Lecture 14. Recursion

Transcription

1 Leture 14 Reursion

2 Announements for Today Prelim 1 Tonight at 5:15 OR 7:30 A D (5:15, Uris G01) E-K (5:15, Statler) L P (7:30, Uris G01) Q-Z (7:30, Statler) Graded y noon on Sun Sores will e in CMS In time for drop date Other Announements Reading: Assignment 3 now graded Mean 93.4, Median 98 Time: 7 hrs, StdDev: 3.5 hrs But only 535 responses Assignment 4 posted Friday Parts 1-3: Can do already Part 4: material from today Due two weeks from today 10/11/18 Reursion 2

3 Reursion Reursive Definition: A definition that is defined in terms of itself Reursive Funtion: A funtion that alls itself (diretly or indiretly) PIP stands for PIP Installs Pakages 10/11/18 Reursion 3

4 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? 10/11/18 Reursion 4

5 Fatorial as a Reursive Funtion def fatorial(n): """Returns: fatorial of n. Pre: n 0 an int""" if n == 0: return 1 n! = n (n-1)! 0! = 1 Base ase(s) return n*fatorial(n-1) Reursive ase What happens if there is no ase ase? 10/11/18 Reursion 5

6 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. 10/11/18 Reursion 6

7 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. 10/11/18 Reursion 7

8 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 Reursive Case a 0 = 1 Base Case a 1 = 1 (another) Base Case Why did we need two ase ases this time? 10/11/18 Reursion 8

9 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 Note differene with ase ase onditional. 10/11/18 Reursion 9

10 Fionai as a Reursive Funtion def fionai(n): """Returns: Fionai no. a n Preondition: n 0 an int""" if n <= 1: return 1 Funtion that alls itself Eah all is new frame Frames require memory alls = memory fionai 3 return (fionai(n-1)+ fionai(n-2)) n 5 fionai 1 fionai 1 n 4 n 3 10/11/18 Reursion 10

11 Fionai: # of Frames vs. # of Calls Fionai is very ineffiient. fi(n) has a stak that is always n But fi(n) makes a lot of redundant alls fi(5) fi(4) fi(3) fi(3) fi(2) fi(2) fi(1) fi(2) fi(1) fi(1) fi(0) fi(1) fi(0) 10/11/18 fi(1) fi(0) Reursion 11

12 Fionai: # of Frames vs. # of Calls Fionai is very ineffiient. fi(n) has a stak that is always n But fi(n) makes a lot of redundant alls Path to end = the all stak fi(4) fi(5) fi(3) fi(3) fi(2) fi(2) fi(1) fi(2) fi(1) fi(1) fi(0) fi(1) fi(0) 10/11/18 fi(1) fi(0) Reursion 12

13 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 This is a topi for more advaned lasses We just want you to understand the tehnique 10/11/18 Reursion 13

14 Reursion is est for Divide and Conquer Goal: Solve prolem P on a piee of data data 10/11/18 Reursion 14

15 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 10/11/18 Reursion 15

16 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! 10/11/18 Reursion 16

17 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' 10/11/18 Reursion 17

18 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 10/11/18 Reursion 18

19 Divide and Conquer Example Count the numer of 'e's in a string: p e n n e Two 'e's Will talk aout how to reak-up later p e n n e Zero 'e's Two 'e's 10/11/18 Reursion 19

20 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 10/11/18 Reursion 20

21 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 s[0] Short-ut for if s[0] == 'e : return 1 else: return 0 s[1:] p e n n e 10/11/18 Reursion 21

22 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 s[0] Short-ut for if s[0] == 'e : return 1 else: return 0 s[1:] p e n n e 10/11/18 Reursion 22

23 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 s[0] Short-ut for if s[0] == 'e : return 1 else: return 0 s[1:] p e n n e 10/11/18 Reursion 23

24 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 s[0] Short-ut for if s[0] == 'e : return 1 else: return 0 s[1:] p e n n e 10/11/18 Reursion 24

25 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 10/11/18 Reursion 25

26 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 10/11/18 Reursion 26

27 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 10/11/18 Reursion 27

28 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 Handle small data left = (s[0]) right = (s[1:]) Break up the data return left+right Comine answers 10/11/18 Reursion 28

29 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 Base Case left = (s[0]) right = (s[1:]) return left+right Reursive Case 10/11/18 Reursion 29

30 Minor Optimization 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:]) Needed seond ase ase to handle s[0] return left+right 10/11/18 Reursion 30

31 Minor Optimization def (s): """Returns: s w/o lanks""" if s == '': return s left = s[0] if s[0] == ' ': left = '' right = (s[1:]) Eliminate the seond ase y omining return left+right Less reursive alls 10/11/18 Reursion 31

32 Following the Reursion a 10/11/18 Reursion 32

33 Following the Reursion a a 10/11/18 Reursion 33

34 Following the Reursion a a a 10/11/18 Reursion 34

35 Following the Reursion a a a 10/11/18 Reursion 35

36 Following the Reursion a a a 10/11/18 Reursion 36

37 Following the Reursion a a a 10/11/18 Reursion 37

38 Following the Reursion a a a 10/11/18 Reursion 38

39 Following the Reursion a a a 10/11/18 Reursion 39

40 Following the Reursion a a a 10/11/18 Reursion 40

41 Following the Reursion a a a 10/11/18 Reursion 41

42 Following the Reursion a a a 10/11/18 Reursion 42

43 Following the Reursion a a a a 10/11/18 Reursion 43

44 Following the Reursion a a a a a 10/11/18 Reursion 44

45 Following the Reursion a a a a a a 10/11/18 Reursion 45

46 Final Modifiation def (s): """Returns: s w/o lanks""" if s == '': return s left = s[0] if s[0] == ' ': left = '' right = (s[1:]) return left+right Real work done here 10/11/18 Reursion 46

47 Final Modifiation def (s): """Returns: s w/o lanks""" if s == '': return s Real work done here left = s if s[0] in string.whitespae left = '' right = (s[1:]) Module string has speial onstants to simplify detetion of whitespae and other haraters. return left+right 10/11/18 Reursion 47

48 Next Time: Breaking Up Reursion 10/11/18 Reursion 48