Merge Sort Goodrich, Tamassia. Merge Sort 1


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1 Merge Sort Merge Sort 1
2 DivideandConquer Divideand 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 Mergesort is a sorting algorithm based on the divideandconquer paradigm It accesses data in a sequential manner (suitable for sorting data on a disk) Merge Sort 2
3 MergeSort Mergesort 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 subarrays 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 MergeSort Tree An execution of mergesort is depicted by a binary tree each node represents a recursive call of mergesort 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 MergeSort The height h of the mergesort 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 mergesort 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 QuickSort Merge Sort 27
28 QuickSort Quicksort is a randomized sorting algorithm based on the divideandconquer 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 quicksort 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, border1); recquicksort(arr, border+1, right); } } Merge Sort 31
32 QuickSort Tree An execution of quicksort is depicted by a binary tree Each node represents a recursive call of quicksort 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 Worstcase Running Time The worst case for quicksort 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 worstcase running time of quicksort is O(n 2 ) depth 0 1 time n n 1 n 1 1 Merge Sort 40
41 Bestcase Running Time The best case for quicksort 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 bestcase running time of quicksort is O(n log n) Merge Sort 41
42 Averagecase Running Time The average case for quicksort: half of the times, the pivot is roughly in the middle Thus, the averagecase running time of quicksort is O(n log n) again Detailed proof in the textbook Merge Sort 42
43 Summary of Sorting Algorithms Algorithm selectionsort insertionsort quicksort mergesort heapsort (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 inplace (no extra memory) slow (good for small inputs) inplace (no extra memory) slow (good for small inputs) inplace (+stack), randomized fast (good for large inputs) sequential data access fast (good for huge inputs) inplace (see G & T for the inplace 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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