1. ©TheMcGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 15 * Recursive Algorithms

2. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Basic Elements of Recursion * A recursive method is a method that contains a statement that makes a call to itself. * Any recursive method will include the following three basic elements: A test to stop or continue the recursion. An end case that terminates the recursion. A recursive call that continues the recursion.

3. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recursive Factorial Method * Factorial: n! = n*(n-1)*(n-2)* … * 2 * 1 public int factorial(int N) { if (N == 1){ return 1; } else { return N * factorial(N-1); }

4. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example1: Recursive Directory Listing * Recursive algorithms may be used for nonnumerical applications. * This example of a recursive algorithm will list the file names of all files in a given directory of a hard disk and its subdirectories.

5. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Preliminaries * We create a new File object by passing the name of a file or a directory: File file = new File(“D:/Java/Projects”); * To get an array of names of files and subdirectories, we use the list method. String[] fileList = file.list();

6. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Directory Listing * For each file in the directory if it is a file print it out otherwise it is a subdirectory do a directory listing

7. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 2: Anagram * We can use a recursive method to derive all anagrams of a given word. * When we find the end case (an anagram), we will print it out.

8. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Anagram Algorithm * How to generate all the anagrams of a word by using recursion.

9. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Finding Anagrams * We find the anagrams by rotating the positions of the letters. * The trick is, when the method is called recursively, we pass a word with the first letter cut off. So the words being passed to successive calls are getting shorter and shorter. * However, we must access all letters of the word in order to print it out.

10. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Anagrams * We solve this problem by passing two parameters: the prefix and the suffix of a word. * In each successive call, the prefix increases by one letter and the suffix decreases by one letter. * When the suffix becomes one letter only, the recursion stops.

11. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 3: Towers of Hanoi * The goal of the Towers of Hanoi puzzle is to move N disks from peg 1 to peg 3: You must move one disk at a time. You must never place a larger disk on top of a smaller disk.

12. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Towers of Hanoi with N=4 disks

13. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Towers of Hanoi * This puzzle can be solved effectively using recursion. * The top N-1 disks must be moved to peg 2, allowing you to then move the largest disk from peg 1 to peg 3. * You can then move the N-1 disks from peg 2 to peg 3.

14. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recursive Sorting Algorithms * There are two commonly used sorting methods that use recursion Merge sort: sort each half of the array and merge the sorted halves together Quicksort: use a pivot value to separate the array into two parts * Both these methods are more efficient that the sorts we’ve seen before N log N instead of N 2

15. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recursion vs. Iteration * Loops and recursion have similar results * Any iterative algorithm could be expressed recursively processing all the elements of a list is equivalent to process the first element process the rest of the list * Iteration is more efficient * Recursion requires a new entry on the run-time stack for each recursive call

16. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. When Not to Use Recursion * Recursion does not always provide an efficient or natural way to express the solution to a problem. public int fibonacci(int N){ if (N == 0 )|| N ==1){ return 1; //end case }else{ //recursive case return fibonacci(N-1) + fibonacci(N- 2); }

17. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fibonacci * Recursive calls to compute fibonacci(5).

18. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recursive Fibonacci * This method is succinct and easy to understand, but it is very inefficient. A nonrecursive version is just as easy to understand.

19. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Non-Recursive Fibonacci public int fibonacci(int N){ int fibN, fibN1, fibN2, cnt; if (N == 0 || N == 1){ return 1; }else{ fibN1 = fibN2 = 1; cnt = 2; while (cnt <= N){ fibN = fibN1 + fibN2; //get next Fib. no. fibN1 = fibN2; fibN2 = fibN; cnt++; } return fibN; }}

20. ©The McGraw-Hill Companies, Inc. Permission required for reproduction or display. When to Use Recursion * In general, use recursion if A recursive solution is natural and easy to understand. A recursive solution does not result in excessive duplicate computation. The equivalent iterative solution is too complex.