1. Introduction to Computer Systems
Lecturer: Steve Maybank
School of Computer Science and Information Systems
Spring 2015
Week 9b: Recursion
24 November 2015
Birkbeck College, U. London

2. Birkbeck College, U. London
Factorial Function
0! = 1
1! = 1
2! = 2x1 = 2
3! = 3x2x1 = 6
4! = 4x3x2x1 = 24
n! = nx((n-1)!)
24 November 2015
Birkbeck College, U. London

3. Birkbeck College, U. London
Pseudocode Version
def fact(n): if(n==0): return 1 else: return n*fact(n-1)
24 November 2015
Birkbeck College, U. London

4. Birkbeck College, U. London
Example
Call fact(3): value returned is 3xfact(2)
Call fact(2): value returned is 2xfact(1)
Call fact(1): value returned is 1xfact(0)
Call fact(0): value returned is 1
24 November 2015
Birkbeck College, U. London

5. Properties of Recursive Algorithms
The code for fact() is recursive because it includes a call to fact()
The included call, fact(n-1) is simpler than the original call fact(n)
The sequence of calls to fac() ends at the base case, fact(0), which returns 1
24 November 2015
Brookshire Section 5.6

6. Non-Recursive Version of fact()
def factNR(n): i=1 f=1 while(i <= n): f = i*f i = i+1 return f
24 November 2015
Birkbeck College, U. London

7. Second Recursive Function
def printTriangle(n): i = 1 while(i <= n): print(‘*’) i = i+1 print(newline) if(n > 1): printTriangle(n-1)
24 November 2015
Birkbeck College, U. London

8. Birkbeck College, U. London
Printed Output
Call printTriangle(4): **** and call printTriangle(3) Call printTriangle(3): *** and call printTriangle(2) Call printTriangle(2): ** and call printTriangle(1) Call printTriangle(1): *
24 November 2015
Birkbeck College, U. London

9. Informal Definition
Recursion occurs when a set of instructions is repeated as a subtask of the original set.
24 November 2015
Brookshear Section 5.5

10. Example of Recursion: Web Pages
readPage(URL)
Page 2 B2 A3
Page 3
24 November 2015
Birkbeck College, U. London

11. Recursive Reading of Web Pages
On encountering a link, follow it immediately.
On reaching the end of a page, return to the previous page.
Order: A1, A2, A3, B2, B1.
24 November 2015
Birkbeck College, U. London

12. Iterative Reading of Web Pages
On encountering a link, remember it.
On reaching the end of a page, choose a remembered link and go to the appropriate page.
Order: A1, B1, A2, B2, A3.
24 November 2015
Birkbeck College, U. London

13. Why does Recursion Matter?
Certain algorithms can be written down very efficiently using recursion.
Many computer languages allow recursion.
The usual implementation of function calls using a stack makes recursion straightforward. See
24 November 2015
Birkbeck College, U. London

14. Brookshear Ch 5 revision problem 42
Example
Produce a recursive algorithm to print the daily salary of a worker who each day is paid twice the previous day’s salary, starting with one penny for the first day’s work.
24 November 2015
Brookshear Ch 5 revision problem 42

15. Greatest Common Divisor
Let m, n be integers such that m >= 0, n >= 0 and (m, n) ≠ (0, 0). Then GCD(m, n) is the largest integer which divides both m and n
GCD(m, n) = GCD(n, m)
GCD(m, 0) = m
Example: GCD(56, 21) = 7
24 November 2015
Birkbeck College, U. London

16. Recursive Property of GCD
Suppose that m >= n and let q be any integer such that q >= 1 and m >= q*n. Then
GCD(m, n) = GCD(m-q*n, n)
Example:
GCD(56, 21) = GCD(56-2*21, 21) = GCD(14, 21)
24 November 2015
Birkbeck College, U. London

17. Recursive Algorithm for GCD
# assume m >= n > 0
RGCD(m, n)
r = m%n; # r=m-q*n
if (r == 0) :
return n
else:
return RGCD(n, r)
24 November 2015
Birkbeck College, U. London

18. Birkbeck College, U. London
Example Run of RGCD
Call RGCD(256, 120):
r = 16, return RGCD(120, 16)
Call RGCD(120, 16):
r = 8, return RGCD(16, 8)
Call RGCD(16, 8):
r = 0, return 8
24 November 2015
Birkbeck College, U. London

19. Recursive Algorithm for Binary Search
# test to see if the element a is in the sorted list L def recursiveBinarySearch(L, a): if(Length(L) == 0): return False j = floor(Length[L]/2) if(L[j] == a): return True if(L[j] > a): L1 = sublist of L from index j+1 to Length(L)-1 else: L1 = sublist of L from index 0 to j-1 return recursivebinarySearch(L1)
24 November 2015
Brookshire, Section 5.5

20. quickSort def quickSort(L): if (Length(L) <= 1): return L
split L into sublists L1 and L2
M1 = quickSort L1
M2 = quickSort L2
M = merge(M1, M2)
return M
24 November 2015
Brookshear Section 5.5

21. Birkbeck College, U. London
merge
def merge(M1, M2): # compare the merge of two
if(Length(M1)==0): # sequential files
return M2
if(Length(M2)==0):
return M1
if(M1[0] <= M2[0]):
m=M1[0]; N1=Drop(M1,1); N2=M2;
else:
m=M2[0]; N1=M1; N2=Drop(M2,1);
return Join({m}, merge(N1, N2))
24 November 2015
Birkbeck College, U. London

22. Example
What sequence of numbers is printed by the following recursive function if it is started with N assigned the value 1?
def exercise(N):
print(N)
if(N < 3):
exercise(N+1)
24 November 2015
Brookshire Section 5.5

23. Examples
What names are interrogated by the binary search when searching for the name Joe in the list Alice, Brenda, Carol, Duane, Evelyn, Fred, George, Henry, Irene, Joe, Karl, Larry, Mary, Nancy and Oliver?
What is the maximum number of entries that must be interrogated when applying the binary search to a list of 200 entries?
24 November 2015
Brookshear, Section 5.5