24. Sorting a List. Topics: Selection Sort. Merge Sort
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1 24. Sorting a List Topics: Selection Sort Merge Sort
2 Our examples will highlight the interplay between functions and lists
3 Sorting a List of Numbers Before: x --> After: x -->
4 We Will First Implement the Method of Selection Sort At the Start: x --> High-Level: for k in range(len(x)-1) Swap x[k] with the smallest value in x[k:]
5 Selection Sort: How It Works Before: x --> Swap x[0] with the smallest value in x[0:]
6 Selection Sort: How It Works Before: x --> Swap x[0] with the smallest value in x[0:] After: x -->
7 Selection Sort: How It Works Before: x --> Swap x[1] with the smallest value in x[1:]
8 Selection Sort: How It Works Before: x --> Swap x[1] with the smallest value in x[1:] After: x -->
9 Selection Sort: How It Works Before: x --> Swap x[2] with the smallest value in x[2:]
10 Selection Sort: How It Works Before: x --> Swap x[2] with the smallest value in x[2:] After: x -->
11 Selection Sort: How It Works Before: x --> Swap x[3] with the smallest value in x[3:]
12 Selection Sort: How It Works Before: x --> Swap x[3] with the smallest value in x[3:] After: x -->
13 Selection Sort: How It Works Before: x --> Swap x[4] with the smallest value in x[4:]
14 Selection Sort: How It Works Before: x --> Swap x[4] with the smallest value in x[4:] After: x -->
15 Selection Sort: Recap
16 The Essential Helper Function: Select(x,i) def Select(x,i): """ Swaps the smallest value in x[i:] with x[i] PreC: x is a list of integers and i is an in that satisfies 0<=i<len(x)""" Does not return anything and it has a list argument
17 How Does it Work? The calling program has a list. E.g., a --> 0 ---> > > > > > 60
18 How Does it Work? The calling program executes Select(a,0) and control passes to Select a --> 0 ---> > > > > > 60
19 How Does Select Work? - Nothing new about the assignment of 0 to i. - But there is no assignment of the list a to x. - Instead x now refers to the same list as a. a --> x > i ---> > > > > > > 60
20 How Does Select Work? If inside Select we have t = x[0]; x[0] = x[2]; x[2] = t it is as if we said t = a[0]; a[0] = a[2]; a[2] = t a --> x > i ---> > > > > > > 60
21 How Does Select Work? It changes the list a in the calling program. We say x and a are aliased. They refer to the same list a --> x > i ---> > > > > > > 60
22 Let s Assume This Is Implemented def Select(x,i): """ Swaps the smallest value in x[i:] with x[i] PreC: x is a list of integers and i is an in that satisfies 0<=i<len(x)"""
23 After this: Initialization Select(a,0) Select(a,1) Select(a,2) Select(a,3) Select(a,4) Select(a,5) The list a looks like this
24 In General We Have This def SelectionSort(a): n = len(a) for k in range(n): Select(a,k)
25 Next Problem Merging Two Sorted Lists into a Single Sorted List
26 Example x-> y-> x and y are input They are sorted z is the output z->
27 Merging Two Sorted Lists x-> ix: 0 y-> ix and iy keep track of where we are in x and y iy: 0 z-> []
28 Merging Two Sorted Lists x-> ix: 0 y-> iy: 0 z-> [] Do we pick from x? x[ix] <= y[iy]???
29 Merge x-> ix: 0 y-> iy: 0 z-> 12 Yes. So update ix
30 Merge x-> ix: 1 y-> iy: 0 z-> 12 Do we pick from x? x[ix] <= y[iy]???
31 Merge x-> ix: 1 y-> iy: 0 z-> iz: No. So update iy
32 Merge x-> ix: 1 y-> iy: 1 z-> Do we pick from x? x[ix] <= y[iy]???
33 Merge x-> ix: 1 y-> iy: 1 z-> Yes. So update ix
34 Merge x-> ix: 2 y-> iy: 1 z-> Do we pick from x? x[ix] <= y[iy]???
35 Merge x-> ix: 2 y-> iy: 1 z-> Yes. So update ix
36 Merge x-> ix: 3 y-> iy: 1 z-> Do we pick from x? x[ix] <= y[iy]???
37 Merge x-> ix: 3 y-> iy: 1 z-> No. So update iy
38 Merge x-> ix: 3 y-> iy: 2 z-> Do we pick from x? x[ix] <= y[iy]???
39 Merge x-> ix: 3 y-> iy: 2 z-> Yes. So update ix.
40 Merge x-> ix: 4 y-> iy: 2 z-> Done with x. Pick from y
41 Merge x-> ix: 4 y-> iy: 2 z-> So update iy
42 Merge x-> ix: 4 y-> iy: 3 z-> Done with x. Pick from y
43 Merge x-> ix: 4 y-> iy: 3 z-> So update iy.
44 Merge x-> ix: 4 y-> iy: 4 z-> Done with x. Pick from y
45 Merge x-> ix: 4 y-> iy: 4 z-> Update iy
46 Merge x-> ix: 4 y-> iy: 5 z-> All Done
47 The Python Implementation
48 def Merge(x,y): n = len(x); m = len(y); ix = 0; iy = 0; z = [] for iz in range(n+m): if Build z up via repeated appending ix>=n: z.append(y[iy]); iy+=1 elif iy>=m: z.append(x[ix]); ix+=1 elif x[ix] <= y[iy]: z.append(x[ix]); ix+=1 elif x[ix] > y[iy]: z.append(y[iy]); iy+=1 return z x-list exhausted y-list exhausted x-value smaller y-value smaller
49 def Merge(x,y): n = len(x); m = len(y); ix = 0; iy = 0; z = [] for iz in range(n+m): if len(x)+len(y) is the total length of the merged list ix>=n: z.append(y[iy]); iy+=1 elif iy>=m: z.append(x[ix]); ix+=1 elif x[ix] <= y[iy]: z.append(x[ix]); ix+=1 elif x[ix] > y[iy]: z.append(y[iy]); iy+=1 return z x-list exhausted y-list exhausted x-value smaller y-value smaller
50 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] Make copies of the Incoming lists while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z
51 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] Build z up via repeated appending while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z
52 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] Every pop reduces the length by 1. The loop shuts down when one of u or v is exhausted while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z
53 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] g gets the popped value and it is appended to z while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z
54 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z Add what is left in u. OK if u is the empty list
55 Implementation Using Pop def Merge(x,y): u = list(x) v = list(y) z = [] while len(u)>0 and len(v)>0 : if u[0]<= v[0]: g = u.pop(0) else: g = v.pop(0) z.append(g) z.extend(u) z.extend(v) return z Add what is left in v. OK if v is the empty list
56 MergeSort Binary Search is an example of a divide and conquer approach top probkem solving. A method for sorting a list that features this strategy is MergeSort
57 Motivation You are asked to sort a list but you have two helpers : H1 and H2. Idea: 1. Split the list in half and have each helper sort one of the halves. 2. Then merge the two sorted lists into a single larger list. This idea can be repeated if H1 has two helpers and H2 has two helpers.
58 Subdivide the Sorting Task H E M G B K A Q F L P D R C J N H E M G B K A Q F L P D R C J N
59 Subdivide Again H E M G B K A Q F L P D R C J N H E M G B K A Q F L P D R C J N
60 And Again H E M G B K A Q F L P D R C J N H E M G B K A Q F L P D R C J N
61 And One Last Time H E M G B K A Q F L P D R C J N
62 Now Merge E H G M B K A Q F L D P C R J N H E M G B K A Q F L P D R C J N
63 And Merge Again E G H M A B K Q D F L P C J N R E H G M B K A Q F L D P C R J N
64 And Again A B E G H K M Q C D F J L N P R E G H M A B K Q D F L P C J N R
65 And One Last Time A B C D E F G H J K L M N P Q R A B E G H K M Q C D F J L N P R
66 Done! A B C D E F G H J K L M N P Q R
67 Done! A B C D E F G H J K L M N P Q R Let s write a function to do this making use of def Merge(x,y): """ Returns a float list that is the merge of sorted lists x and y. PreC: x and y are lists of floats that are sorted from small to big. """
68 8 Merges Producing length-2 lists E H G M B K A Q F L D P C R J N H E M G B K A Q F L P D R C J N
69 Handcoding the n =16 case A0 = Merge(a[0],a[1]) A1 = Merge(a[2],a[3]) A2 = Merge(a[4],a[5]) A3 = Merge(a[6],a[7]) A4 = Merge(a[8],a[9]) A5 = Merge(a[10],a[11]) A6 = Merge(a[12],a[13]) A7 = Merge(a[14],a[15])
70 4 Merges Producing Length-4 lists E G H M A B K Q D F L P C J N R E H G M B K A Q F L D P C R J N
71 Handcoding the n =16 case B0 = Merge(A0,A1) B1 = Merge(A2,A3) B2 = Merge(A4,A5) B3 = Merge(A6,A7)
72 2 Merges Producing Length-8 Lists A B E G H K M Q C D F J L N P R E G H M A B K Q D F L P C J N R
73 Handcoding the n =16 case C0 = Merge(B0,B1) C1 = Merge(B2,B3)
74 1 Merge Producing a Length-16 List A B C D E F G H J K L M N P Q R A B E G H K M Q C D F J L N P R
75 All Done! D0 = Merge(C0,C1) For general n, it can be handled using recursion.
76 Recursive Merge Sort def MergeSort(a): n = length(a) if n==1: return a else: m = n/2 u0 = list(a[:m]) u1 = list(a[m:]) y0 = MergeSort(u0) y1 = MergeSort(u1) return Merge(y0,y1) A function can call Itself!
77 Back To Merge Sort
78 Recursive Merge Sort def MergeSort(a): n = length(a) if n==1: return a else: m = n/2 u0 = list(a[:m]) u1 = list(a[m:]) y0 = MergeSort(u0) y1 = MergeSort(u1) return Merge(y0,y1) A function can call Itself!
79 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
80 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
81 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
82 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
83 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
84 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
85 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
86 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
87 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which
88 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
89 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
90 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
91 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
92 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
93 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: A Sorted List is produced at each : Let s look at the order in which lists are sorted.
94 A Schematic 0:16 0:8 8:16 0:4 4:8 8:12 12:16 0:2 2:4 4:6 6:8 8:10 10:12 12:14 14: All Done!
95 Some Conclusions Infinite recursion (like infinite loops) can happen so careful reasoning is required. Will we reach the base case? In MergeSort, a recursive call always involves a list that is shorter than the input list. So eventually we reach the len(a)==1 base case.
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