Lecture Objectives

1. Lecture Objectives • Learn how to think recursively • Trace a recursive method • Prove a recursive method is correct • Write recursive algorithms and methods for searching arrays CS340

2. Recursion • Recursion can solve many programming problems that are difficult to solve linearly • In artificial intelligence, recursion is used to write programs that • play games of chess • prove mathematical theorems • recognize patterns • Recursive algorithms can • compute factorials • compute a greatest common divisor • process data structures (strings, arrays, linked lists, etc.) • search efficiently using a binary search • find a path through a maze, and more CS340

3. Recursive Thinking CS340

4. Recursive Thinking • Recursion reduces a problem into one or more simpler versions of itself CS340

5. Recursive Thinking (cont.) CS340

6. Recursive Thinking (cont.) Recursion of Process Nested Dreams A child couldn't sleep, so her mother told a story about a little frog, who couldn't sleep, so the frog's mother told a story about a little bear, who couldn't sleep, so bear's mother told a story about a little weasel ...who fell asleep. ...and the little bear fell asleep; ...and the little frog fell asleep; ...and the child fell asleep. CS340

7. Recursive Thinking (cont.) • Consider searching for a target value in an array • With elements sorted in increasing order • Compare the target to the middle element • If the middle element does not match the target • search either the elements before the middle element • or the elements after the middle element • Instead of searching n elements, we search n/2 elements CS340

8. Recursive Thinking (cont.) Recursive Algorithm to Search an Array if the array is empty return -1 as the search result else if the middle element matches the target return the subscript of the middle element as the result else if the target is less than the middle element recursively search the array elements before the middle element and return the result else recursively search the array elements after the middle element and return the result CS340

9. Steps to Design a Recursive Algorithm • Base case: • for a small value of n, it can be solved directly • Recursive case(s) • Smaller versions of the same problem • Algorithmic steps: • Identify the base case and provide a solution to it • Reduce the problem to smaller versions of itself • Move towards the base case using smaller versions CS340

10. Recursive Algorithm for Finding the Length of a String if the string is empty (has no characters) the length is 0 else the length is 1 plus the length of the string that excludes the first character CS340

11. Recursive Algorithm for Finding the Length of a String (cont.) /** Recursive method length @param str The string @return The length of the string */ public static int length(String str) { if (str == null || str.equals("")) return 0; else return 1 + length(str.substring(1)); } CS340

12. Proving that a Recursive Method is Correct • Proof by induction • Prove the theorem base case is true • Assuming that the theorem is true for n, prove it is true for n+1 • Recursive proof is similar to induction. Verify that • The base case is recognized and solved correctly • Each recursive case makes progress towards the base case • If all smaller problems are solved correctly, then the original problem also is solved correctly CS340

13. Tracing a Recursive Method • Unwinding the recursion is the process of • returning from recursive calls • computing the partial results CS340

14. Run-Time Stack and Activation Frames • Java maintains a run-time stack • Activation frames save information in the run-time stack • The activation frame contains storage for • method arguments • local variables (if any) • the return address of the instruction that called the method • Whenever a new method is called (recursive or not), Java pushes a new activation frame into the run-time stack CS340

15. Run-Time Stack and Activation Frames (cont.) CS340

16. Run-Time Stack and Activation Frames CS340

17. Recursive Algorithm for Printing String Characters in Reverse /** Recursive method printCharsReverse post: The argument string is displayed in reverse, one character per line @param str The string */ public static void printCharsReverse(String str) { if (str == null || str.equals("")) return; else { printCharsReverse(str.substring(1)); System.out.println(str.charAt(0)); } } CS340

18. Recursive Algorithm for Finding Duplicate characters /** Recursive method removeDuplicates post: The word without duplicates @param word The string */ public String removeDuplicates(String word) { if(word == null || word.length() <= 1) return word; else if( word.charAt(0) == word.charAt(1) ) return removeDuplicates(word.substring(1, word.length())); else return word.charAt(0) + removeDuplicates(word.substring(1, word.length())); } CS340

19. Recursive Definitions of Mathematical Formulas CS340

20. Recursive Definitions of Mathematical Formulas • Mathematicians often use recursive definitions of formulas • Examples include: • factorials • powers • greatest common divisors (gcd) CS340

21. Factorial of n: n! • The factorial of n, or n! is defined as follows: 0! = 1 n! = n x (n -1)! (n > 0) • The base case: n equal to 0 • The second formula is a recursive definition CS340

22. Factorial of n: n! (cont.) • The recursive definition can be expressed by the following algorithm: ifn equals 0 n! is 1 else n! = n x (n – 1)! • The last step can be implemented as: return n * factorial(n – 1); CS340

23. Factorial of n: n! (cont.) public static int factorial(int n) { if (n == 0) return 1; else return n * factorial(n – 1); } CS340

24. Infinite Recursion and Stack Overflow • Call factorialwith a negative argument, what will happen? StackOverflowException CS340

25. Recursive Algorithm for Calculating xn Eclipse example CS340

26. Recursive Algorithm for Calculating gcd • The greatest common divisor (gcd) of two numbers is the largest integer that divides both numbers • The gcd of 20 and 15 is 5 • The gcd of 36 and 24 is 12 • The gcd of 38 and 18 is 2 CS340

27. Recursive Algorithm for Calculating gcd (cont.) • Given 2 positive integers m and n (m > n) if n is a divisor of m gcd(m, n) = n else gcd (m, n) = gcd (n, m % n) CS340

28. Recursive Algorithm for Calculating gcd (cont.) /** Recursive gcd method (in RecursiveMethods.java). pre: m > 0 and n > 0 @param m The larger number @param n The smaller number @return Greatest common divisor of m and n */ public static double gcd(int m, int n) { if (m % n == 0) return n; else if (m < n) return gcd(n, m); // Transpose arguments. else return gcd(n, m % n); } CS340

29. Recursion Versus Iteration • Similarities between recursion and iteration • Iteration • Loop repetition condition determines whether to repeat the loop body or • exit from the loop • Recursion • The condition usually tests for a base case • You can always write an iterative solution to a recursion • But can you always write a recursion for an iteration problem? • Which one is best? CS340

30. Recursion Versus Iteration • Difference between iteration and recursion is: • with iteration, each step clearly leads onto the next, like stepping stones across a river, • in recursion, each step replicates itself at a smaller scale, so that all of them combined together eventually solve the problem. CS340

31. Iterative factorial Method /** Iterative factorial method. pre: n >= 0 @param n The integer whose factorial is being computed @return n! */ public static int factorialIter(int n) { int result = 1; for (int k = 1; k <= n; k++) result = result * k; return result; } CS340

32. Efficiency of Recursion • Recursive methods often have slower execution times • Why? • What about memory and recursion vs iteration? • Why would we ever prefer recursion? CS340

33. Fibonacci Numbers • Fibonacci numbers were invented to model the growth of a rabbit colony fib1 = 1 fib2 = 1 fibn = fibn-1 + fibn-2 • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … CS340

34. An Exponential Recursive fibonacci Method CS340

35. Efficiency of Recursion: Exponential fibonacci Inefficient CS340

36. An O(n) Recursive fibonacci Method CS340

37. An O(n) Recursive fibonacci Method (cont.) • Non-recursive wrapper method: /** Wrapper method for calculating Fibonacci numbers (in RecursiveMethods.java). pre: n >= 1 @param n The position of the desired Fibonacci number @return The value of the nth Fibonacci number */ public static int fibonacciStart(int n) { return fibo(1, 0, n); } CS340

38. Efficiency of Recursion: O(n) fibonacci Efficient CS340

39. Efficiency of Recursion: O(n) fibonacci • Method fibo is an example of tail recursion or last-line recursion • Why is it more effective? CS340

40. Recursive Array Search CS340

41. Recursive Array Search • Simplest way to search is a linear search • Start with the first element • Examine one element at a time • End with the last • (n + 1)/2 elements examined in a linear search • If the target is not in the list, n elements are examined • What is the complexity? CS340

42. Recursive Array Search (cont.) • Base cases for recursive search: • Empty array, target can not be found; result is -1 • First element of the array being searched = target? • Yes? Then result is the subscript of first element • No? Then recursive step searches the rest of the array, excluding the first element CS340

43. Algorithm for Recursive Linear Array Search ifthe array is empty the result is –1 else if the first element matches the target the result is the subscript of the first element else search the array excluding the first element and return the result CS340

44. Implementation of Recursive Linear Search CS340

45. Implementation of Recursive Linear Search (cont.) • A non-recursive wrapper method: /** Wrapper for recursive linear search method @param items The array being searched @param target The object being searched for @return The subscript of target if found; otherwise -1 */ public static int linearSearch(Object[] items, Object target) { return linearSearch(items, target, 0); } CS340

46. Implementation of Recursive Linear Search (cont.) CS340

47. Design of a Binary Search Algorithm • Array has to be sorted • Base cases • The array is empty • Element being examined matches the target • Compares the middle element for a match with the target • Excludes the half of the array within which the target cannot lie CS340

48. Binary Search Algorithm (cont.) ifthe array is empty return –1 as the search result else if the middle element matches the target return the subscript of the middle element as the result else if the target is less than the middle element recursively search the array elements before the middle element and return the result else recursively search the array elements after the middle element and return the result CS340

49. Binary Search Algorithm target First call England China Denmark England Greece Japan Russia USA first = 0 last = 6 middle = 3 CS340

50. Binary Search Algorithm (cont.) target Second call England China Denmark England Greece Japan Russia USA first = 0 last = 2 middle = 1 CS340