1. Recursion
Lakshmish Ramaswamy

2. Basics of Recursion
A recursive method makes a call to itself directly or indirectly
Is this not endless circular logic?
Call is for a simpler instance
Several real-world applications are recursive
Dictionary lookup for definitions
Scanning files on a computer
Computer languages

3. A Simple Example Calculate sum of first N integers (S(N))
We know S(1) = 1
S(N) can be expressed as S(N) = S(N-1) + N
S(2) = 2 + S(1)
S(3) = S(2) + 3
S(N) = N(N+1)/2 <---- Closed form
Closed form are often complex
Recursive expressions might be simpler

4. Recursive Method for S(N)

5. Points to Note
Function calls a clone of itself (note the difference in parameters)
Only one clone is active at any instance
Computer does all the book keeping for you
Base case – Instance that can be solved without recursion
S(1) is the base case
Recursive call should make progress towards base case

6. Two Rules Always have at least one base case solved without recursion
Any recursive call should make progress towards base case

7. Second Example Printing numbers in any base
Simpler case – Printing number in decimal form
No method that can print an integer
Method that can print a single character
Digits are treated as characters
print(2457) can be expressed as print(245) followed by printing 7 (via character printing method)

8. Decimal Printing Method
Identify the base condition?
How is the progress done?

9. Printing Number in any Base
Exception if base > 16
Infinite loop if base is 1
Notice identical call to the method (No progress)

10. Robust Implementation

11. How is Recursion Supported
Method invocation requires bookkeeping
Recursion is no different (may be a special case)
Activation records for implementing methods
Activation records hold important information like parameter values & local variables

12. Stacks for Activation Records
Stack is a good data structure for reversing order
First-In-Last-Out paradigm
Why stacks?
Notice that methods return in reverse order of invocation
Activation record at top is that of currently active method
When a method is called, its activation record is pushed on the stack
Top record is popped when the method returns

13. Stacks and Recursive Methods
Each method invocation results in an activation record
The base case is the last one to be pushed in and the first one to pop out

14. Fibonacci Numbers Definition Very interesting properties
FN = FN-1 + FN-2
F1 = 1
F0 = 0
Very interesting properties
Sum of first N Fibonacci numbers < FN+2
Sum of squares of two consecutive Fibonacci numbers is also a Fibonacci number
Natural recursive implementation

15. Simple but Inefficient Implementation

16. What is the Problem?
Each call to fib(n-1) and fib(n-2) results in calling fib(n-3)
Keeps getting worse – Observe how many times F2, F1 and F0 are called
Lesson – Avoid duplicate work of solving same instance in separate duplicate calls

17. Factorial Computation