1. Recursion Gordon College CPS212
Adapted from Nyhoff: ADTs, Data Structures, and Problem Solving

2. Recursive Functions
Recursive programming is based off of Recursive Formulas
Definition of Recursive Formula
a formula that is used to determine the next term of a sequence using one or more of the preceding terms.
Example of Recursive Formula
The recursive formula for the sequence 5, 20, 80, 320, ... is an = 4 an-1
How would you program such a thing?

3. Program Solution int series(int n) { } if (n == 1) return 5;
return 4 * series(n-1); }
Base case
Recursive case

4. Recursion
A function is defined recursively if it has the following two parts:
An anchor or base case
The function is defined for one or more specific values of the parameter(s)
An inductive or recursive case
The function's value for current parameter(s) is defined in terms of previously defined function values and/or parameter(s)
How would you normally write a powers – OK let’s say you can’t use the powers operator…use a loop with so that it can be variable!!

5. Recursive Example
Consider a recursive power function double power (double x, unsigned n) { if ( n == 0 ) return 1.0; else return x * power (x, n-1); }
Which is the anchor?
Which is the inductive or recursive part?
How does the anchor keep it from going forever?
No loops used here: simply call: x = power(3.0,3);

6. Recursive Example Note the results of a call Recursive calls
Resolution of the calls
Let’s check out the run-time stack for this example

7. Recursive Factorial another example
n! = 1 x 2 x …x n, for n > 0
n! = (n – 1)! X n
5! = 5 x 4! 120
4! = 4 x 3! 24
3! = 3 x 2! 6
2! = 2 x 1! 2
1! =
This is born to be used for recursion…see if you can construct the function to perform this recursively??
How would this be written conventionally??
int fact(int n) {
if (n==1) return 1;
return n * fact(n – 1); }

8. Another Example of Recursion
Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34 f1 = 1, f2 = 1 … fn = fn -2 + fn -1
A recursive function double Fib (unsigned n) { if (n <= 2) return 1; else return Fib (n – 1) + Fib (n – 2); }
Add the previous two in the series to get the next.
A loop does nicely in this case

9. A Bad Use of Recursion Why is this inefficient?
Note the recursion tree
Remember each function call is overhead…
This is can be done in a much more efficient manner using a normal algorithm – with a loop.

10. Uses of Recursion Binary Search See source code
Note results of recursive call
Recursive functions need not return values via a return statement – but may instead return values via the parameters or may return no values.

11. Uses of Recursion Palindrome checker
A palindrome has same value with characters reversed racecar
Recursive algorithm for an integer palindrome checker
If numDigits <= 1 return true
Else check first and last digits num/10numDigits-1 and num % 10
if they do not match return false
If they match, check more digits Apply algorithm recursively to: (num % 10numDigits-1)/10 and numDigits - 2
first digit – /10^5 = / =
Last digit % 10 = remainder is 4
New number ( % )/10 = 4 remainder is 56654/10  5665
Length of integer: ceil(log10(n))

12. Recursion Example: Towers of Hanoi
Think Recursive algorithm
Task
Move disks from left peg to right peg
When disk moved, must be placed on a peg
Only one disk (top disk on a peg) moved at a time
Larger disk may never be placed on a smaller disk

13. Recursion Example: Towers of Hanoi
Identify base case: If there is one disk move from A to C
Inductive solution for n > 1 disks
Move topmost n – 1 disks from A to B, using C for temporary storage
Move final disk remaining on A to C
Move the n – 1 disk from B to C using A for temporary storage

14. Towers of Hanoi: Solution
void hanoi(int n, const string& initNeedle,
const string& endNeedle, const string& tempNeedle) {
if (n == 1)
cout << "move " << initNeedle << " to "
<< endNeedle << endl;
else
hanoi(n-1,initNeedle,tempNeedle,endNeedle);
hanoi(n-1,tempNeedle,endNeedle,initNeedle); }

15. Towers of Hanoi: Solution
string beginneedle = "A", middleneedle = "B", endneedle = "C";
hanoi(3, beginneedle, endneedle, middleneedle);
output
The solution for n = 3
move A to C
move A to B
move C to B
move B to A
move B to C

16. Recursion Example: Towers of Hanoi
Note the graphical steps to the solution
Stopped 2/27/06

17. Recursion Example: Parsing
Examples so far are direct recursion
Function calls itself directly
Indirect recursion occurs when
A function calls other functions
Some chain of function calls eventually results in a call to original function again
An example of this is the problem of processing arithmetic expressions

18. Recursion Example: Parsing
Parser is part of the compiler
Input to a compiler is characters
Broken up into meaningful groups
Identifiers, reserved words, constants, operators
These units are called tokens
Recognized by lexical analyzer
Syntax rules applied

19. Recursion Example: Parsing
Parser generates a parse tree using the tokens according to rules below:
An expression: term + term | term – term | term
A term: factor * factor | factor / factor | factor
A factor: ( expression ) | letter | digit
Note the indirect recursion

20. Implementing Recursion: The Runtime Stack
Activation record created for each function call
Activation records placed on run-time stack
Recursive calls generate stack of similar activation records

21. Implementing Recursion: The Runtime Stack
When base case reached and successive calls resolved
Activation records are popped off the stack