MergeSort.ppt algorithm History and Examples
- 1. 1 Merge Sort • Review of Sorting • Merge Sort
- 2. 2 Sorting Algorithms • Selection Sort uses a priority queue P implemented with an unsorted sequence: – Phase 1: the insertion of an item into P takes O(1) time; overall O(n) time for inserting n items. – Phase 2: removing an item takes time proportional to the number of elements in P, which is O(n): overall O(n2 ) – Time Complexity: O(n2 )
- 3. 3 Sorting Algorithms (cont.) • Insertion Sort is performed on a priority queue P which is a sorted sequence: – Phase 1: the first insertItem takes O(1), the second O(2), until the last insertItem takes O(n): overall O(n2 ) – Phase 2: removing an item takes O(1) time; overall O(n). – Time Complexity: O(n2 ) • Heap Sort uses a priority queue K which is a heap. – insertItem and removeMin each take O(logk), k being the number of elements in the heap at a given time. – Phase 1: n elements inserted: O (nlogn) time – Phase 2: n elements removed: O (nlogn) time. – Time Complexity: O (nlog n)
- 4. 4 Divide-and-Conquer • Divide and Conquer is more than just a military strategy; it is also a method of algorithm design that has created such efficient algorithms as Merge Sort. • In terms or algorithms, this method has three distinct steps: – Divide: If the input size is too large to deal with in a straightforward manner, divide the data into two or more disjoint subsets. – Recur: Use divide and conquer to solve the subproblems associated with the data subsets. – Conquer: Take the solutions to the subproblems and “merge” these solutions into a solution for the original problem.
- 5. 5 Merge-Sort • Algorithm: – Divide: If S has at leas two elements (nothing needs to be done if S has zero or one elements), remove all the elements from S and put them into two sequences, S1 and S2, each containing about half of the elements of S. (i.e. S1 contains the first n/2 elements and S2 contains the remaining n/2 elements. – Recur: Recursive sort sequences S1 and S2. – Conquer: Put back the elements into S by merging the sorted sequences S1 and S2 into a unique sorted sequence. • Merge Sort Tree: – Take a binary tree T – Each node of T represents a recursive call of the merge sort algorithm. – We associate with each node v of T a the set of input passed to the invocation v represents. – The external nodes are associated with individual elements of S, upon which no recursion is called.
- 6. 6 Merge-Sort
- 18. 18 Merging Two Sequences (cont.)
- 19. 19 Merging Two Sequences (cont.)
- 20. 20 Merging Two Sequences (cont.)
- 21. 21 Merging Two Sequences (cont.)
- 22. 22 Merging Two Sequences (cont.)