1. CSE 501N Fall ‘09 12: Recursion and Recursive Algorithms 8 October 2009 Nick Leidenfrost

2. 2 Lecture Outline Lab 4 questions Recursion Recursive Algorithms

3. 3 Recursion Recursion is a fundamental programming technique that can provide an elegant solution certain kinds of problems We will look at  thinking & programming in a recursive manner  the correct use of recursion  recursion examples

4. 4 Recursive Definitions A recursive definition is one which uses the word or concept being defined in the definition itself A recursive computation solves a problem by using the solution of the same problem with simpler values For recursion to terminate, there must be special cases for the simplest inputs  These are often called “base cases” When defining an English word, a recursive definition is often not helpful but in other situations, a recursive definition can be an appropriate way to express a concept

5. 5 Recursion Two key requirements for recursion success:  Every recursive call must simplify the computation in some way  There must be special cases to handle the simplest computations directly Base cases

6. 6 Recursive Definitions Consider the following list of numbers: 24, 88, 40, 37 Such a list can be defined recursively as follows: That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST The concept of a LIST is used to define itself A LIST is a: number or a: number comma LIST

7. 7 Recursive Definitions The recursive part of the LIST definition is used several times, terminating with the non-recursive part: number comma LIST 24, 88, 40, 37 number comma LIST 88, 40, 37 number comma LIST 40, 37 number 37

8. 8 Infinite Recursion All recursive definitions have to have a non- recursive part  (The base case) If they didn't, there would be no way to terminate the recursive path  Such a definition would cause infinite recursion This problem is similar to an infinite loop, but the non-terminating "loop" is part of the definition itself

9. 9 Recursive Definitions N!, for any positive integer N, is defined to be the product of all integers between 1 and N inclusive This definition can be expressed recursively as: 1! = 1 N! = N * (N-1)! A factorial is defined in terms of another factorial Eventually, the base case of 1! is reached

10. 10 Recursive Definitions 5! 5 * 4! 4 * 3! 3 * 2! 2 * 1! 1 2 6 24 120

11. 11 Recursive Programming A method in Java can invoke itself  Called a recursive method The code of a recursive method must be structured to handle both the base case and the recursive case Each call to the method sets up a new execution environment, with new parameters and local variables As with any method call, when the method completes, control returns to the method that invoked it (which may be an earlier invocation of itself)

12. 12 Recursive Programming Consider the problem of computing the sum of all the numbers between 1 and any positive integer N This problem can be recursively defined as:

13. 13 Recursive Programming // This method returns the sum of 1 to num public int summation (int num) { int result; if (num == 1) result = 1; else result = num + summation(n-1); return result; }

14. 14 Recursive Programming main sum sum(3) sum sum(1) sum sum(2) result = 1 result = 3 result = 6

15. 15 Recursive Programming Note that just because we can use recursion to solve a problem, doesn't mean we should For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version You must carefully decide whether recursion is the correct technique for any problem

16. 16 Indirect Recursion A method invoking itself is considered to be direct recursion A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again This is called indirect recursion, and requires all the same care as direct recursion It is often more difficult to trace and debug

17. 17 Indirect Recursion m1m2m3 m1m2m3 m1m2m3

18. 18 Detecting Palindromes How can we determine whether a String is a palindrome or not?  Iterative solution?  Recursive solution?

19. 19 Detecting Palindromes String text = “able was I ere I saw elba”; public boolean isPalindrome (String text, int start, int end) { if (start == end) return true; else if (text.getCharAt(start) == text.getCharAt(end)) { return isPalindrome(start++, end--); } else { return false; } isPalindrome(text, start, end) returns whether the substring of text from start to end is a palindrome

20. 20 Pascal’s Triangle 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 [ Example on Board ] int computePascalTriangleValue (int col, int row) { // … }

21. 21 Tiled Pictures Consider the task of repeatedly displaying a set of images in a mosaic  Three quadrants contain individual images  Upper-left quadrant repeats pattern The base case is reached when the area for the images shrinks to a certain size

22. 22 Tiled Pictures

23. 23 Tiled Pictures private final int MIN = 20; public void drawPictures(int size, Graphics page) { page.drawImage(everest, 0, size/2, size/2, size/2); page.drawImage(goat, size/2, 0, size/2, size/2); page.drawImage(world, size/2, size/2, size/2, size/2); if (size > MIN) { drawPictures(size/2, page); }

24. 24 Fractals A fractal is a geometric shape made up of the same pattern repeated in different sizes and orientations The Koch Snowflake is a particular fractal that begins with an equilateral triangle To get a higher order of the fractal, the sides of the triangle are replaced with angled line segments

25. 25 Koch Snowflakes Becomes

26. 26 Koch Snowflakes

27. 27 Koch Snowflakes

28. 28 Conclusion Questions? Midterm in class on Tuesday  Email me questions if you have them I will be in lab now