Lecture 26. Sequence Algorithms (Continued)

Transcription

1 Lecture 26 Sequence Algoritms (Continued)

2 Announcements for Tis Lecture Assignment & Lab A6 is not graded yet Done early next wee A7 due Mon, Dec. 4 But extensions possible Just as for one! But mae good effort Lab Today: Office Hours Get elp on A7 paddle Anyone can go to any lab Next Wee Last Wee of Class! Finis sorting algoritms Special final lecture Lab eld, but is optional Unless only ave 10 labs Also use lab time on A7 Details about te exam Multiple review sessions 11/22/16 Sequences (Continued) 2

3 Recall: Horizontal Notation 0 len(b) b <= sorted >= Example of an assertion about an sequence b. It asserts tat: 1. b[0.. 1] is sorted (i.e. its values are in ascending order) 2. Everyting in b[0.. 1] is everyting in b[..len(b) 1] b 0 Given index of te first element of a segment and index of te element tat follows tat segment, te number of values in te segment is. b[.. 1] as elements in it. +1 (+1) = 1 11/22/16 Sequences (Continued) 3

4 Partition Algoritm Given a sequence b[..] wit some value x in b[]: pre: b x? Swap elements of b[..] and store in j to trutify post: i i+1 post: b <= x x >= x inv: b i j <= x x? >= x Agrees wit precondition wen i =, j = +1 Agrees wit postcondition wen j = i+1 11/22/16 Sequences (Continued) 4

5 Partition Algoritm Implementation def partition(b,, ): """Partition list b[..] around a pivot x = b[]""" i = ; j = +1; x = b[] # invariant: b[..i-1] < x, b[i] = x, b[j..] >= x wile i < j-1: if b[i+1] >= x: # Move to end of bloc. swap(b,i+1,j-1) j = j - 1 else: # b[i+1] < x swap(b,i,i+1) i = i + 1 # post: b[..i-1] < x, b[i] is x, and b[i+1..] >= x return i partition(b,,), not partition(b[:+1]) Remember, slicing always copies te list! We want to partition te original list 11/22/16 Sequences (Continued) 5

6 Partition Algoritm Implementation def partition(b,, ): """Partition list b[..] around a pivot x = b[]""" i = ; j = +1; x = b[] # invariant: b[..i-1] < x, b[i] = x, b[j..] >= x wile i < j-1: if b[i+1] >= x: # Move to end of bloc. swap(b,i+1,j-1) j = j - 1 else: # b[i+1] < x swap(b,i,i+1) i = i + 1 # post: b[..i-1] < x, b[i] is x, and b[i+1..] >= x return i <= x x? >= x i i+1 j /22/16 Sequences (Continued) 6

7 Partition Algoritm Implementation def partition(b,, ): """Partition list b[..] around a pivot x = b[]""" i = ; j = +1; x = b[] # invariant: b[..i-1] < x, b[i] = x, b[j..] >= x wile i < j-1: if b[i+1] >= x: # Move to end of bloc. swap(b,i+1,j-1) j = j - 1 else: # b[i+1] < x swap(b,i,i+1) i = i + 1 # post: b[..i-1] < x, b[i] is x, and b[i+1..] >= x return i <= x x? >= x i i+1 j i i+1 j /22/16 Sequences (Continued) 7

8 Partition Algoritm Implementation def partition(b,, ): """Partition list b[..] around a pivot x = b[]""" i = ; j = +1; x = b[] # invariant: b[..i-1] < x, b[i] = x, b[j..] >= x wile i < j-1: if b[i+1] >= x: # Move to end of bloc. swap(b,i+1,j-1) j = j - 1 else: # b[i+1] < x swap(b,i,i+1) i = i + 1 # post: b[..i-1] < x, b[i] is x, and b[i+1..] >= x return i <= x x? >= x i i+1 j i i+1 j i j /22/16 Sequences (Continued) 8

9 Partition Algoritm Implementation def partition(b,, ): """Partition list b[..] around a pivot x = b[]""" i = ; j = +1; x = b[] # invariant: b[..i-1] < x, b[i] = x, b[j..] >= x wile i < j-1: if b[i+1] >= x: # Move to end of bloc. swap(b,i+1,j-1) j = j - 1 else: # b[i+1] < x swap(b,i,i+1) i = i + 1 # post: b[..i-1] < x, b[i] is x, and b[i+1..] >= x return i <= x x? >= x i i+1 j i i+1 j i j i j /22/16 Sequences (Continued) 9

10 Dutc National Flag Variant Sequence of integer values red = negatives, wite = 0, blues = positive Only rearrange part of te list, not all pre: b post: b? < 0 = 0 > 0 t i j inv: b < 0? = 0 > 0 11/22/16 Sequences (Continued) 10

11 Dutc National Flag Variant Sequence of integer values red = negatives, wite = 0, blues = positive Only rearrange part of te list, not all pre: b? post: b < 0 = 0 > 0 t i j inv: b < 0? = 0 > 0 pre: t =, i = +1, j = post: t = i 11/22/16 Sequences (Continued) 11

12 Dutc National Flag Algoritm def dnf(b,, ): """Returns: partition points as a tuple (i,j)""" t = ; i = +1, j = ; # inv: b[..t-1] < 0, b[t..i-1]?, b[i..j] = 0, b[j+1..] > 0 wile t < i: if b[i-1] < 0: swap(b,i-1,t) t = t+1 elif b[i-1] == 0: else: i = i-1 swap(b,i-1,j) i = i-1; j = j-1 # post: b[..i-1] < 0, b[i..j] = 0, b[j+1..] > 0 return (i, j) < 0? = 0 > 0 t i j /22/16 Sequences (Continued) 12

13 Dutc National Flag Algoritm def dnf(b,, ): """Returns: partition points as a tuple (i,j)""" t = ; i = +1, j = ; # inv: b[..t-1] < 0, b[t..i-1]?, b[i..j] = 0, b[j+1..] > 0 wile t < i: if b[i-1] < 0: swap(b,i-1,t) t = t+1 elif b[i-1] == 0: else: i = i-1 swap(b,i-1,j) i = i-1; j = j-1 # post: b[..i-1] < 0, b[i..j] = 0, b[j+1..] > 0 return (i, j) < 0? = 0 > 0 t i j t i j /22/16 Sequences (Continued) 13

14 Dutc National Flag Algoritm def dnf(b,, ): """Returns: partition points as a tuple (i,j)""" t = ; i = +1, j = ; # inv: b[..t-1] < 0, b[t..i-1]?, b[i..j] = 0, b[j+1..] > 0 wile t < i: if b[i-1] < 0: swap(b,i-1,t) t = t+1 elif b[i-1] == 0: else: i = i-1 swap(b,i-1,j) i = i-1; j = j-1 # post: b[..i-1] < 0, b[i..j] = 0, b[j+1..] > 0 return (i, j) < 0? = 0 > 0 t i j t i j t i j /22/16 Sequences (Continued) 14

15 Dutc National Flag Algoritm def dnf(b,, ): """Returns: partition points as a tuple (i,j)""" t = ; i = +1, j = ; # inv: b[..t-1] < 0, b[t..i-1]?, b[i..j] = 0, b[j+1..] > 0 wile t < i: if b[i-1] < 0: swap(b,i-1,t) t = t+1 elif b[i-1] == 0: else: i = i-1 swap(b,i-1,j) i = i-1; j = j-1 # post: b[..i-1] < 0, b[i..j] = 0, b[j+1..] > 0 return (i, j) < 0? = 0 > 0 t i j t i j t i j t j /22/16 Sequences (Continued) 15

16 Now we ave four colors! Flag of Mauritius Negatives: red = odd, purple = even Positives: yellow = odd, green = even pre: b? post: b inv: b < 0 odd < 0 even 0 odd 0 even r s i t < 0, o < 0, e 0, o? 0, e 11/22/16 Sequences (Continued) 16

17 Flag of Mauritius < 0, o < 0, e 0, o? 0, e r s i t r s i t One swap is not good enoug 11/22/16 Sequences (Continued) 17

18 Flag of Mauritius < 0, o < 0, e 0, o? 0, e r s i t r s i t Need two swaps for two spaces 11/22/16 Sequences (Continued) 18

19 Flag of Mauritius < 0, o < 0, e 0, o? 0, e r s i t r s i t And adjust te loop variables 11/22/16 Sequences (Continued) 19

20 Flag of Mauritius < 0, o < 0, e 0, o? 0, e r s i t r s i t See algoritms.py for Pyton code r s i t /22/16 Sequences (Continued) 20

21 Flag of Mauritius < 0, o < 0, e 0, o? 0, e r s i t r s i t See algoritms.py for Pyton code r s i t r s i t /22/16 Sequences (Continued) 21

22 Linear Searc Vague: Find first occurrence of v in b[..-1]. 11/22/16 Sequences (Continued) 22

23 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] 11/22/16 Sequences (Continued) 23

24 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] pre: b? post: b i v not ere v? 11/22/16 Sequences (Continued) 24

25 Linear Searc Vague: Find first occurrence of v in b[..-1]. Better: Store an integer in i to trutify result condition post: post: 1. v is not in b[..i-1] 2. i = OR v = b[i] pre: b? post: b i v not ere v? OR b v not ere i 11/22/16 Sequences (Continued) 25

26 pre: b Linear Searc? post: b i v not ere v? OR b v not ere i inv: b i v not ere? 11/22/16 Sequences (Continued) 26

27 Linear Searc def linear_searc(b,v,,): """Returns: first occurrence of v in b[..-1]""" # Store in i index of te first v in b[..-1] 1. Does te initialization mae inv true? i = # invariant: v is not in b[0..i-1] wile i < and b[i]!= v: i = i + 1 # post: v is not in b[..i-1] # i >= or b[i] == v return i if i < else -1 Analyzing te Loop 2. Is post true wen inv is true and condition is false? 3. Does te repetend mae progress? 4. Does te repetend eep te invariant inv true? 11/22/16 Sequences (Continued) 27

28 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. 11/22/16 Sequences (Continued) 28

29 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. Better: Precondition: b[..-1] is sorted (in ascending order). Postcondition: b[..i] <= v and v < b[i+1..-1] Below, te array is in non-descending order: pre: b? post: b i <= v > v 11/22/16 Sequences (Continued) 29

30 Binary Searc Vague: Loo for v in sorted sequence segment b[..]. Better: Precondition: b[..-1] is sorted (in ascending order). Postcondition: b[..i] <= v and v < b[i+1..-1] Below, te array is in non-descending order: pre: b post: b inv: b <= v? i > v i j < v? > v Called binary searc because eac iteration of te loop cuts te array segment still to be processed in alf 11/22/16 Sequences (Continued) 30

31 Extras Not Covered in Class 11/22/16 Sequences (Continued) 31

32 Loaded Dice Sequence p of lengt n represents n-sided die Contents of p sum to 1 p[] is probability die rolls te number weigted d6, favoring 5, 6 Goal: Want to roll te die Generate random number r between 0 and 1 Pic p[i] suc tat p[i-1] < r p[i] /22/16 Sequences (Continued) 32

33 Loaded Dice Want: Value i suc tat p[i-1] < r <= p[i] pre: b post: b 0 n? 0 i n r > sum r <= sum 0 i n inv: b r > sum? Same as precondition if i = 0 Postcondition is invariant + false loop condition 11/22/16 Sequences (Continued) 33

34 Loaded Dice def roll(p): """Returns: randint in 0..len(p)-1; i returned wit prob. p[i] Precondition: p list of positive floats tat sum to 1.""" r = random.random() # r in [0,1) # Tin of interval [0,1] divided into segments of size p[i] # Store into i te segment number in wic r falls. i = 0; sum_of = p[0] # inv: r >= sum of p[0].. p[i 1]; pend = sum of p[0].. p[i] wile r >= sum_of: sum_of = sum_of + p[i+1] i = i + 1 # post: sum of p[0].. p[i 1] <= r < sum of p[0].. p[i] return i r < sum Analyzing te Loop 1. Does te initialization mae inv true? 2. Is post true wen inv is true and condition is false? 3. Does te repetend mae progress? 4. Does te repetend eep inv true? 0 r is not ere p[0] p[1] p[i] pend p[n 1] inv 1 11/22/16 Sequences (Continued) 34 0 p[0] p[1] p[i] r p[n 1] post 1

35 Reversing a Sequence pre: b not reversed post: b reversed cange: b into b i j inv: b swapped not reversed swapped 11/22/16 Sequences (Continued) 35