Notes

Outline

Week 4 Recursive Algorithms

Concept of Recursion We have already seen that a method in a class can call other methods, either in the same or other classes • However a method in a class can also call itself • This self-referential behavior is known as recursion • Recursion is an extremely powerful technique for expressing certain complex programming tasks, as it provides a very natural way to decompose problems • Despite this, there are costs associated with recursion and the careful programmer will always be aware of these

Recursion The factorial of N  is the product of the first N positive integers:   N * (N – 1) * (N – 2 ) * . . . * 2 * 1 The factorial of N can be defined recursively as

Recursive Method An recursive method  is a method that contains a statement (or statements) that makes a call to itself. Implementing the factorial of N recursively will result in the following method.

What happens in the method call?

Local variables Each recursive call to the method creates a “new copy” of the method – all the local variables are created again • The compiler keeps track of them all so that the programmer cannot mix them up factorial(4)has a parameter n equal to 4 factorial(3) has its own parameter n, equal to 3 factorial(2) has its own parameter n, equal to 2 factorial(1) has its own parameter n, equal to 1 • This means that you should not use recursion to replace straightforward iteration (like factorial), particularly if the loops are performed thousands or millions of times.

Components of recursive definition • Every valid recursive method definition needs two parts – the base case and the recursive part • The base case gives the value that the method should return for some specified parameters – Usually the base case represents some sort of “trivial” case, such as a zero parameter or an empty list etc. • The recursive part expresses the value to be returned in terms of another call to the same method, but with different parameters – To work correctly, the “different parameters” must be closer to the base case (in some sense)

Order is IMPORTANT! public long factorial(int n) { return n*factorial(n-1); if (n == 0) { return 1; } } You must put the base case before the recursive part of the definition or bad things will happen – in this case a stack overflow will result!

More than one base case There is no need to have just one base case • The (mathematical) definition of the Fibonacci numbers is a recursive definition with two base cases • This gives the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Be careful public int fibonacci(int n) { if (n == 0 || n == 1) { return 1; } else { return fibonacci(n-1)+fibonacci(n-2); } } • This looks fine, but in fact is disastrously slow because it unnecessarily repeats some calculations over and over again!

Directory Listing Here’s a sample recursive method for non-numerical application. The routine lists the names of all files in a given directory and its subdirectories.

Anagram Here’s the second sample recursive method for nonnumerical application. The routine lists all anagrams of a given word.

Anagram Solution: Rotate and Recurse The basic idea is to recurse on a sub-word after every rotation. Here’s how:

Anagram Method

Example What does the method funny return, in terms of its argument? This code does compile and run. Recall that substr(a, b)creates a string from the original, using indices from A to b-1 .    private static String funny( String thing ) {         String result = thing;         int last = thing.length() - 1;         if ( last > 0 ) {             result = thing.charAt( last )                      + funny( thing.substring( 0, last ) );         } Show the output that would result from executing the following instruction in the "main" method: System.out.println( funny("hello") );

Example Write a recursive or non-recursive method powerOfTwo that computes 2 to the power of n.  You may assume that n is a positive integer.  You are not allowed to use the Math class.