Recursion. CSCI 112: Programming in C

Transcription

1 Recursion CSCI 112: Programming in C 1

2 What is recursion? Recursion looks at a large problem as a bunch of smaller versions of the same problem. 2 2

3 What is recursion? Recursion looks at a large problem as a bunch of smaller versions of the same problem. To get to the final solution, we have to solve the smaller problems first. 3 3

4 Recursion in the real world: art 4 4

5 Recursion in the real world: vegetables 5 5

6 Recursion in the real world: snowflakes 6 6

7 Recursion in the real world: nesting dolls 7 7

8 When do we use recursion? We don t know how to directly solve a complex problem, but we do know how to solve a simplified case of the problem. We can redefine the complex problem in terms of solutions to the simple problem. By continuing to reduce the problem, we ll eventually end up at all simple cases of the problem! 8 8

9 Recursive programming The basic format for coding recursively: if this is a simple case solve it else redefine the problem using recursion 9 9

10 Recursive programming The basic format for coding recursively: if this is a simple case solve it else redefine the problem using recursion base case 10 10

11 The base case The base case is the point at which we can solve the problem directly I.e., there s no need to make any more recursive calls 11 11

12 Example: Removing flowers from a vase We only know how to remove a single flower from a vase. How de we empty a vase with n flowers? 12 12

13 Example: Removing flowers from a vase We only know how to remove a single flower from a vase. How de we empty a vase with n flowers? We remove one flower, then recursively remove n-1 flowers 13 13

14 Example: Removing flowers from a vase We only know how to remove a single flower from a vase. How de we empty a vase with n flowers? We remove one flower, then recursively remove n-1 flowers 14 14

15 Example: Removing flowers from a vase void removeflowers(int flowers) { if (flowers == 1) { Remove single flower } } else { Remove single flower removeflowers(flowers 1) } 15 15

16 Example: Removing flowers from a vase void removeflowers(int flowers) { if (flowers == 1) { Remove single flower } } else { Remove single flower removeflowers(flowers 1) } base case: this is the simple case we know how to handle directly If we still have more than 1 flower, then we remove 1 and call the function again, with 1 less flower 16 16

17 Coding example: calculating factorials Recall that n! = n n 1 n We could tackle this iteratively, with a for loop. Or, we could think of this problem as follows, and use recursion: n! = n n 1! 17 17

18 Coding example: calculating factorials Recall that n! = n n 1 n We could tackle this iteratively, with a for loop. Or, we could think of this problem as follows, and use recursion: n! = n n 1! 18 18

19 Coding example: calculating factorials Recall that n! = n n 1 n We could tackle this iteratively, with a for loop. Or, we could think of this problem as follows, and use recursion: n! = n n 1! Recursive call! 19 19

20 Coding example: calculating factorials Let s setup a recursive function: n! = n n 1! int factorial(int n) What s the base case for this recursion? That is, what s the value of n for which calculating the factorial is dead-simple? 20 20

21 Coding example: calculating factorials Let s setup a recursive function: n! = n n 1! int factorial(int n) What s the base case for this recursion? That is, what s the value of n for which calculating the factorial is dead-simple? That s right, when n = 1. (Since 1! = 1)

22 Coding example: finding GCD The GCD (great common divisor/denominator) is the largest integer which can evenly divide two other integers. For example, the GCD of 10 and 15 is

23 Coding example: finding GCD The GCD (great common divisor/denominator) is the largest integer which can evenly divide two other integers. For example, the GCD of 10 and 15 is 5. An efficient method of find the GCD of two integers is the Euclidean Algorithm: 23 23

24 Coding example: finding GCD An efficient method of find the GCD of two integers is the Euclidean Algorithm: What s the base case (two numbers for which we instantly know the GCD)? 24 24

25 Coding example: finding GCD An efficient method of find the GCD of two integers is the Euclidean Algorithm: What s the base case (two numbers for which we instantly know the GCD)? If (m % n == 0), the GCD of m is???? 25 25

26 Coding example: finding GCD An efficient method of find the GCD of two integers is the Euclidean Algorithm: What s the base case (two numbers for which we instantly know the GCD)? If (m % n == 0), the GCD of m is???? That s right, it s n 26 26