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1 Merge Sort Merge Sort 1

2 Divide-and-Conquer Divide-and conquer is a general algorithm design paradigm: Divide: divide the input data S in two disjoint subsets S 1 and S 2 Recur: solve the subproblems associated with S 1 and S 2 Conquer: combine the solutions for S 1 and S 2 into a solution for S The base case for the recursion are subproblems of size 0 or 1 Merge-sort is a sorting algorithm based on the divideand-conquer paradigm It accesses data in a sequential manner (suitable for sorting data on a disk) Merge Sort 2

3 Merge-Sort Merge-sort on an input sequence S with n elements consists of three steps: Divide: partition S into two sequences S 1 and S 2 of about n/2 elements each Recur: recursively sort S 1 and S 2 Conquer: merge S 1 and S 2 into a unique sorted sequence Merge Sort 3

4 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 4

5 Merging sorted arrays array A: array B: lowptr highptr workspace: 17 j Merge Sort 5

6 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 6

7 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 7

8 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 8

9 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 9

10 Merging sorted arrays array A: array B: 70 lowptr highptr workspace: j Merge Sort 10

11 Merging sorted arrays array A: array B: lowptr highptr workspace: j Merge Sort 11

12 Merging halves of an array Pass boundaries of sub-arrays to the algorithm instead of new arrays: l m m+1 r lowptr highptr Merge Sort 12

13 Implementation public static void recmergesort (int[] arr, int[] workspace, int l, int r) { if (l == r) { return; } else { int m = (l+r) / 2; recmergesort(arr, workspace, l, m); recmergesort(arr, workspace, m+1, r); merge(arr, workspace, l, m+1, r); } } Merge Sort 13

14 Merge-Sort Tree An execution of merge-sort is depicted by a binary tree each node represents a recursive call of merge-sort and stores unsorted sequence before the execution and its partition sorted sequence at the end of the execution the root is the initial call the leaves are calls on subsequences of size 0 or Merge Sort 14

15 Execution Example Partition Merge Sort 15

16 Execution Example (cont.) Recursive call, partition Merge Sort 16

17 Execution Example (cont.) Recursive call, partition Merge Sort 17

18 Execution Example (cont.) Recursive call, base case Merge Sort 18

19 Execution Example (cont.) Recursive call, base case Merge Sort 19

20 Execution Example (cont.) Merge Merge Sort 20

21 Execution Example (cont.) Recursive call,, base case, merge Merge Sort 21

22 Execution Example (cont.) Merge Merge Sort 22

23 Execution Example (cont.) Recursive call,, merge, merge Merge Sort 23

24 Execution Example (cont.) Merge Merge Sort 24

25 Analysis of Merge-Sort The height h of the merge-sort tree is O(log n) at each recursive call we divide in half the sequence, The overall amount or work done at the nodes of depth i is O(n) we partition and merge 2 i sequences of size n/2 i we make 2 i+1 recursive calls Thus, the total running time of merge-sort is O(n log n) depth 0 1 i #seqs i size n n/2 n/2 i Merge Sort 25

26 Using merge sort Fast sorting method for arrays: this is what Java s Arrays.sort uses, for example Good for sorting data in external memory because works with adjacent indices in the array Not so good with lists: relies on constant time access to the middle of the sequence Merge Sort 26

27 Quick-Sort Merge Sort 27

28 Quick-Sort Quick-sort is a randomized sorting algorithm based on the divide-and-conquer paradigm: x Divide: pick a random element x (called pivot) and partition S into x L elements less than x E elements equal x L E G G elements greater than x Recur: sort L and G Conquer: join L, E and G x Merge Sort 28

29 Partition of lists (or using extra workspace) We partition an input sequence as follows: We remove, in turn, each element y from S and We insert y into L, E or G, depending on the result of the comparison with the pivot x Each insertion and removal is at the beginning or at the end of a sequence, and hence takes O(1) time Thus, the partition step of quick-sort takes O(n) time Merge Sort 29

30 Partitioning arrays Perform the partition using two indices to split S into L and E+G. j k Repeat until j and k cross: Scan j to the right until finding an element > x. Scan k to the left until finding an element < x. Swap elements at indices j and k (pivot = 6) j k Merge Sort 30

31 Implementation public static void recquicksort(int[] arr, int left, int right) { if (right - left <= 0) return; else { int border = partition(arr, left, right); recquicksort(arr, left, border-1); recquicksort(arr, border+1, right); } } Merge Sort 31

32 Quick-Sort Tree An execution of quick-sort is depicted by a binary tree Each node represents a recursive call of quick-sort and stores Unsorted sequence before the execution and its pivot Sorted sequence at the end of the execution The root is the initial call The leaves are calls on subsequences of size 0 or Merge Sort 32

33 Execution Example Pivot selection Merge Sort 33

34 Execution Example (cont.) Partition, recursive call, pivot selection Merge Sort 34

35 Execution Example (cont.) Partition, recursive call, base case Merge Sort 35

36 Execution Example (cont.) Recursive call,, base case, join Merge Sort 36

37 Execution Example (cont.) Recursive call, pivot selection Merge Sort 37

38 Execution Example (cont.) Partition,, recursive call, base case Merge Sort 38

39 Execution Example (cont.) Join, join Merge Sort 39

40 Worst-case Running Time The worst case for quick-sort occurs when the pivot is the unique minimum or maximum element One of L and G has size n 1 and the other has size 0 The running time is proportional to the sum n + (n 1) Thus, the worst-case running time of quick-sort is O(n 2 ) depth 0 1 time n n 1 n 1 1 Merge Sort 40

41 Best-case Running Time The best case for quick-sort occurs when the pivot is the median element The L and G parts are equal the sequence is split in halves, like in merge sort Thus, the best-case running time of quick-sort is O(n log n) Merge Sort 41

42 Average-case Running Time The average case for quick-sort: half of the times, the pivot is roughly in the middle Thus, the average-case running time of quick-sort is O(n log n) again Detailed proof in the textbook Merge Sort 42

43 Summary of Sorting Algorithms Algorithm selection-sort insertion-sort quick-sort merge-sort heap-sort (later in the module) Time O(n 2 ) O(n 2 ) O(n log n) expected O(n log n) O(n log n) Notes in-place (no extra memory) slow (good for small inputs) in-place (no extra memory) slow (good for small inputs) in-place (+stack), randomized fast (good for large inputs) sequential data access fast (good for huge inputs) in-place (see G & T for the in-place version) fast (good for large inputs) Merge Sort 43

44 Comparison sorting A sorting algorithm is a comparison sorting algorithm if it uses comparisons between elements in the sequence to determine in which order to place them Examples of comparison sorts: bubble sort, selection sort, insertion sort, heap sort, merge sort, quicksort. Example of not a comparison sort: bucket sort (Ch. 11.4). Runs in O(n), relies on knowing the range of values in the sequence (e.g. integers between 1 and 1000). Merge Sort 44

45 Lower bound for comparison sort We can model sorting which depends on comparisons between elements as a binary decision tree. At each node, a comparison between two elements is made; there are two possible outcomes and we find out a bit more about the correct order of items in the array. Finally arrive at full information about the correct order of the items in the array. Merge Sort 45

46 Comparison sorting a 2 < a 3? yes a 1,,a n : don t know what the correct order is. a 1 < a 2? yes no Correct order is (a 1,,a n ) (a 2, a 1,...,a n ) (total of n! possibilities) Merge Sort 46

47 How many comparisons? If a binary tree has n! leaves, than the minimal number of levels (assuming the tree is perfect) is (log 2 n!) +1. This shows that O(n log n) sorting algorithms are essentially optimal (log 2 n! is not equal to n log 2 n, but has the same growth rate modulo some hairy constants). Merge Sort 47

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