Algorithmic Analysis and Sorting, Part Two
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1 Algorithmic Analysis and Sorting, Part Two
2 Friday Four Square! 4:15PM, Outside Gates
3 An Initial Idea: Selection Sort
4 An Initial Idea: Selection Sort
5 An Initial Idea: Selection Sort
6 An Initial Idea: Selection Sort
7 An Initial Idea: Selection Sort
8 An Initial Idea: Selection Sort
9 An Initial Idea: Selection Sort
10 An Initial Idea: Selection Sort
11 An Initial Idea: Selection Sort
12 An Initial Idea: Selection Sort
13 An Initial Idea: Selection Sort
14 An Initial Idea: Selection Sort
15 An Initial Idea: Selection Sort
16 An Initial Idea: Selection Sort
17 An Initial Idea: Selection Sort
18 An Initial Idea: Selection Sort
19 An Initial Idea: Selection Sort
20 An Initial Idea: Selection Sort
21 An Initial Idea: Selection Sort
22 An Initial Idea: Selection Sort
23 An Initial Idea: Selection Sort
24 An Initial Idea: Selection Sort
25 An Initial Idea: Selection Sort
26 Notes on Selection Sort Selection sort is O(n2 ) in the worst case. Selection sort is O(n2 ) in the best case. Notation: Selection sort is Θ(n2 ).
27 Another Idea: Insertion Sort
28 Another Idea: Insertion Sort
29 Another Idea: Insertion Sort
30 Another Idea: Insertion Sort
31 Another Idea: Insertion Sort
32 Another Idea: Insertion Sort
33 Another Idea: Insertion Sort
34 Another Idea: Insertion Sort
35 Another Idea: Insertion Sort
36 Another Idea: Insertion Sort
37 Another Idea: Insertion Sort
38 Another Idea: Insertion Sort
39 Another Idea: Insertion Sort
40 Another Idea: Insertion Sort
41 Another Idea: Insertion Sort
42 Another Idea: Insertion Sort
43 Another Idea: Insertion Sort
44 Another Idea: Insertion Sort
45 Another Idea: Insertion Sort
46 Another Idea: Insertion Sort
47 Another Idea: Insertion Sort
48 Another Idea: Insertion Sort
49 Another Idea: Insertion Sort
50 Another Idea: Insertion Sort
51 Another Idea: Insertion Sort
52 Insertion Sort in Code void InsertionSort(Vector<int>& v) { for (int i = 0; i < v.size(); i++) { for (int j = i 1; j >= 0; j--) { if (v[j] < v[j + 1]) break; swap(v[j], v[j + 1]); } } }
53 How Fast is Insertion Sort?
54 How Fast is Insertion Sort?
55 How Fast is Insertion Sort?
56 How Fast is Insertion Sort?
57 How Fast is Insertion Sort?
58 How Fast is Insertion Sort?
59 How Fast is Insertion Sort?
60 How Fast is Insertion Sort?
61 How Fast is Insertion Sort?
62 How Fast is Insertion Sort?
63 How Fast is Insertion Sort?
64 How Fast is Insertion Sort?
65 How Fast is Insertion Sort?
66 How Fast is Insertion Sort?
67 How Fast is Insertion Sort? Work done: O(n)
68 How Fast is Insertion Sort?
69 How Fast is Insertion Sort?
70 How Fast is Insertion Sort?
71 How Fast is Insertion Sort?
72 How Fast is Insertion Sort?
73 How Fast is Insertion Sort?
74 How Fast is Insertion Sort?
75 How Fast is Insertion Sort?
76 How Fast is Insertion Sort?
77 How Fast is Insertion Sort?
78 How Fast is Insertion Sort?
79 How Fast is Insertion Sort?
80 How Fast is Insertion Sort?
81 How Fast is Insertion Sort?
82 How Fast is Insertion Sort?
83 How Fast is Insertion Sort?
84 How Fast is Insertion Sort?
85 How Fast is Insertion Sort?
86 How Fast is Insertion Sort?
87 How Fast is Insertion Sort?
88 How Fast is Insertion Sort?
89 How Fast is Insertion Sort?
90 How Fast is Insertion Sort?
91 How Fast is Insertion Sort?
92 How Fast is Insertion Sort?
93 How Fast is Insertion Sort?
94 How Fast is Insertion Sort?
95 How Fast is Insertion Sort?
96 How Fast is Insertion Sort?
97 How Fast is Insertion Sort?
98 How Fast is Insertion Sort? Work done: O(n 2 )
99 Notes on Insertion Sort Insertion sort is O(n) in the best case. Insertion sort is O(n2 ) in the worst case. On average, insertion sort is roughly twice as fast as selection sort. In a random array, every element is about halfway away from where it belongs. So for the kth element, about (k 1) / 2 other elements must be looked at.
100 Selection Sort vs Insertion Sort Size Selection Sort Insertion Sort
101 Thinking About O(n 2 ) T(n)
102 Thinking About O(n 2 ) T(n)
103 Thinking About O(n 2 ) T(n) T(½n) ¼T(n) T(½n) ¼T(n)
104 Thinking About O(n 2 ) T(n) T(½n) ¼T(n) T(½n) ¼T(n)
105 Thinking About O(n 2 ) T(n) T(½n) ¼T(n) T(½n) ¼T(n) It It takes takes roughly roughly ½T(n) ½T(n) to to sort sort each each half half separately! separately!
106 The Key Insight: Merge
107 The Key Insight: Merge
108 The Key Insight: Merge
109 The Key Insight: Merge
110 The Key Insight: Merge
111 The Key Insight: Merge
112 The Key Insight: Merge
113 The Key Insight: Merge
114 The Key Insight: Merge
115 The Key Insight: Merge
116 The Key Insight: Merge
117 The Key Insight: Merge
118 The Key Insight: Merge
119 The Key Insight: Merge
120 The Key Insight: Merge
121 The Key Insight: Merge
122 The Key Insight: Merge
123 The Key Insight: Merge
124 The Key Insight: Merge
125 The Key Insight: Merge
126 The Key Insight: Merge
127 The Key Insight: Merge
128 Split Sort
129 Split Sort void SplitSort(Vector<int>& v) { Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int j = v.size() / 2; j < v.size(); j++) right += v[i]; InsertionSort(left); InsertionSort(right); } Merge(left, right, v);
130 Performance Comparison Size Selection Sort Insertion Sort Split Sort
131 Performance Comparison Size Selection Sort Insertion Sort Split Sort
132 A Better Idea Splitting the input in half and merging halves the work. So why not split into four? Or eight? Question: What happens if we never stop splitting?
133
134
135
136
137
138
139 High-Level Idea A recursive sorting algorithm! Base case: An empty or single-element list is already sorted. Recursive step: Break the list in half and recursively sort each part. Use merge to combine them back into a single sorted list. This algorithm is called mergesort.
140 Code for Mergesort
141 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
142 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
143 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
144 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
145 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
146 What is the complexity of mergesort?
147 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
148 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
149 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
150 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); } /* Combine them together. */ Merge(left, right, v);
151 Code for Mergesort void Mergesort(Vector<int>& v) { /* Base case: 0- or 1-element lists are already sorted. */ if (v.size() <= 1) return; /* Split v into two subvectors. */ Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; } /* Recursively sort these arrays. */ Mergesort(left); Mergesort(right); /* Combine them together. */ Merge(left, right, v); How How do do we we analyze this this step? step?
152 A Graphical Intuition O(n) O(n) O(n) O(n) O(n)
153 A Graphical Intuition O(n) O(n) O(n) O(n) O(n) How many levels are there?
154 Slicing and Dicing After zero recursive calls: N After one recursive call: N / 2 After two recursive calls: N / 4 After three recursive calls: N / 8 After k recursive calls: N / 2 k
155 Cutting in Half After k recursive calls, there are N / 2 k elements left. Mergesort stops recursing when there are zero or one elements left. Solving for the number of levels: N / 2 k = 1 N = 2 k log 2 N = k So mergesort recurses log 2 N levels deep.
156 A Graphical Intuition O(n) O(n) O(n) O(n) O(n) O(n log n)
157 Mergesort Times Size Selection Sort Insertion Sort Split Sort Mergesort
158 Mergesort Times Size Selection Sort Insertion Sort Split Sort Mergesort
159 Analysis of Mergesort Mergesort is O(n log n). This is asymptotically better than O(n2 ) Can we do better? In general, no: comparison-based sorts cannot have a worst-case runtime better than O(n log n). We can only get faster by a constant factor!
160 Growth Rates O(n) O(n log n) O(n^2)
161 A Trivial Observation
162 A Trivial Observation
163 A Trivial Observation
164 A Trivial Observation
165 A Trivial Observation
166 A Trivial Observation
167 So What? This idea leads to a particularly clever sorting algorithm. Idea: Pick an element from the array. Put the smaller elements on one side. Put the bigger elements on the other side. Recursively sort each half. But how do we do the middle two steps?
168 Partitioning Pick a pivot element. Move everything less than the pivot to the left of the pivot. Move everything greater than the pivot to the right of the pivot. Good news: O(n) algorithm exists! Bad news: it's a bit tricky...
169 The Partition Algorithm
170 The Partition Algorithm
171 The Partition Algorithm
172 The Partition Algorithm
173 The Partition Algorithm
174 The Partition Algorithm
175 The Partition Algorithm
176 The Partition Algorithm
177 The Partition Algorithm
178 The Partition Algorithm
179 The Partition Algorithm
180 The Partition Algorithm
181 The Partition Algorithm
182 The Partition Algorithm
183 The Partition Algorithm
184 The Partition Algorithm
185 The Partition Algorithm
186 The Partition Algorithm
187 The Partition Algorithm
188 The Partition Algorithm
189 Code for Partition
190 Code for Partition int Partition(Vector<int>& v, int low, int high) { int pivot = v[low]; int left = low + 1, right = high; while (left < right) { while (left < right && v[right] >= pivot) --right; while (left < right && v[left] < pivot) ++left; if (left < right) swap(v[left], v[right]); } if (pivot < v[right]) return low; } swap (v[low], v[right]); return right;
191 A Partition-Based Sort Idea: Partition the array around some element. Recursively sort the left and right halves. This works extremely quickly. In fact... the algorithm is called quicksort.
192 Quicksort
193 Quicksort void Quicksort(Vector<int>& v, int low, int high) { if (low >= high) return; } int partitionpoint = Partition(v, low, high); Quicksort(v, low, partitionpoint 1); Quicksort(v, partitionpoint + 1, high);
194 How fast is quicksort?
195 It depends on our choice of pivot.
196 Suppose we get lucky...
197 Suppose we get lucky...
198 Suppose we get lucky...
199 Suppose we get lucky...
200 Suppose we get lucky... O(n log n)
201 Suppose we get sorta lucky...
202 Suppose we get sorta lucky...
203 Suppose we get sorta lucky...
204 Suppose we get sorta lucky...
205 Suppose we get sorta lucky...
206 Suppose we get sorta lucky... O(n log n)
207 Suppose we get unlucky
208 Suppose we get unlucky
209 Suppose we get unlucky
210 Suppose we get unlucky
211 Suppose we get unlucky...
212 Suppose we get unlucky n n - 1 n - 2 n
213 Suppose we get unlucky n n - 1 n - 2 n O(n 2 )
214 Quicksort is Strange In most cases, quicksort is O(n log n). However, in the worst case, quicksort is O(n2 ). How can you avoid this? Pick better pivots! Pick the median. Can be done in O(n), but expensive O(n). Pick the median-of-three. Better than nothing, but still can hit worst case. Pick randomly. Extremely low probability of O(n 2 ).
215 Quicksort is Fast Although quicksort is O(n2 ) in the worst case, it is one of the fastest known sorting algorithms. O(n2 ) behavior is extremely unlikely with random pivots; runtime is usually a very good O(n log n). It's hard to argue with the numbers...
216 Timing Quicksort Size Selection Sort Insertion Sort Split Sort Mergesort Quicksort
217 Timing Quicksort Size Selection Sort Insertion Sort Split Sort Mergesort Quicksort
218 An Interesting Observation Big-O notation talks about long-term growth, but says nothing about small inputs. For small inputs, insertion sort can be better than mergesort or quicksort. Why? Mergesort and quicksort both have high overhead per call. Insertion sort is extremely simple.
219 Hybrid Sorts Combine multiple sorting algorithms together to get advantages of each. Insertion sort is good on small inputs. Merge sort is good on large inputs. Can we combine them?
220 Hybrid Mergesort
221 Hybrid Mergesort void HybridMergesort(Vector<int>& v) { if (v.size() <= 8) { InsertionSort(v); } else { Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; HybridMergesort(left); HybridMergesort(right); } } Merge(left, right, v);
222 Hybrid Mergesort void HybridMergesort(Vector<int>& v) { if (v.size() <= 8) { InsertionSort(v); } else { Vector<int> left, right; for (int i = 0; i < v.size() / 2; i++) left += v[i]; for (int i = v.size() / 2; i < v.size(); i++) right += v[i]; HybridMergesort(left); HybridMergesort(right); } } Merge(left, right, v);
223 Runtime for Hybrid Mergesort Size Mergesort Hybrid Mergesort Quicksort
224 Hybrid Sorts in Practice Introspective Sort (Introsort) Based on quicksort, insertion sort, and heapsort. Heapsort is Θ(n log n) and a bit faster than mergesort. Uses quicksort, then switches to heapsort if it looks like the algorithm is degenerating to O(n 2 ). Uses insertion sort for small inputs. Gains the raw speed of quicksort without any of the drawbacks.
225 Runtime for Introsort Size Mergesort Hybrid Mergesort Quicksort Introsort
226 Runtime for Introsort Size Mergesort Hybrid Mergesort Quicksort Introsort
227 We've spent all of our time talking about fast and efficient sorting algorithms.
228 However, we have neglected to find slow and inefficient sorting algorithms.
229
230 Introducing Bogosort Intuition: Suppose you want to sort a deck of cards. Check if it's sorted. If so, you're done. Otherwise, shuffle the deck and repeat. This is O(n n!) in the average case. Could run for much longer...
231 Further Topics in Sorting Heapsort Extremely fast Θ(n log n) sorting algorithm. Radix Sort Sorts integers in O(n log U) by working one digit at a time, where U is the largest integer being sorted. Used by old IBM sorting machines. Bead Sort Ingenious sorting algorithm for integers. Uses gravity... how cool is that?
Algorithmic Analysis and Sorting, Part Two. CS106B Winter
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