1. TUTORIAL 8 Generative Recursion

2. Reminders  Deadline of midterm remark  Assignment 7 due Wed, Mar 18 th

3. Tips for python  Don’t leave lines in the middle of your code  Up arrow key in interactions window  Make sure you are using Python 3  Do not add any special characters in the code

4. Generative recursion  CQ: which one is a generative recursion?

5. Generative recursion  CQ: which one is a generative recursion?

6. Generative recursion  CQ: what does recursion3([1,2,2,1]) and recursion3([1,2,1]) produce?  A) True  B) False  C) None

7. Generative recursion  CQ: what does recursion3([1,2,2,1]) and recursion3([1,2,1]) produce?  A) True  B) False  C) None

8. Generative Recursion  Sorting algorithms  Log(n) and nlog(n) runtimes  Sort vs sorted functions  Insertion, selection, merge  https://www.youtube.com/watch?v=kPRA0W1kECg https://www.youtube.com/watch?v=kPRA0W1kECg

9. Question 1 - smaller Write a function smaller that consumes a string containing only numeric characters and produces the smallest digit in the string. The produced digit should be in the form of a string. For example, smaller(“4325”) => “2” smaller(“1”) => “1” smaller(“2325”) => “2”

10. Solution method: - If there is only one digit, produce that digit - Compare the first digit with the last digit - Remove the larger of the two digits from the string and repeat the process - If there is a tie, remove the last digit from the string and repeat the process YEAH!!

11. Question 2 - Quicksort Consider a different way of sorting a list L of distinct integers: - Let n be the first element of the list - Let lst1 be all the elements in the list smaller than n - Let lst2 be all the elements in the list larger than n - Sort lst1 and lst2 - Add lst1 and lst2 together with n in the middle

12. Example quicksort([2,3,1,4,0])  quicksort([1,0]) + [2] + quicksort([3,4])  (quicksort([0]) + [1]) + [2] + ([3] +quicksort([4]))  ([0] + [1])+ [2] + ([3] + [4])  [0,1] + [2] +[3,4]  [0,1,2,3,4]

13. Questions 6 – sum_columns  Consider a new type: Table. A Table is a (listof (listof Int)), which is nonempty, and in which each list corresponds to a row of a Table. It is assumed that each row is nonempty and each row has the same number of entries as every other row.  Write a function sum_columns that consumes a Table t, and produces a list containing the columns sums for t.

14. Example  t0 = [[1]] t1 = [[1,2,3]] t2 = [[1],[2],[3]] t3 = [[1,2],[3,4],[5,6],[7,8]]  sum_columns(t0) => [1] sum_columns(t1) => [1,2,3] sum_columns(t3) => [16, 20]

15. Question 4 – binary_gcd Consider a different algorithm for computing the greatest common divisors of two positive integers m and n: - If either m or n are 0, produce the other number - If both numbers are even, produce 2*binary_gcd(m/2,n/2) - If m is even and n is odd, produce binary_gcd(m/2,n) - If n is even and m is odd, produce binary_gcd(m,n/2) - If both m and n are odd and m>=n, produce binary_gcd((m- n)/2,n) - If both m and n are odd and m<n, produce binary_gcd((n- m)/2,m)

16. Question 3 – is_balanced We define a string to be “balanced” based on the following criteria: - Any string not containing any brackets (any of: (,),{,},[,] ) is considered balanced - If s is a balanced string then (s), {s}, and [s] are also balanced strings - If s is a balanced string and t is a string not containing any brackets, then s+t is a balanced string and so is t+s - For example, “(Wo)rd” is balanced as well as “rd{Wo}”

17. Examples  Balanced strings:  “”  “(hi)”  “wrong{}thing”  “[cool(brackets)]”  “{[([{()}])]}”  Unbalanced strings:  “NO)PE”  “{no)”

18. Question 5 – skip_value Given a list L of positive integers, what is the skip- value of the list? - If L is empty, the skip value is 0 - If L is nonempty: - Add one to the current skip value - Move ahead in the list by n (where n is the first element in the list) and repeat the process

19. Example 1 skip_value([1,1,1])  1+skip_value([1,1])  1+(1+skip_value([1]))  1+(1+(1+skip_value([])))  1+(1+(1+0))  3

20. Example 2 skip_value([2,100,3,1,1,1])  1+skip_value([3,1,1,1])  1+(1+skip_value([1]))  1+(1+(1+skip_value([])))  1+(1+(1+0))  3