1. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Starting Out with Programming Logic & Design Second Edition by Tony Gaddis Chapter 13: Recursion

2. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-2 Chapter Topics 13.1 Introduction to Recursion 13.2 Problem Solving with Recursion 13.3 Examples of Recursive Algorithms

3. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-3 13.1 Introduction to Recursion A recursive module is a module that calls itself –When this happens, it becomes like an infinite loop because there may be no way to break out –Depth of Recursion is the number of times that a module calls itself –Recursion should be written so that it can eventually break away This can be done with an If statement

4. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-4 13.1 Introduction to Recursion

5. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-5 13.2 Problem Solving with Recursion A problem can be solved with recursion if it can be broken down into successive smaller problems that are identical to the overall problems –This process is never required, as a loop can do the same thing –It is generally less efficient to use than loops because it causes more overhead (use of system resources such as memory)

6. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-6 13.2 Problem Solving with Recursion How it works –If the problem can be solved now, then the module solves it and ends –If not, then the module reduces it to a smaller but similar problem and calls itself to solve the smaller problem –A Base Case is where a problem can be solved without recursion –A Recursive Case is where recursion is used to solve the problem

7. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-7 13.2 Problem Solving with Recursion Using recursion to calculate the factorial of a number –A factorial is defined as n! whereas n is the number you want to solve –4! or “four factorial” mean 1*2*3*4 = 24 –5! or “five factorial” means 1*2*3*4*5 = 120 –0! is always 1 Factorials are often solved using recursion

8. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-8 13.2 Problem Solving with Recursion Continued… User enters a number which determines how many the function calls itself

9. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-9 13.2 Problem Solving with Recursion n is equal number

10. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-10 13.2 Problem Solving with Recursion Inside Program 13-3 –Inside the function, if n is 0, then the function returns a 1, as the problem is solved –Else, Return n * factorial(n-1) is processed and the function is called again –While the Else does return a value, it does not do that until the value of factorial(n-1) is solved

11. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-11 13.2 Problem Solving with Recursion Figure 13-4 The value of n and the return value during each call of the function

12. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-12 13.3 Examples of Recursive Algorithms Summing a Range of Array Elements with Recursion Continued…

13. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13.3 Examples of Recursive Algorithms 13-13 Summing a Range of Array Elements with Recursion

14. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-14 13.3 Examples of Recursive Algorithms Inside Program 13-4 –start and end represent the array range –Return array[start] + rangeSum(array, start+1), end) This continuously returns the value of the first element in the range plus the sum of the rest of the elements in the range It only breaks out when start is greater than end start must be incremented

15. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-15 13.3 Examples of Recursive Algorithms The Fibonacci Series Continued… The Fibonacci sequence (also known as Leonardo Pisano or Fibonacci ) is named for Leonardo Pisano, an Italian mathematician who lived from 1170 - 1250. The Fibonacci sequence was used to illustrate a problem based on a pair of breeding rabbits: "How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?" The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34... MonthF n = F n-1 + F n -2 # of Rabbit pairs Month 1 1 Month 20 + 11 Month 31 + 12 Month 41 + 23 Month 52 + 35 Month 63 + 58 Month 75 + 813 Month 88 + 1321 Month 913 + 2134

16. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-16 13.3 Examples of Recursive Algorithms The Fibonacci Series Continued…

17. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-17 13.3 Examples of Recursive Algorithms The Fibonacci Series

18. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-18 13.3 Examples of Recursive Algorithms Inside Program 13-5 –The Fibonacci numbers are 0,1,1,2,3,5,8,13,21… –After the second number, each number in the series is the sum of the two previous numbers –The recursive function continuously processes the calculation until the limit is reached as defined in the for loop in the main module

19. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-19 13.3 Examples of Recursive Algorithms Additional examples that can be solved with recursion –The Greatest Common Divisor –A Recursive Binary Search –The Towers of Hanoi

20. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13-20 13.3 Examples of Recursive Algorithms Recursion vs. Looping –Reasons not to use recursion They are certainly less efficient than iterative algorithms because of the overhead Harder to discern what is going on with recursion –Why use recursion The speed and amount of memory available to modern computers diminishes the overhead factor –The decision is primarily a design choice

21. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley #Program 13 ‐ 1 (recursion_demo) #This program has a recursive function. def main(): #By passing the argument 5 to the message #function we are telling it to display the #message five times. message(5) def message(times): print times if (times > 0): print 'This is a recursive function' message(times - 1)#calls itself minus one time #Call the main function. main() Python Code for Program 13-1

22. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley #Program 13 ‐ 2 (factorial) # This program uses recursion to calculate # the factorial of a number. def main(): # Get a number from the user. #This is the Python version of Program 13 ‐ 3 in the textbook page 108 number = input('Enter a nonnegative integer: ') # Get the factorial of the number. fact = factorial(number) # Display the factorial. print 'The factorial of', number, 'is', fact # The factorial function uses recursion to # calculate the factorial of its argument, # which is assumed to be nonnegative. def factorial(num): if num == 0: return 1 else: print num return num * factorial(num - 1) # Call the main function. main() Python Code for Program 13-2