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Recursion
COMP104 Lecture 35 / Slide 2
Recursion: Example 0

What does the following program do?
#include <iostream>
using namespace std;
int fac(int n){
// Assume n >= 0
int product;
if(n <= 1)
return 1;
product = n * fac(n-1);
return product;
}
void main(){ // driver function
int number;
while(true){
cout << "Enter integer (negative to stop): ";
cin >> number;
if(number < 0)
break;
cout << fac(number) << endl;
}
}
COMP104 Lecture 35 / Slide 3
Recursion: Example 0

Assume the number typed is 3.
fac(3) :
3 <= 1 ?
product3 = 3 * fac(2)
fac(2) :
2 <= 1 ?
product2 = 2 * fac(1)
fac(1) :
1 <= 1 ?
return 1
product2 = 2 * 1 = 2
return product2
product3 = 3 * 2 = 6
return product3
fac(3) has the value 6
No.
No.
Yes.
COMP104 Lecture 35 / Slide 4
Recursion
 Recursion
is one way to decompose a task into
smaller subtasks.
 At least one of the subtasks is a smaller
example of the same task.
 The smallest example of the same task has a
non-recursive solution.
Example: The factorial function
n! = n * (n-1) * (n-2) * ... * 1
or
n! = n * (n-1)! and 1! = 1
COMP104 Lecture 35 / Slide 5
Recursion
A recursive
solution may be simpler to
write (once you get used to the idea) than
a non-recursive solution.
But a recursive solution may not be as
efficient as a non-recursive solution of the
same problem.
COMP104 Lecture 35 / Slide 6
Writing Recursive Function
 A recursive
function always consists of two
parts:

basis case (stopping criterion)
e.g.

inductive case
e.g.
 It
F(n) = 1 when n = 1
F(n) = n * F(n-1)
has the same mathematical meaning as
induction
COMP104 Lecture 35 / Slide 7
Iterative Factorial
// Non-recursive factorial function
// Compute the factorial using a loop
int fac(int n){
// Assume n >= 0
int k, product;
if(n <=1)
return 1;
product = 1;
for(k=1; k<=n; k++)
product*= k;
return product;
}
COMP104 Lecture 35 / Slide 8
Other Recursive Applications
Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
where each number is the sum of the preceding
two.
 Recursive



definition:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2)
COMP104 Lecture 35 / Slide 9
Other Recursive Applications
Binary
search:
 Compare
search element with middle
element of the array:
 If not equal, then apply binary search to
half of the array (if not empty) where the
search element would be.
COMP104 Lecture 35 / Slide 10
Recursion General Form
 How
to write recursively?
int rec(1-2 parameters){
if(stopping condition)
return stopping value;
// second stopping condition if needed
return value/rec(revised parameters)
+-*/ rec(revised parameters);
}
COMP104 Lecture 35 / Slide 11
Recursion: Example 1
to write exp(int x, int y)
recursively?
How
int exp(int x, int y){
if(y==0)
return 1;
return x * exp(x, y-1);
}
COMP104 Lecture 35 / Slide 12
Recursion: Example 2
 Write
a recursive function that takes a double
array and its size as input and returns the sum of
the array:
double asum(int a[], int size){
if(size==0)
return 0;
return asum(a, size-1)+a[size-1];
}
COMP104 Lecture 35 / Slide 13
Recursion: Example 3
 Write
a recursive function that takes a double
array and its size as input and returns the product
of the array:
double aprod(int a[], int size){
if(size==0)
return 1;
return aprod(a, size-1)*a[size-1];
}
COMP104 Lecture 35 / Slide 14
Recursion: Example 4
Write a recursive function that counts the number of zero
digits in a non-negative integer
 zeros(10200) returns 3

int zeros(int n){
if(n==0)
return 1;
if(n < 10)
return 0;
if(n%10 == 0)
return 1 + zeros(n/10);
else
return zeros(n/10);
}
COMP104 Lecture 35 / Slide 15
Recursion: Example 5
Write
a recursive function to determine how
many factors m are part of n. For example,
if n=48 and m=4, then the result is 2
(48=4*4*3).
int factors(int n, int m){
if(n%m != 0)
return 0;
return 1 + factors(n/m, m);
}