Unit 8: Recursion. Dave Abel. April 8th, 2016

Transcription

1 Unit 8: Recursion Dave Abel April 8th,

2 Outline For Today Quick Recap and Factorial Warmup Recursion, Theory Prefix notation Recursive Searching! Recursive Sorting! 2

3 Recursion: Recap Definition: a process, program, or object is said to be recursive if it involves repeated self-reference. 3

4 Recursion: Recap Definition: a process, program, or object is said to be recursive if it involves repeated self-reference. In general, recursive entities can be described as: - A simple step - A recursive step Recursion can be infinite For recursion to be finite, we need a base case. 4

5 Recursive Algorithms Factorial Word Length Is a word a palindrome? Fibonacci 5

6 Recursion: Factorial factorial(4) 6

7 Recursion: Factorial factorial(4) = 4 * factorial(3) 7

8 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) 8

9 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) 9

10 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 10

11 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 11

12 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * 1 12

13 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * 2 13

14 Recursion: Factorial factorial(4) = 4 * 6 14

15 Recursion: Factorial factorial(4) = 24 15

16 Recursion: Factorial factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 16

17 Recursion! So some algorithms can be recursive. 17

18 Recursion! computing length of word So some algorithms can be recursive. computing factorial 18

19 Recursion! computing length of word is word a palindrome? So some algorithms can be recursive. linear search computing factorial 19

20 Cool Connection to Theory Things that can be computed, period. SOLVE Things a regular computer can compute before the sun goes supernova Things a domino computer could compute before the sun goes supernova 20

21 Cool Connection to Theory Things that can be computed, period. 21

22 Cool Connection to Theory Things that can be computed, period. Computations that can be represented recursively 22

23 Cool Connection to Theory factorial Things that can be computed, period. Computations that can be represented recursively length of word 23

24 Cool Connection to Theory factorial Things that can be computed, period. Computations that can be represented recursively length of word Q: How do these two bubbles relate? 24

25 Cool Connection to Theory factorial Things that can be computed, period. Computations that can be represented length of recursively word Q: How do these two bubbles relate? A: They re identical 25

26 Cool Connection to Theory factorial Things that can be computed, period. Computations that can be represented length of recursively word If something can t be computed, it also can t be represented recursively Q: How do these two bubbles relate? A: They re identical 26

27 Prefix Notation Idea: another way of writing arithmetic that fits naturally into recursive solutions Put the operator at the beginning: becomes (5 + 7) * (3 + 2) becomes * * becomes (34 + 2) * 10 27

28 Clicker Question! Q: What is the result of: * ? 28

29 Clicker Question! Q: What is the regular notation of: * * 2 4 2? [A] (3*7) + (2*4) * 2 [C] (3+7) * (2+4) + 2 [B] (3+7) * (2+4) * 2 [D] (3+7) * ((2*4) + 2) [E] I m confused. 29

30 Our First Problem: Search Problem Specification: Input: - a collection of objects, call it Basket - a specific object, call it Snozzberry Output: - True if Snozzberry is in Basket. - False if Snozzberry is not in Basket 30

31 Recursive Solution! Q: Can we do linear search recursively? Sure! Recursive Linear Search: - Is our list empty? If so, return false! - Is the first item in the list our item? - If not, run Recursive Linear Search on the rest of the list! 31

32 Recursive Solution! Q: Can we do binary search recursively? Sure! This one is perhaps more naturally recursive Recursive Binary Search: - Is our list empty? If so, return false! - Check the middle item, is it our item? If so, return True. - If not, run Binary Search on the correct half of the list. 32

33 Our Second Problem: Sorting Problem Specification: Input: - a collection of orderable objects, call it Basket Output: - Basket, where each item is in order. 33

34 Recursive Solution! Q: Can we do sort recursively? Sure! There are many ways. Lets talk about one. Merge Sort: - Split the list in half, merge sort each half. - Merge the two together. 34

35 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 35

36 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 36

37 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 37

38 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 38

39 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 39

40 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 40

41 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 41

42 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 42

43 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 43

44 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 44

45 Merge Sort Sorting a length 1 list is our trivial case! - Split the list in half, merge sort each half. - Merge the two together. 45

46 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 46

47 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 47

48 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 48

49 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 49

50 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 50

51 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 51

52 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 52

53 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 53

54 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 54

55 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 55

56 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 56

57 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 57

58 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 58

59 Merge Sort Split the list in half, merge sort each half. - Merge the two together. 59

60 Merge Sort Neat Fact: growth rate of Merge Sort is N*log(N), fastest possible sort! - Split the list in half, merge sort each half. - Merge the two together. 60

61 The Reading! 61

62 Recursive Algorithms Factorial Word Length Is a word a palindrome? Fibonacci Search, Sort 62

63 Recursive Algorithms Factorial Word Length Is a word a palindrome? Fibonacci Search, Sort Recursion order is important 63

64 Recursion: Recap Definition: a process, program, or object is said to be recursive if it involves repeated self-referencesub bullet one In general, recursive entities can be described as: - A simple step - A recursive step - Recursion can be infinite - For recursion to be finite, we need a base case. - Problems: Fibonacci, Factorial, Searching, Sorting, and more. - Prefix Notation: becomes