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Chapter 15

*Recursive Algorithms

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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.

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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);}

}

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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.

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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();

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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

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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.

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Anagram Algorithm

*How to generate all the anagrams of a word by using recursion.

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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.

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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.

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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.

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Towers of Hanoi with N=4 disks

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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.

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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 N2

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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

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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);

}}

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Fibonacci

*Recursive calls to compute fibonacci(5).

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Recursive Fibonacci

*This method is succinct and easy to understand, but it is very inefficient. A nonrecursive version is just as easy to understand.

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Non-Recursive Fibonaccipublic 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; }}

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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.