Lesson 24: Recursive Algorithms #1 (W07D3)

Transcription

1 Lesson 24: Recursive Algorithms #1 (W07D3) Balboa High School Michael Ferraro October 2, / 52

2 Do Now public static int mysteryfcn(int n) { //precondition: n > 0 int result = 1; while ( n >= 1 ) { result *= n; n--; } } return result; Consider the code above for the mystery function above, mysteryfcn(). 1 What does mysteryfcn(3) return? Make a table! 2 How about mysteryfcn(5)? 3 What s a more fitting name for mysteryfcn()? 2 / 52

3 Aim Students will begin working with recursive algorithms, learning their two key attributes and tracing the execution of examples. 3 / 52

4 Demonstration (1 of 2) Watch as I write a program to... evaluate an infinite sum involving alternating ± terms ± via if()/else powers of 1 infinite loops via 1==1 true 4 / 52

5 Demonstration (2 of 2) Watch as I write a program to... compute an estimate of π using Euler s 1 infinite sum: π 2 6 = solve for π Scanner to ask for # of iterations 1 Leonhard Euler was a famous Swiss mathematician. See here. 5 / 52

6 Check-In Re: PS #4a Due date: Monday, 5 October 2015, before 5th Period Extra help over coming days: Room 319 during lunch and study hall Room 124 when afterschool help announced Today s material is related to PS #4b Where are you getting stuck in 3.5 (book problems)? 6 / 52

7 What is Recursion? Recursion, in programming, happens when a procedure 2 accomplishes a task by calling upon itself. Some problems lend themselves to being solved in such a way, while others are more easily solved using iterative 3 algorithms. Any method that can be written recursively can be written iteratively though doing so may not be trivial! 2 Think method in Java. 3 Think while() loops. 7 / 52

8 Example of Recursion: Factorial Recall the definition of factorial: n! = n (n 1) (n 2) / 52

9 Example of Recursion: Factorial Recall the definition of factorial: n! = n (n 1) (n 2) Ex: 5! = = / 52

10 Example of Recursion: Factorial Recall the definition of factorial: n! = n (n 1) (n 2) Ex: 5! = = 120 You can write an iterative procedure to figure out n!: public static int factorialiter(int n) { int result = 1; while ( n >= 1 ) { result *= n; n--; } } return result; 10 / 52

11 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 4! 11 / 52

12 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 3!) 12 / 52

13 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 (3 2!)) 13 / 52

14 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 (3 (2 1!))) 14 / 52

15 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 (3 (2 (1)))) 15 / 52

16 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 (3 (2))) 16 / 52

17 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (4 (6)) 17 / 52

18 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 (24) 18 / 52

19 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = / 52

20 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 4! in general terms: n! = n (n 1)! 20 / 52

21 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 4! in general terms: n! = n (n 1)! Let s watch this single-stepping video of a broken recursive factorial function. Save this Scratch program file to your Desktop. Start cloud-based Scratch and upload the downloaded program file. Fix the problem so the final value of 5! is / 52

22 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 4! in general terms: n! = n (n 1)! Consider this recursive implementation of a factorial procedure in Java: public static int factorialrec(int n) { return n * factorialrec(n-1); } 22 / 52

23 Example of Recursion: Factorial But you can think of factorial in terms of itself: 5! = 5 4! in general terms: n! = n (n 1)! Consider this recursive implementation of a factorial procedure in Java: public static int factorialrec(int n) { return n * factorialrec(n-1); } Download RecursiveFactorial.java from here, import into a new project called Lesson24, and run it. What happens? Why? 23 / 52

24 How factorialrec() Works Now 24 / 52

25 How factorialrec() Works Now 25 / 52

26 How factorialrec() Works Now 26 / 52

27 How factorialrec() Works Now 27 / 52

28 How factorialrec() Works Now 28 / 52

29 How factorialrec() Works Now 29 / 52

30 How factorialrec() Works Now 30 / 52

31 How factorialrec() Works Now 31 / 52

32 The Two Elements of Recursion All recursive procedures... 1 Make calls to themselves Ex: return n * factorialrec(n-1); 32 / 52

33 The Two Elements of Recursion All recursive procedures... 1 Make calls to themselves Ex: return n * factorialrec(n-1); 2 Have a base case, which acts to stop the recursion this was missing from factorialrec.java! 33 / 52

34 How factorialrec() Should Work 34 / 52

35 How factorialrec() Should Work 35 / 52

36 How factorialrec() Should Work 36 / 52

37 How factorialrec() Should Work 37 / 52

38 How factorialrec() Should Work 38 / 52

39 How factorialrec() Should Work 39 / 52

40 How factorialrec() Should Work 40 / 52

41 How factorialrec() Should Work 41 / 52

42 How factorialrec() Should Work 42 / 52

43 How factorialrec() Should Work 43 / 52

44 Determine the Base Case Whenever factorialrec() is called, check the value of n: if n = 1, simply return 1 otherwise, return as we did before 44 / 52

45 Determine the Base Case Whenever factorialrec() is called, check the value of n: if n = 1, simply return 1 otherwise, return as we did before Update factorialrec(): public static int factorialrec(int n) { if ( n == 1 ) { return 1; } return n * factorialrec(n - 1); } 45 / 52

46 A Simple Example Using a piece of paper, trace the execution of the code below and state its output. public static int simplerecex(int a) { if ( a == 0 ) { return 0; } return 2 + simplerecex(a - 1); } public static void main(string[] args) { int result = simplerecex(4); System.out.println("result is " + result); } 46 / 52

47 Sum of Squares, Part II How could you think of findsumofsquares() in recursive terms? findsumofsquares(n) = n 2 47 / 52

48 Sum of Squares, Part II How could you think of findsumofsquares() in recursive terms? findsumofsquares(n) = n 2 First, consider running the sum in reverse: findsumofsquares(n) = n 2 + (n 1) / 52

49 Sum of Squares, Part II How could you think of findsumofsquares() in recursive terms? findsumofsquares(n) = n 2 First, consider running the sum in reverse: findsumofsquares(n) = n 2 + (n 1) Rewrite the problem in terms of itself: findsumofsquares(n) = n 2 + findsumofsquares(n-1) 49 / 52

50 Sum of Squares, Part II How could you think of findsumofsquares() in recursive terms? findsumofsquares(n) = n 2 First, consider running the sum in reverse: findsumofsquares(n) = n 2 + (n 1) Rewrite the problem in terms of itself: findsumofsquares(n) = n 2 + findsumofsquares(n-1) Base case: What s the least value of n for which we should compute/evaluate findsumofsquares(n)? 50 / 52

51 Sum of Squares, Part II How could you think of findsumofsquares() in recursive terms? findsumofsquares(n) = n 2 First, consider running the sum in reverse: findsumofsquares(n) = n 2 + (n 1) Rewrite the problem in terms of itself: findsumofsquares(n) = n 2 + findsumofsquares(n-1) Base case: What s the least value of n for which we should compute/evaluate findsumofsquares(n)? Next class: You ll work in pairs to write a recursive version of findsumofsquares(). 51 / 52

52 HW Work on completing PS #4a 52 / 52