CS 1110: Introduction to Computing Using Python Recursion
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1 CS 1110: Introdution to Computing Using Python Leture 15 Reursion [Andersen, Gries, Lee, Marshner, Van Loan, White]
2 Announements: Prelim 1 Graded and released Mean: 81 out of 104 (78%) Can pik up your exam in homework handak room Need Cornell ID Suggest printing your netid on paper Do not disuss exam with people taking makeups. Regrade requests: we will send to you 10/13/16 Reursion 2
3 Announements: Assignment 3 Released. Due: Thursday, Marh 30 th, 11:59pm Reommendation: follow milestone deadlines. You MUST aknowledge help from others We run software analyzers to detet similar programs Have had some aademi integrity violations so far Not a reursion assignment! 10/13/16 Reursion 3
4 Out. Not a reursion la! Announement: La 8 10/13/16 Reursion 4
5 Reursion Reursive Definition: A definition that is defined in terms of itself 10/13/16 Reursion 5
6 A Mathematial Example: Fatorial Non-reursive definition: n! = n n = n (n-1 2 1) Reursive definition: n! = n (n-1)! for n 0 0! = 1 Reursive ase Base ase What happens if there is no ase ase? 10/13/16 Reursion 6
7 Reursion Reursive Definition: A definition that is defined in terms of itself Reursive Funtion: A funtion that alls itself (diretly or indiretly) 10/13/16 Reursion 7
8 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1 n 3 3 return n*fatorial(n-1) Call: fatorial(3) 10/13/16 Reursion 8
9 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 3 return n*fatorial(n-1) Call: fatorial(3) 10/13/16 Reursion 9
10 Reursion 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 3 return n*fatorial(n-1) Call: fatorial(3) Now what? Eah all is a new frame. 10/13/16 Reursion 10
11 What happens next? fatorial 1, 3 n 3 A: fatorial 1, 3 n 3 fatorial 1 n 2 C: CORRECT ERASE FRAME fatorial 1 n 3 B: fatorial 1, 3, 1 n 3 2 D: fatorial 1, 3, 1 n 3 2 fatorial 1 n 2 10/13/16 Reursion 11
12 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1 n 2 3 return n*fatorial(n-1) Call: fatorial(3) 10/13/16 Reursion 12
13 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 3 return n*fatorial(n-1) Call: fatorial(3) 10/13/16 Reursion 13
14 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1 3 return n*fatorial(n-1) n 1 Call: fatorial(3) 10/13/16 Reursion 14
15 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) n 1 Call: fatorial(3) 10/13/16 Reursion 15
16 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) n 1 Call: fatorial(3) fatorial n /13/16 Reursion 16
17 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) n 1 Call: fatorial(3) fatorial n 0 1, 2 10/13/16 Reursion 17
18 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 18
19 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 19
20 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 20
21 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 21
22 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 RETURN 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 22
23 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 fatorial 1, 3 n 2 RETURN 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 23
24 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 RETURN 6 fatorial 1, 3 n 2 RETURN 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 24
25 Reursive Call Frames 1 2 def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 fatorial 1, 3 n 3 RETURN 6 fatorial 1, 3 n 2 RETURN 2 fatorial 1, 3 3 return n*fatorial(n-1) Call: fatorial(3) n 1 RETURN 1 fatorial 1, 2 n 0 RETURN 1 10/13/16 Reursion 25
26 Example: Fionnai Sequene Sequene of numers: 1, 1, 2, 3, 5, 8, 13,... a 0 a 1 a 2 a 3 a 4 a 5 a 6 Get the next numer y adding previous two What is a 8? A: a 8 = 21 B: a 8 = 29 C: a 8 = 34 D: None of these. 10/13/16 Reursion 26
27 Example: Fionnai Sequene Sequene of numers: 1, 1, 2, 3, 5, 8, 13,... a 0 a 1 a 2 a 3 a 4 a 5 a 6 Get the next numer y adding previous two What is a 8? A: a 8 = 21 B: a 8 = 29 C: a 8 = 34 orret D: None of these. 10/13/16 Reursion 27
28 Example: Fionnai Sequene Sequene of numers: 1, 1, 2, 3, 5, 8, 13,... a 0 a 1 a 2 a 3 a 4 a 5 a 6 Get the next numer y adding previous two What is a 8? Reursive definition: a n = a n-1 + a n-2 a 0 = 1 a 1 = 1 Reursive Case Base Case (another) Base Case Why did we need two ase ases this time? 10/13/16 Reursion 28
29 Fionai as a Reursive Funtion def fionai(n): """Returns: Fionai no. a n Preondition: n 0 an int""" if n <= 1: return 1 Base ase(s) return (fionai(n-1)+ fionai(n-2)) Reursive ase Handles oth ase ases in one onditional. 10/13/16 Reursion 29
30 Fionai as a Reursive Funtion def fionai(n): """Returns: Fionai no. a n Preondition: n 0 an int""" if n <= 1: return 1 return (fionai(n-1)+ fionai(n-2)) fionai 3 n 5 fionai 1 fionai 1 n 4 n 3 10/13/16 Reursion 30
31 Reursion vs Iteration Reursion is provaly equivalent to iteration Iteration inludes for-loop and while-loop (later) Anything an do in one, an do in the other But some things are easier with reursion And some things are easier with iteration Will not teah you when to hoose reursion We just want you to understand the tehnique 10/13/16 Reursion 31
32 Reursion is est for Divide and Conquer Goal: Solve prolem P on a piee of data data 10/13/16 Reursion 32
33 Reursion is est for Divide and Conquer Goal: Solve prolem P on a piee of data data Idea: Split data into two parts and solve prolem data 1 data 2 Solve Prolem P Solve Prolem P 10/13/16 Reursion 33
34 Reursion is est for Divide and Conquer Goal: Solve prolem P on a piee of data data Idea: Split data into two parts and solve prolem data 1 data 2 Solve Prolem P Solve Prolem P Comine Answer! 10/13/16 Reursion 34
35 Divide and Conquer Example Count the numer of 'e's in a string: p e n n e Two 'e's p e n n e One 'e' One 'e' 10/13/16 Reursion 35
36 Divide and Conquer Example Count the numer of 'e's in a string: p e n n e Two 'e's p e n n e Zero 'e's Two 'e's 10/13/16 Reursion 36
37 Three Steps for Divide and Conquer 1. Deide what to do on small data Some data annot e roken up Have to ompute this answer diretly 2. Deide how to reak up your data Both halves should e smaller than whole Often no wrong way to do this (next leture) 3. Deide how to omine your answers Assume the smaller answers are orret Comining them should give igger answer 10/13/16 Reursion 37
38 Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Comine the result return left+right Short-ut for s[0] if s[0] == 'e : return 1 else: return 0 p e n n s[1:] e 10/13/16 Reursion 38
39 Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Comine the result return left+right Short-ut for s[0] if s[0] == 'e : return 1 else: return 0 p e n n s[1:] e 10/13/16 Reursion 39
40 Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Comine the result return left+right Short-ut for s[0] if s[0] == 'e : return 1 else: return 0 p e n n s[1:] e 10/13/16 Reursion 40
![Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2.](/docs-images/74/70071297/images/41-0.jpg "Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3.")
41 Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Comine the result return left+right Short-ut for s[0] if s[0] == 'e : return 1 else: return 0 p e n n s[1:] e 10/13/16 Reursion 41
42 Divide and Conquer Example def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 Base Case # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Comine the result return left+right Reursive Case 10/13/16 Reursion 42
43 Exerise: Remove Blanks from a String def (s): """Returns: s ut with its lanks removed""" 1. Deide what to do on small data If it is the empty string, nothing to do if s == '': return s If it is a single harater, delete it if a lank if s == ' ': # There is a spae here return '' # Empty string else: return s 10/13/16 Reursion 43
44 Exerise: Remove Blanks from a String def (s): """Returns: s ut with its lanks removed""" 2. Deide how to reak it up left = (s[0]) # A string with no lanks right = (s[1:]) # A string with no lanks 3. Deide how to omine the answer return left+right # String onatenation 10/13/16 Reursion 44
45 Putting it All Together def (s): """Returns: s w/o lanks""" if s == '': return s elif len(s) == 1: return '' if s[0] == ' ' else s left = (s[0]) right = (s[1:]) return left+right Handle small data Break up the data Comine answers 10/13/16 Reursion 45
46 Putting it All Together def (s): """Returns: s w/o lanks""" if s == '': return s elif len(s) == 1: return '' if s[0] == ' ' else s left = (s[0]) right = (s[1:]) return left+right Base Case Reursive Case 10/13/16 Reursion 46
47 Following the Reursion a 10/13/16 Reursion 47
48 Following the Reursion a a stops (ase ase) a stops (ase ase) 10/13/16 Reursion 48
49 Following the Reursion a a 10/13/16 Reursion 49
50 Following the Reursion a a a 10/13/16 Reursion 50
51 Following the Reursion a a a 10/13/16 Reursion 51
52 Following the Reursion a a a 10/13/16 Reursion 52
53 Following the Reursion a a a 10/13/16 Reursion 53
54 Following the Reursion a a a
55 Following the Reursion a a a
56 Following the Reursion a a a
57 Following the Reursion a a a
58 Following the Reursion a a a
59 Following the Reursion a a a a
60 Following the Reursion a a a a a
61 Following the Reursion a a a a a a
62 Tower of Hanoi Three towers: left, middle, and right n disks of unique sizes on left Goal: move all disks from left to right Cannot put a larger disk on top of a smaller disk left middle right 10/13/16 Reursion 62
63 1 Dis 1. Move from left to right 1 left middle right 10/13/16 Reursion 63
64 1 Dis 1. Move from left to right left middle right 1 10/13/16 Reursion 64
65 2 Diss 1. Move from left to middle 1 2 left middle right 10/13/16 Reursion 65
66 2 Diss 1. Move from left to middle 2. Move from left to right 2 1 left middle right 10/13/16 Reursion 66
67 2 Diss 1. Move from left to middle 2. Move from left to right 3. Move from middle to right 1 left middle right 2 10/13/16 Reursion 67
68 2 Diss 1. Move from left to middle 2. Move from left to right 3. Move from middle to right 1 2 left middle right 10/13/16 Reursion 68
69 3 Diss 1. Move from left to right left middle right 10/13/16 Reursion 70
70 3 Diss 1. Move from left to right 2. Move from left to middle left middle right 10/13/16 Reursion 71
71 3 Diss 1. Move from left to right 2. Move from left to middle 3. Move from right to middle left middle right 10/13/16 Reursion 72
72 1 3 2 left middle right 3 Diss 1. Move from left to right 2. Move from left to middle 3. Move from right to middle 4. Move from left to right 10/13/16 Reursion 73
73 3 Diss 1 2 left middle right 3 1. Move from left to right 2. Move from left to middle 3. Move from right to middle 4. Move from left to right 5. Move from middle to left 10/13/16 Reursion 74
74 3 Diss left middle right 1. Move from left to right 2. Move from left to middle 3. Move from right to middle 4. Move from left to right 5. Move from middle to left 6. Move from middle to right 10/13/16 Reursion 75
75 3 Diss left middle right 1. Move from left to right 2. Move from left to middle 3. Move from right to middle 4. Move from left to right 5. Move from middle to left 6. Move from middle to right 7. Move from left to right 10/13/16 Reursion 76
76 3 Diss 1 2 left middle right 3 1. Move from left to right 2. Move from left to middle 3. Move from right to middle 4. Move from left to right 5. Move from middle to left 6. Move from middle to right 7. Move from left to right 10/13/16 Reursion 77
77 4 Diss: High-level Idea left middle right 10/13/16 Reursion 78
78 4 Diss: High-level Idea Plan: move top three disks from left to middle left middle right 10/13/16 Reursion 79
79 4 Diss: High-level Idea left middle right Plan: move top three disks from left to middle Move: largest disk from left to right 10/13/16 Reursion 80
80 4 Diss: High-level Idea Plan: move top three disks from left to middle Move: largest disk from left to right Plan: move top three disks from middle to right left middle right 10/13/16 Reursion 81
81 4 Diss: High-level Idea Plan: move disks 1, 2, and 3 from left to middle left middle right 10/13/16 Reursion 82
82 4 Diss: High-level Idea Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right left middle right 10/13/16 Reursion 83
83 4 4 Diss: High-level Idea left middle right Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right 10/13/16 Reursion 84
84 4 Diss: High-level Idea left middle right Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle 10/13/16 Reursion 85
85 4 Diss: High-level Idea left middle right Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle 10/13/16 Reursion 86
86 4 Diss: High-level Idea left middle right Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle Move: disk 4 from left to right 10/13/16 Reursion 87
87 4 Diss: High-level Idea left middle right Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle Move: disk 4 from left to right Plan: move disks 1, 2, and 3 from middle to right 10/13/16 88
88 4 Diss: High-level Idea Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle left middle right Move: disk 4 from left to right Plan: move disks 1, 2, and 3 from middle to right 10/13/16 89
89 Oservation: Plans within a Plan High-level plan Low-level plan Plan: move disks 1, 2, and 3 from left to middle Plan: move disks 1 and 2 from left to right Move: disk 3 from left to right Plan: move disks 1 and 2 from right to middle Move: disk 4 from left to right Plan: move disks 1, 2, and 3 from middle to right 10/13/16 90
90 General Pattern To move n disks from soure to target: disks 1,, n-1 4 disk n soure other disks? other target (soure, other, and target an e any permutation of left, middle and right) 1. Plan: move disks 1,, n-1 from soure to other 2. Move: disk n to from soure to target 3. Plan: move disks 1,, n-1 from other to target 10/13/16 Reursion 91
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