Sorting. CMPS 2200 Fall Carola Wenk Slides courtesy of Charles Leiserson with small changes by Carola Wenk
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1 CMPS 2200 Fall 2014 Sorting Carola Wenk Slides courtesy of Charles Leiserson with small changes by Carola Wenk 11/11/14 CMPS 2200 Intro. to Algorithms 1
2 How fast can we sort? All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements. E.g., insertion sort, merge sort, heapsort. The best worst-case running time that we ve seen for comparison sorting is O(n log n). Is O(n log n) the best we can do? Decision trees can help us answer this question. Déjà vu from CMPS 1500: des/theory%20and%20frontiers%20of%20computer%20science_1.pdf 11/11/14 CMPS 2200 Intro. to Algorithms 2
3 Decision-tree model A decision tree models the execution of any comparison sorting algorithm: One tree per input size n. The tree contains all possible comparisons (= if-branches) that could be executed for any input of size n. The tree contains all comparisons along all possible instruction traces (= control flows) for all inputs of size n. For one input, only one path to a leaf is executed. Running time = length of the path taken. Worst-case running time = height of tree. 11/11/14 CMPS 2200 Intro. to Algorithms
4 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 :a 2 a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a a 2 a 1 a a 2 :a insert a 2 a 1 a a 2 a a 1 a 2 a 2 a a 1 a a 2 a 1 Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 4
5 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 :a 2 a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a a 2 a 1 a a 2 :a insert a 2 a 1 a a 2 a a 1 a 2 a 2 a a 1 a a 2 a 1 Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 5
6 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 :a a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a a 2 a 1 a a 2 :a insert a 2 a 1 a a 2 a a 1 a 2 a 2 a a 1 a a 2 a 1 Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 6
7 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 :a 2 a 1 a 2 a insert a a a 1 a 2 a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a 9 6 a 2 a 1 a a 2 a 1 a a 2 :a insert a 2 a 1 a a 2 a a 1 a 2 a 2 a a 1 a a 2 a 1 Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 7
8 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 :a 2 a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a a 2 a 1 a a 2 :a 4 6 insert a 2 a 1 a a 2 a a 1 a 2 a 2 a a 1 a a 2 a 1 Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 8
9 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 a a 2 a a 1 a 2 a 1 :a 2 a 2 a 1 a a 2 :a a 2 a a 1 a a 2 a Each internal node is labeled a i :a j for i, j {1, 2,, n}. The left subtree shows subsequent comparisons if a i a j. The right subtree shows subsequent comparisons if a i a j. 11/11/14 CMPS 2200 Intro. to Algorithms 9 a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a insert a 2
10 Decision-tree for insertion sort Sort a 1, a 2, a a 1 a 2 a a 1 :a a 1 a a 2 a a 1 a 2 a 1 :a 2 a 2 a 1 a a 2 :a a 2 a a 1 a a 2 a 1 11/11/14 CMPS 2200 Intro. to Algorithms 10 a 1 a 2 a insert a a a 2 :a a 1 :a 2 a 1 a insert a a 1 a 2 a a 1 a 2 a a 2 a 1 a 46 9 insert a 2 Each leaf contains a permutation,,, (n) to indicate that the ordering a (1) a (2) a (n) has been established.
11 Decision-tree model A decision tree models the execution of any comparison sorting algorithm: One tree per input size n. The tree contains all possible comparisons (= if-branches) that could be executed for any input of size n. The tree contains all comparisons along all possible instruction traces (= control flows) for all inputs of size n. For one input, only one path to a leaf is executed. Running time = length of the path taken. Worst-case running time = height of tree. 11/11/14 CMPS 2200 Intro. to Algorithms 11
12 Lower bound for comparison sorting Theorem. Any decision tree that can sort n elements must have height (n log n). Proof. The tree must contain n! leaves, since there are n! possible permutations. For a binary tree of height-h holds that #leaves 2 h. Thus, n! 2 h. h log(n!) log ((n/2) n/2 ) = n/2 log n/2 h (n log n). (log is mono. increasing) 11/11/14 CMPS 2200 Intro. to Algorithms 12
13 Lower bound for comparison sorting Corollary. Mergesort is an asymptotically optimal comparison sorting algorithm. 11/11/14 CMPS 2200 Intro. to Algorithms 1
14 Sorting in linear time Counting sort: No comparisons between elements. Input: A[0.. n-1], where A[ j]{0, 1, 2,, k-1}. Output: B[0.. n-1], sorted. Auxiliary storage: C[0.. k-1]. 11/11/14 CMPS 2200 Intro. to Algorithms 14
15 Counting sort for (i =0; i < k; i++) C[i] = 0 for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 for (i =1; i < k; i++) C[i] = C[i] + C[i 1] for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 // C[i] == {key = i} // C[i] == {key i} 11/11/14 CMPS 2200 Intro. to Algorithms 15
16 Counting-sort example C: B: 11/11/14 CMPS 2200 Intro. to Algorithms 16
17 Loop C: B: 1. for (i =0; i < k; i++) C[i] = 0 11/11/14 CMPS 2200 Intro. to Algorithms 17
18 Loop C: B: 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == {key = i} 11/11/14 CMPS 2200 Intro. to Algorithms 18
19 Loop C: B: 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == {key = i} 11/11/14 CMPS 2200 Intro. to Algorithms 19
20 Loop C: B: 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == {key = i} 11/11/14 CMPS 2200 Intro. to Algorithms 20
21 Loop C: B: 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == {key = i} 11/11/14 CMPS 2200 Intro. to Algorithms 21
22 Loop C: B: 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == {key = i} 11/11/14 CMPS 2200 Intro. to Algorithms 22
23 Loop C: B: C': for (i =1; i < k; i++) C[i] = C[i] + C[i 1] // C[i] == {key i} 11/11/14 CMPS 2200 Intro. to Algorithms 2
24 Loop C: B: C': for (i =1; i < k; i++) C[i] = C[i] + C[i 1] // C[i] == {key i} 11/11/14 CMPS 2200 Intro. to Algorithms 24
25 Loop C: B: C': for (i =1; i < k; i++) C[i] = C[i] + C[i 1] // C[i] == {key i} 11/11/14 CMPS 2200 Intro. to Algorithms 25
26 Loop C: B: 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 26
27 Loop C: B: 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 27
28 Loop C: B: 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 28
29 Loop C: B: 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 29
30 Loop C: B: 2 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 0
31 Loop C: B: 2 2 C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 1
32 Loop C: B: C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 2
33 Loop C: B: C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms
34 Loop C: B: C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 4
35 Loop C: B: C': for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 5
36 Analysis (k) (n) (k) (n) (n + k) 1. for (i =0; i < k; i++) C[i] = 0 2. for (j =0; i < n; j++) C[A[ j]] = C[A[ j]] + 1. for (i =1; i < k; i++) C[i] = C[i] + C[i 1] 4. for (j = n-1; i 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] 1 11/11/14 CMPS 2200 Intro. to Algorithms 6
37 Running time If k = O(n), then counting sort takes (n) time. But, sorting takes (n log n) time! Where s the fallacy? Answer: Comparison sorting takes (n log n) time. Counting sort is not a comparison sort. In fact, not a single comparison between elements occurs! 11/11/14 CMPS 2200 Intro. to Algorithms 7
38 Stable sorting Counting sort is a stable sort: it preserves the input order among equal elements. B: Exercise: What other sorts have this property? 11/11/14 CMPS 2200 Intro. to Algorithms 8
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