CSCI 121: Recursive Functions & Procedures

Size: px

Start display at page:

Download "CSCI 121: Recursive Functions & Procedures"

Rachel Manning
5 years ago
Views:

1 CSCI 121: Recursive Functions & Procedures

2 Sorting quizzes Every time I give a quiz, I need to enter grades into my gradebook. My grading sheet is organized alphabetically. So I sort the papers alphabetically, too.

3 My sorting algorithm: DEFINE SORT_THE_STACK(stack_of_papers): 1. Split the papers into two piles, of roughly equal size. This gives me a stack_of_papers1 and a stack_of_papers2 2. Sort each of those stacks from A to Z. 3. Merge the two sorted stacks into the full sorted stack.

4 My sorting algorithm: DEFINE SORT_THE_STACK(stack_of_papers): 1. Split the papers into two piles, of roughly equal size. This gives me a stack_of_papers1 and a stack_of_papers2 2. Sort each of those stacks from A to Z. Maybe I give the piles to my TAs and have them do it 3. Merge the two sorted stacks into the full sorted stack.

5 My sorting algorithm: DEFINE SORT_THE_STACK(stack_of_papers): 1. Split the papers into two piles, of roughly equal size. This gives me a stack_of_papers1 and a stack_of_papers2 2. Sort each of those stacks from A to Z. (????) 3. Merge the two sorted stacks into the full sorted stack.

6 My sorting algorithm: DEFINE SORT_THE_STACK(stack_of_papers): 1. Split the papers into two piles, of roughly equal size. This gives me a stack_of_papers1 and a stack_of_papers2 2. Sort each of those stacks from A to Z. That is, First execute SORT_THE_STACK(stack_of_papers1) and then execute SORT_THE_STACK(stack_of_papers2) 3. Merge the two sorted stacks into the full sorted stack.

7 My sorting algorithm: DEFINE SORT_THE_STACK(stack_of_papers): 1. Split the papers into two piles, of roughly equal size. This gives me a stack_of_papers1 and a stack_of_papers2 2. Sort each of those stacks from A to Z. That is, First execute SORT_THE_STACK(stack_of_papers1) and then execute SORT_THE_STACK(stack_of_papers2) 3. Merge the two sorted stacks into the full sorted stack. This use of SORT_THE_STACK within itself is recursion.

8 Recursive calculations The nth factorial is the product of 1 to n. n! = 1 * 2 * 3 * * (n-1) * n In Python, we can compute this with a loop: factorial = 1 i = 1 while i <= n: factorial = factorial * i i = i + 1

9 Recursive calculations The nth factorial is the product of 1 to n. n! = 1 * 2 * 3 * * (n-1) * n In Python, we can compute this with a loop: factorial = 1 i = 1 while i <= n-1: factorial = factorial * i i = i + 1 factorial = factorial * n

10 Recursive calculations The nth factorial is the product of 1 to n. n! = 1 * 2 * 3 * * (n-1) * n In Python, we can compute this with a loop: # compute (n-1)! factorial = 1 i = 1 while i <= n-1: factorial = factorial * i i = i + 1 # compute n! by just multiplying the result above by n factorial = factorial * n

11 Recursive calculations DEFINE: 1! := 1 DEFINE, for n >1: n! := ((n-1)!) * n In this definition, 1 is called the BASE CASE. Then cases for n >1 are called the RECURSIVE CASES.

12 Recursive functions DEFINE: f(1) := 1 DEFINE, for n > 1: f(n) := f(n-1) * n Here is the Python code for that function: def f(n): if n == 1: return 1 else: return f(n-1) * n

13 Recursive function evaluation Here is the Python code for that function: def f(n): if n == 1: return 1 else: return f(n-1) * n # base case # recursive case How does Python execute/evaluate the following? >>> f(5)

14 Recursive procedures Suppose I m in charge of having a count from 1 to 100, shouted out loud. To do this work, I do the following: 1. delegate to someone else the task of shouting out a count from 1 to wait for them to finish their task (wait to hear 99!!!! ) 3. shout 100!!!! Voilà!

15 Recursive procedures Here is that recursive procedure, in Python: def output_count_up_to(n): output_count_up_to(n-1) print(str(n) +!!!! )

16 Recursive procedures Here is that recursive procedure, in Python: def output_count_up_to(n): output_count_up_to(n-1) print(str(n) +!!!! ) WHOOPS! What happened?

17 Recursive procedures Here is the correct recursive procedure, in Python: def output_count_up_to(n): if n > 1: output_count_up_to(n-1) print(str(n) +!!!! )

18 Recursive procedures Here is the correct recursive procedure, in Python: def output_count_up_to(n): if n > 1: output_count_up_to(n-1) print(str(n) +!!!! ) >>> output_count_up_to(5) 1!!!! 2!!!! 3!!!! 4!!!! 5!!!!

19 Recursive procedures Here is the correct recursive procedure, in Python: def output_count_up_to(n): if n > 1: output_count_up_to(n-1) print(str(n) +!!!! ) >>> output_count_up_to(5) 1!!!! <==== This is output_count_up_to(4) 2!!!! <==== 3!!!! <==== 4!!!! <==== 5!!!!

20 Fibonacci sequence, revisited DEFINE: F 1 := 1 F 2 := 1 DEFINE, for n > 2: F n := F n-2 + F n-1 The Python code with a loop: def fib(n): curr,next = 1,1 count = 1 while count < n: curr,next = next,curr+next count = count + 1 return count

21 Fibonacci sequence, revisited DEFINE: F 1 := 1 F 2 := 1 DEFINE, for n > 2: F n := F n-2 + F n-1 The Python code written as a recursive function: def fib(n): if n <= 2: return 1 else: return fib(n-2) + fib(n-1)

22 Fibonacci sequence, revisited The Python code written as a recursive function: def fib(n): if n <= 2: return 1 else: return fib(n-2) + fib(n-1) How does Python execute/evaluate the following? >>> fib(4)

23 Let s play with Fibonacci def fib_list(n): if n == 1: return [1] elif n == 2: return [2] else: fs = fib_list(n-1) prior = fs[-2] last = fs[-1] return fs + [prior + last]

24 Let s play with Fibonacci def fib_string(n): if n <= 2: return 1 else: fs2 = fib_string(n-2) fs1 = fib_string(n-1) return ( + fs2 + )+( + fs1 + )

RECURSION 7. 1 Recursion COMPUTER SCIENCE 61A. October 15, 2012

RECURSION 7. 1 Recursion COMPUTER SCIENCE 61A. October 15, 2012 RECURSION 7 COMPUTER SCIENCE 61A October 15, 2012 1 Recursion We say a procedure is recursive if it calls itself in its body. Below is an example of a recursive procedure to find the factorial of a positive