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Merge Sort

7 2  9 4  2 4 7 9. 7  2  2 7. 9  4  4 9. 7  7. 2  2. 9  9. 4  4. Merge Sort. Divide-and-conquer ( 分治演算法 ) is a general algorithm design paradigm: Divide : divide the input data S in two disjoint subsets S 1 and S 2

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Merge Sort

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  1. 7 2  9 4 2 4 7 9 7  2 2 7 9  4 4 9 7 7 2 2 9 9 4 4 Merge Sort Merge Sort

  2. Divide-and-conquer (分治演算法) is a general algorithm design paradigm: Divide: divide the input data S in two disjoint subsets S1and S2 Conquer: solve the subproblems associated with S1and S2 Combine: combine the solutions for S1and S2 into a solution for S Merge-sort: A sorting algorithm based on divide and conquer Like heap-sort It uses a comparator It has O(n log n) running time Unlike heap-sort It does not use an auxiliary priority queue It accesses data in a sequential/local manner (suitable to sort data on a disk) Divide-and-Conquer (§ 10.1.1) 化整為零 各個擊破 Some don’t! Good for external sorting Merge Sort

  3. Merge Sort • Merge sort • A divide-and-conquer algorithm • Invented by John von Neumann in 1945 約翰·馮·紐曼（John von Neumann，1903年12月28日－1957年2月8日），出生於匈牙利的美國籍猶太人數學家，現代電腦創始人之一。他在電腦科學、經濟、物理學中的量子力學及幾乎所有數學領域都作過重大貢獻，被譽為「電腦之父」。(圖及說明摘自維基百科) Merge Sort

  4. Three steps of merge-sort on an input sequence S with n elements: Divide: partition S into two sequences S1and S2 of about n/2 elements each Conquer: recursively sort S1and S2 Combine: merge S1and S2 into a sorted sequence Merge-Sort (§ 10.1) AlgorithmmergeSort(S, C) Inputsequence S with n elements, comparator C Outputsequence S sorted according to C ifS.size() > 1 (S1, S2)partition(S, n/2) mergeSort(S1, C) mergeSort(S2, C) Smerge(S1, S2) Key step! Merge Sort

  5. Merging Two Sorted Sequences Algorithmmerge(A, B) Inputsequences A and B withn/2 elements each Outputsorted sequence of A  B S empty sequence whileA.empty() B.empty() ifA.front()<B.front() S.addBack(A.front()); A.eraseFront(); else S.addBack(B.front()); B.eraseFront(); whileA.empty() S.addBack(A.front()); A.eraseFront(); whileB.empty() S.addBack(B.front()); B.eraseFront(); return S • Merging two sorted sequences (implemented as linked lists) with n/2 elements each, takes O(n) time. Merge Sort

  6. To Merge 2 Sorted Sequences Properties Need extra space to store the sorted results  Not an in-place sort Total time = O(|A|+|B|) = O(m+n) A B Also good for singly linked lists 6

  7. 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 1 7 2  9 4 2 4 7 9 7  2 2 7 9  4 4 9 7 7 2 2 9 9 4 4 Merge Sort

  8. 7 2 9 4  2 4 7 9 3 8 6 1  1 3 8 6 7 2  2 7 9 4  4 9 3 8  3 8 6 1  1 6 7  7 2  2 9  9 4  4 3  3 8  8 6  6 1  1 Execution Example • Partition 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 Merge Sort

  9. 7 2  2 7 9 4  4 9 3 8  3 8 6 1  1 6 7  7 2  2 9  9 4  4 3  3 8  8 6  6 1  1 Execution Example (cont.) • Recursive call, partition 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 Merge Sort

  10. 7  7 2  2 9  9 4  4 3  3 8  8 6  6 1  1 Execution Example (cont.) • Recursive call, partition 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 7  2 2 7 9 4  4 9 3 8  3 8 6 1  1 6 Merge Sort

  11. 7  2 2 7 9 4  4 9 3 8  3 8 6 1  1 6 Execution Example (cont.) • Recursive call, base case 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 77 2  2 9  9 4  4 3  3 8  8 6  6 1  1 Merge Sort

  12. Execution Example (cont.) • Recursive call, base case 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 7  2 2 7 9 4  4 9 3 8  3 8 6 1  1 6 77 22 9  9 4  4 3  3 8  8 6  6 1  1 Merge Sort

  13. Execution Example (cont.) • Merge 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 7  22 7 9 4  4 9 3 8  3 8 6 1  1 6 77 22 9  9 4  4 3  3 8  8 6  6 1  1 Merge Sort

  14. Execution Example (cont.) • Recursive call, …, base case, merge 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 4 2 4 7 9 3 8 6 1  1 3 8 6 7  22 7 9 4  4 9 3 8  3 8 6 1  1 6 77 22 9 9 4 4 3  3 8  8 6  6 1  1 Merge Sort

  15. Execution Example (cont.) • Merge 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 42 4 7 9 3 8 6 1  1 3 8 6 7  22 7 9 4  4 9 3 8  3 8 6 1  1 6 77 22 9 9 4 4 3  3 8  8 6  6 1  1 Merge Sort

  16. Execution Example (cont.) • Recursive call, …, merge, merge 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 42 4 7 9 3 8 6 1  1 3 6 8 7  22 7 9 4  4 9 3 8 3 8 6 1  1 6 77 22 9 9 4 4 33 88 66 11 Merge Sort

  17. Execution Example (cont.) • Merge 7 2 9 4  3 8 6 11 2 3 4 6 7 8 9 7 2  9 42 4 7 9 3 8 6 1  1 3 6 8 7  22 7 9 4  4 9 3 8 3 8 6 1  1 6 77 22 9 9 4 4 33 88 66 11 Merge Sort

  18. Merge Sort: Example divide merge 18

  19. Why Merge Sort Is Stable? 50 63 Same relative positions after merging!

  20. Resources on Merge Sort • Numerous resources on merge sort • Wiki • Animation by sorting a vector • Animation by dots • Youtube • Detailed explanation with pseudo code Merge Sort

  21. 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 2i sequences of size n/2i • we make 2i+1 recursive calls • Thus, the total running time of merge-sort is O(n log n) O(n) O(n/2) O(n/4) Merge Sort

  22. Summary of Sorting Algorithms Merge Sort

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