1. Chapter 1 Introduction Recursion
Section 1.3

2. Recursion Recursive function Base Case Recursive Call
A function that is defined in terms of itself
E.g. f(0)=0 and f(x) = 2f(x-1)+x2
Implementation in C or C++
Base Case
Recursive Call

3. Recursion Examples Example: evaluation of f(x) = 2f(x-1)+x2
What proof technique will you use to prove g(N) = N2?
g(1) = 1
g(k) = 2k g(k-1), k > 1
Bookkeeping
Backtrack

4. Recursion and the Program Stack
void reverse() {
char ch;
cin.get(ch);
if(ch != ‘\n’)
reverse();
cout.put(ch); }
Input: ABC’\n’
ch = ‘\n’
(to 204)
Line 201
ch = ‘C’
(to 204)
Line 202
Output: C
ch = ‘B’
(to 204)
Line 203
Output: B
Line 204
ch = ‘A’
(to main)
Output: A 4 4

5. Write Recursive Code p(x, 0) = 1 p(x, n) = x·p(x, n-1), n > 0
(to 203)
Return value =
float power(float x, unsigned int n ) {
if(n==0)
return 1.0;
else
return x*power(x, n-1); }
main: y = 1 + 3*power(2.0,2); 1
x = 2
n = 1
(to 203)
Return value =
Line 201
Line 202 2
x = 2
n = 2
(to 100)
Return value =
Line 203 4
Line 100 5 5

6. Ensuring Recursive Programs Terminate
What happens if you make the call f(-1) in the implementation of f(x) = 2f(x-1)+x2 ?
Another non-terminating recursive function
What happens on
call to bad(1)?
And bad(2)?
And bad(3)?

7. Making sure recursive programs terminate (contd.)
Two rules for writing correct recursive programs
Define a base case
This is what terminates recursion
Make progress
At each step make progress towards the base case
More rules for writing clean and efficient recursive programs
Design rule
Assume that all recursive calls work.
Compound interest rule
Never duplicate work by making same recursive call twice!
E.g. Fibonacci series f(i) = f(i-1) + f(i-2). Why??

8. Example Code
Code examples are in the following directory on the CS file server
~cop4530/fall11/examples/Lec2
Implementation of Fibonacci numbers
examples/Lec2/fibonacci_recursive.cpp
examples/Lec2/fibonacci_nonrecursive.cpp
Compare the running times of these programs
Experiment with the other code examples
Can you explain what reverse2.cpp does?