1. Recursion in Java The answer to life’s greatest mysteries are on the last slide

2. Recursion  Recursion is a technique widely used in programming  In fact some programming languages have no loops, only recursion  Recursion is a key element to advanced studies such as computational intelligence  Beginning programmers usually groan at recursion… real programmers love it!

3. Recursion in Java  A method is recursive if it calls itself.  There are two key things in recursion  Base step: ends the recursion  Recursive step: breaks down the problem to smaller sub problems  Before you code anything, you need to find those two steps… otherwise you may not understand the problem properly

4. Example  Here is a simple recursive java function for called countdown: public void countDown( int value ) { if( value == 0 ) System.out.println( “Blast Off!!!” );//base case, no recursion else{ System.out.println( value ); countDown( value – 1);//recursive call of small problem }} LETS TRACE countDown( 5 ) BY HAND

5. Why Recursion?  Fact: All recursion can be re written as a non recursive form with a loop  Some algorithms are exceedingly complex as a loop, yet very elegant in a recursive form.  Easier to understand  less “bugs” = better programs  Let’s take a look at Paint for a minute…

6. The Flood Fill Algorithm  The recursion is tricky to follow, but look at how simple the code is! public void floodFill( int x, int y, Colour clicked, Colour change ) { if( outOfBounds(x,y) || getColour(x,y) == change || getColour(x,y) != clicked ) return;//nothing to do – base case setColour(x,y,change);//colour is the same as was clicked, change it //Recursive call on neighbours floodFill( x + 1, y, clicked, colour ); floodFill( x, y + 1, clicked, colour ); floodFill( x - 1, y, clicked, colour ); floodFill( x, y - 1, clicked, colour ); }

7. Getting the hang of it?  Let’s try a couple together  Base step  Recursive step  Sum of n numbers  Ex. 1 + 2 + 3 + 4 + 5 = 15  Factorials  0! = 1  3! = 3 x 2 x 1 = 3 x 2!  100! = 100x99x98x..x3x2x1 = 100 x 99!

8. Still Need Practice?  Here are two other famous recursive examples:  Fibonacci numbers  1 st number is 1, 2 nd number is 1  Every number after 2 is the sum of the previous two: 1,1,2,3,5,8,13,…  Euclid’s GCF (from ~300BC)  http://en.wikipedia.org/wiki/Euclidean_algorithm http://en.wikipedia.org/wiki/Euclidean_algorithm  Take the larger of the two and divide it by the smaller  If the remainder is zero, return the smaller number  Otherwise take the GCF of smaller and remainder  Example let’s try 60 and 105

9. Pascal’s Triangle 1 11 121 1331 14641 …

10. Pascal’s Triangle Cont’  What’s the recursive pattern in Pascal’s triangle?  Hint: Number each row and column  What is/are the base case(s)?  What is the recursive case?

11. Performance: Loops vs. Recursion  Why not always use recursion if it makes our code easier to understand?  The key reason is performance  Recursion can quickly overflow the computer’s memory  Each method call has its own “stack” in memory when it is called (holds variables, loops, etc.)  When you make a recursive call, another copy of the method is stored in memory to preserve the state when it returns.  With loops, the variables are reused  Last I had heard, Sun’s java compiler will translate tail recursion into a loop for you  Tail recursion means the recursive call is the last piece of code in the method (nothing is computed after it returns)

12. Recursion in Memory  Consider a call to factorial( 3 ):  factorial(3)  3 * factorial(2)  2 * factorial(1)  2 * factorial(1)  1 * factorial(0) 1 Recursion ends, return to last caller: 1 * 1 = 1  2 * 1 = 2  3 * 2 = 6

13. Recursion in Memory  With a loop it would look like this: public int factorial( int n ) { int fac = 1; for( int counter = 2; counter <= n; counter++ ) fac *= counter; return fac; } A call to factorial( 3 ) is now just: 1 * 2 * 3 = 6

14. The H-Fractal  I’ll code a fractal with you so you can see just how powerful recursion is for problem solving  It would probably take me all day to code this with a loop  I can do it with recursion in about 5 minutes

15. Exercises  Create a recursive method to calculate the sum of squares:  For example, sumSquares( 3 )  Is 1 + 4 + 9 = 14

16. Exercises  Create a recursive method to calculate positive integer exponents  Ex.  findExponent( 3, 2 )  Returns 9

17. Exercises  Create a recursive method to determine if the values in an array are increasing  You need an extra parameter telling you which index to start testing from  isIncr( int[] data, int start )  isIncr( {1,3,5,9},0 ) returns true  isIncr( {3,1,2,4},0 ) returns false

18. Exercises  Create a recursive method to sum the values in an array:  sum( int[] data, int start )  sum( {4,2,5}, 0 )  Returns 11

19. Exercises  Create a recursive method called reverse that reverses the elements in an array  Reverse( int[] data )  Reverse( {1,3,5,2,7 } )  {7,2,5,3,1}

20. Exercises – Reverse Cont’  Sometimes when we are using recursion, we will write a helper method to hold more parameters.  In the previous example to reverse an array you should write a helper method called:  ReverseHelper( int[] data, int start, int end )  When you call Reverse( int[] data ) use the helper method as follows (it does all the work)  ReverseHelper( data, 0, data.length)

21. Exercises  Create a recursive method to determine whether or not a word is a palindrome or not.  A palindrome is a word spelled the same forwards and backwards  For example Ogopogo  public boolean isPalidrome(String word)

22. Exercises  To convert an integer to base 2 (binary) here is an algorithm:  public String toBinary( int n )  If the number is zero, return 0  Otherwise,  Compute the quotient and remainder when dividing by 2  The binary value of the number given is the binary value of the quotient followed by the remainder

23. Binary Numbers  Ex. Binary value of 13: CalculationQuotientRemainder 13/261 6/230 3/211 1/201 0return 0 13 = 01101 in binary

24. Bonus Graphics - Fractals  Three fractals which can be created using recursion are:  Sierpinski’s Gasket  Koch’s Snowflake  Dragon Fractal  Create a recursive graphics program to display any of these fractals at a given level  Demo will be done in class for each fractal and how they can be obtained

25. Life’s Greatest Mysteries Revealed!  The answer to life’s greatest mysteries are on the first slide