إن الله لا يغير ما بقوم حتى يغيروا ما بأنفسهم

إن الله لا يغير ما بقوم حتى يغيروا ما بأنفسهم

إذا لم تعمل على تحقيق حلمك... فإن شخصا ما سيقوم بتوظيفك لتعمل على تحقيق حلمه هو !!

CS214 – Data StructuresLecture 6&7: Quadratic Sorting Slides by Mohamed El-Ramly, PhD Basheer Youssef ,PhD

http://en.wikipedia.org/wiki/List_of_mathematical_series

Agenda 0 Why Sorting? Quadratic Sorting algorithms Selection Sort Insertion Sort Bubble Sort Included but read yourself Theoretical Boundaries of Sorting Problem STL support for sorting Sub-quadratic Sorting Merge Sort I Shell Sort Quick Sort Heap Sort

Divide and Conquer • Base Case, solve the problem directly if it is small enough • Divide the problem into two or more similar and smaller subproblems • Recursively solve the subproblems • Combine solutions to the subproblems Chapter 9: Sorting

Quadratic Sorting Algorithms • Take long time • Are O(n2), which is quite slow • Theoritical limits show that we can do better. • It is possible to achieve O(n log(n)), at least theoritically. Chapter 9: Sorting

Shell Sort: A Better Insertion Sort • Shell sort is a type of insertion sort but with O(n^(3/2)) or better performance • Named after its discoverer, Donald Shell • the first sub quadratic sorting algorithm • Divide and conquer approach to insertion sort • Instead of sorting the entire array, sort many smaller subarrays using insertion sort before sorting the entire array Chapter 9: Sorting

Shell Sort • A variation of the insertion sort • But faster than O(n2) • Done by sorting subarrays of equally spaced indices • Instead of moving to an adjacent location an element moves several locations away • Results in an almost sorted array • This array sorted efficiently with ordinary insertion sort • Wanted to stop moving data small distances (in the case of insertion sort and bubble sort) and stop making swaps that are not helpful (in the case of selection sort) • Start with sub arrays created by looking at data that is far apart and then reduce the gap size Chapter 9: Sorting

Chapter 9: Sorting

Shell Sort Donald Shell suggested that the initial separation between indices be n/2 and halve this value at each pass until it is 1. An array has 13 elements, and the subarrays formed by grouping elements whose indices are 6 apart. Chapter 9: Sorting

Shell Sort The subarrays after they are sorted, and the array that contains them. Chapter 9: Sorting

Shell Sort The subarrays by grouping elements whose indices are 3 apart Chapter 9: Sorting

Shell Sort The subarrays after they are sorted, and the array that contains them. Chapter 9: Sorting

Efficiency of Shell Sort • Efficiency is O(n2) for worst case • If n is a power of 2 • Average-case behavior is O(n1.5) • Shell sort uses insertion sort repeatedly. • Initial sorts are much smaller, the later sorts are on arrays that are partially sorted, the final sort is on an array that is almost entirely sorted. Chapter 9: Sorting

ShellSort in practice 46 2 83 41 102 5 17 31 64 49 18 Gap of five. Sort sub array with 46, 5, and 1852 83 41 102 1817 31 64 49 46 Gap still five. Sort sub array with 2 and 175 283 41 102 18 1731 64 49 46 Gap still five. Sort sub array with 83 and 315 2 3141 102 18 17 8364 49 46 Gap still five Sort sub array with 41 and 645 2 31 4110218 17 83 644946 Gap still five. Sort sub array with 102 and 495 2 31 41 49 18 17 83 64 102 46 Continued on next slide: Chapter 9: Sorting

Completed Shellsort 5 2 31 41 49 18 17 83 64 102 46 Gap now 2: Sort sub array with 5 31 49 17 64 465 2 17 41 31 18 46 83 49 102 64 Gap still 2: Sort sub array with 2 41 18 83 1025 2 17 18 31 41 46 83 49 102 64 Gap of 1 (Insertion sort) 2 5 17 18 31 41 46 49 64 83 102 Array sorted Chapter 9: Sorting

Pseudocode of ShellSort Chapter 9: Sorting

Java ShellSort Code public static void shellsort(Comparable[] list){ Comparable temp; boolean swap; for(int gap = list.length / 2; gap > 0; gap /= 2) for(int i = gap; i < list.length; i++) { Comparable tmp = list[i]; int j = i; for( ; j >= gap && tmp.compareTo( list[j - gap] ) < 0; j -= gap ) list[ j ] = list[ j - gap ]; list[ j ] = tmp; }} // See C++ version in Drozdek’s Book Chapter 9: Sorting

Chapter 9: Sorting

Merge Sort • A merge is a common data processing operation that is performed on two sequences of data with the following characteristics • Both sequences contain items with a common compareTo method • The objects in both sequences are ordered in accordance with this compareTo method Chapter 9: Sorting

Merge Algorithm • Merge Algorithm • Access the first item from both sequences • While not finished with either sequence • Compare the current items from the two sequences, copy the smaller current item to the output sequence, and access the next item from the input sequence whose item was copied • Copy any remaining items from the first sequence to the output sequence • Copy any remaining items from the second sequence to the output sequence Chapter 9: Sorting

Merge Merge Sort: Idea Divide into two halves A FirstPart SecondPart Recursively sort SecondPart FirstPart A is sorted! Chapter 9: Sorting

Merge Sort: Algorithm • Merge-Sort(A, left, right) • if left ≥ right return • else • middle ←(left+right)/2 • Merge-Sort(A, left, middle) • Merge-Sort(A, middle+1, right) • Merge(A, left, middle, right) Recursive Call Chapter 9: Sorting

Merge-Sort: Merge Sorted A: merge Sorted SecondPart SortedFirstPart A: A[right] A[left] A[middle] Chapter 9: Sorting

5 15 28 30 6 10 14 5 2 3 7 8 1 4 5 6 R: L: 3 5 15 28 6 10 14 22 Temporary Arrays Merge-Sort: Merge Example A: Chapter 9: Sorting

Merge-Sort: Merge Example A: 3 1 5 15 28 30 6 10 14 k=0 R: L: 3 2 15 3 28 7 30 8 1 6 10 4 14 5 22 6 i=0 j=0 Chapter 9: Sorting

Merge-Sort: Merge Example A: 3 1 2 15 28 30 6 10 14 k=2 R: L: 2 3 7 8 6 1 4 10 14 5 22 6 i=1 j=1 Chapter 9: Sorting

Merge-Sort: Merge Example A: 1 2 3 6 10 14 4 k=3 R: L: 2 3 7 8 1 6 10 4 5 14 22 6 j=1 i=2 Chapter 9: Sorting

Merge-Sort: Merge Example A: 1 2 3 4 6 10 14 5 k=4 R: L: 2 3 7 8 6 1 10 4 5 14 22 6 i=2 j=2 Chapter 9: Sorting

Merge-Sort: Merge Example A: 6 1 2 3 4 5 6 10 14 k=5 R: L: 2 3 7 8 1 6 10 4 5 14 22 6 i=2 j=3 Chapter 9: Sorting

Merge-Sort: Merge Example A: 7 1 2 3 4 5 6 14 k=6 R: L: 2 3 7 8 1 6 4 10 5 14 6 22 i=2 j=4 Chapter 9: Sorting

Merge-Sort: Merge Example A: 1 2 3 4 5 6 7 8 k=8 R: L: 2 3 3 5 15 7 8 28 6 1 10 4 14 5 6 22 j=4 i=4 Chapter 9: Sorting

Merge(A, left, middle, right) • n1 ← middle – left + 1 • n2 ← right – middle • create array L[n1], R[n2] • for i ← 0 to n1-1 do L[i] ← A[left +i] • for j ← 0 to n2-1 do R[j] ← A[middle+j] • k ← i ← j ← 0 • while i < n1 & j < n2 • if L[i] < R[j] • A[k++] ← L[i++] • else • A[k++] ← R[j++] • while i < n1 • A[k++] ← L[i++] • while j < n2 • A[k++] ← R[j++] n = n1+n2 Space: n Time : cn for some constant c Chapter 9: Sorting

Merge-Sort(A, 0, 7) Divide A: 3 7 5 1 6 2 8 4 6 2 8 4 3 7 5 1 Chapter 9: Sorting

Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 0, 3) A: 3 7 5 1 8 4 6 2 6 2 8 4 Chapter 9: Sorting

Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 0, 1) A: 3 7 5 1 8 4 2 6 6 2 Chapter 9: Sorting

Merge-Sort(A, 0, 7) , base case Merge-Sort(A, 0, 0) A: 3 7 5 1 8 4 2 6 Chapter 9: Sorting

Merge-Sort(A, 0, 7) Merge-Sort(A, 0, 0), return A: 3 7 5 1 8 4 2 6 Chapter 9: Sorting

Merge-Sort(A, 0, 7) , base case Merge-Sort(A, 1, 1) A: 3 7 5 1 8 4 6 2 Chapter 9: Sorting

Merge-Sort(A, 0, 7) Merge(A, 0, 0, 1) A: 3 7 5 1 8 4 2 6 Chapter 9: Sorting

Merge-Sort(A, 0, 7) , divide Merge-Sort(A, 2, 3) A: 3 7 5 1 2 6 4 8 8 4 Chapter 9: Sorting

Merge-Sort(A, 0, 7) Merge-Sort(A, 2, 2), base case A: 3 7 5 1 2 6 4 8 Chapter 9: Sorting

Merge-Sort(A, 0, 7) Merge-Sort(A, 2, 2),return A: 3 7 5 1 2 6 4 8 Chapter 9: Sorting

Merge-Sort(A, 0, 7) Merge-Sort(A, 3, 3), base case A: 2 6 8 4 Chapter 9: Sorting

إن الله لا يغير ما بقوم حتى يغيروا ما بأنفسهم

إن الله لا يغير ما بقوم حتى يغيروا ما بأنفسهم

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