery(30);
} // end main
public static void mystery (int n)
if (n < 1)
System.out.print (n);
else
mystery(n/2);
System.out.print (", " + n); }
} // end mystery

8. Examples A collection of M numbers is defined recursively by
2 and 3 belong to M
If X and Y belong to M, so does X*Y
Which of the following belong to M? 6 9 16 21 26 54 72
218

9. Recursively defined sets of strings
A collection S of strings of characters is defined recursively by
a and b belong to S
If X belongs to S, so does Xb
Which of the following belong to S? a ab
aba
aaab
bbbbb

10. Selection Sort
Simulate the execution of algorithm SelectionSort on the following list L; write the list after every exchange that changes the list.
4, 10, -6, 2, 5

11. Answer

12. Binary Search
The binary search algorithm is used with the following list; x has the value “flour”
Name the elements against which x is compared.
butter, chocolate, eggs, flour, shortening, sugar

13. Algorithm for recursive Binary Search
BinarySearch (list L; integer i; integer j, itemtype x)
// searches sorted list L from L[i] to l[j] for item x
If i > j then
write (“not found”)
else
find the index k of the middle item in the list L[i] to L[j]
if x = middle item then
write (“found”)
if x < middle item then
BinarySearch (L, I k-1, x)
BinarySearch (L, k+1, j, x)
end if
end function BinarySearch

14. Recursion with logical formulas
Rule 1: Any statement letter is a wff
Rule 2: If P and Q are wffs, so are (P Λ Q), (P V Q), (P → Q), P’ , and (P ↔ Q)
Example
A, B, C are all wffs by rule 1
(A Λ B) and (C’) are both wffs by rule 2
((A Λ B) → (C’)) is a wff, by rule 2
(((A Λ B) → (C’))’)
((A Λ B) → C’)’

15. Recursion in family relations
James’s parents are ancestors of James

16. System.out.println ();
A computer virus that spreads by way of messages is planted in 3 machines the first day. Each day, each infected machine from the day before infects 5 new machines. On the second day, a software solution is found to counteract the virus, and 1 machine is cleaned. Each day thereafter, 6 times as many machines are cleaned as were cleaned the day before. How many days will it be before the effects of the virus are completely gone?