An Introduction to Sorting

1. An Introduction to Sorting Chapter 9

2. Chapter Contents • Selection Sort • Iterative Selection Sort • Recursive Selection Sort • The Efficiency of Selection Sort • Insertion Sort • Iterative Insertion Sort • Recursive Insertion Sort • The Efficiency of Insertion Sort • Insertion Sort of a Chain of Linked Nodes • Shell Sort • The Java Code • The Efficiency of Shell Sort • Comparing the Algorithms

3. Selection Sort • Sorting: Arrange things into either ascending or descending order • Task: rearrange books on shelf by height • Shortest book on the left • Approach: • Look at books, select shortest book • Swap with first book • Look at remaining books, select shortest • Swap with second book • Repeat …

4. Selection Sort Before and after exchanging shortest book and the first book.

5. Selection Sort A selection sort of an array of integers into ascending order.

6. Iterative Selection Sort • Iterative algorithm for selection sort Algorithm selectionSort(a, n) // Sorts the first n elements of an array a. for (index = 0; index < n - 1; index++){ indexOfNextSmallest = the index of the smallest value amonga[index], a[index+1], . . . , a[n-1]Interchange the values of a[index] and a[indexOfNextSmallest] // Assertion: a[0] £ a[1] £ . . . £ a[index], and these are the smallest// of the original array elements. // The remaining array elements begin at a[index+1].}

7. Recursive Selection Sort • Recursive algorithm for selection sort Algorithm selectionSort(a, first, last) // Sorts the array elements a[first] through a[last] recursively. if (first < last){ indexOfNextSmallest = the index of the smallest value amonga[first], a[first+1], . . . , a[last]Interchange the values of a[first] and a[indexOfNextSmallest] // Assertion: a[0] £ a[1] £ . . . £ a[first] and these are the smallest// of the original array elements. // The remaining array elements begin at a[first+1].selectionSort(a, first+1, last)}

8. The Efficiency of Selection Sort • Iterative method for loop executes n – 1 times • For each of n – 1 calls, the indexOfSmallest is invoked, last is n-1, and first ranges from 0 to n-2. • For each indexOfSmallest, compares last – first times • Total operations: (n – 1) + (n – 2) + …+ 1 = n(n – 1)/2 = O(n2) • It does not depends on the nature of the data in the array. • Recursive selection sort performs same operations • Also O(n2)

9. Insertion Sort • If only one book, it is sorted. • Consider the second book, if shorter than first one • Remove second book • Slide first book to right • Insert removed book into first slot • Then look at third book, if it is shorter than 2nd book • Remove 3rd book • Slide 2nd book to right • Compare with the 1st book, if is taller than 3rd, slide 1st to right, insert the 3rd book into first slot

10. Insertion Sort The placement of the third book during an insertion sort.

11. Insertion Sort • Partitions the array into two parts. One part is sorted and initially contains the first element. • The second part contains the remaining elements. • Removes the first element from the unsorted part and inserts it into its proper sorted position within the sorted part by comparing with element from the end of sorted part and toward its beginning. • The sorted part keeps expanding and unsorted part keeps shrinking by one element at each pass

12. Iterative Insertion Sort • Iterative algorithm for insertion sort Algorithm insertionSort(a, first, last) // Sorts the array elements a[first] through a[last] iteratively. for (unsorted = first+1 through last){ firstUnsorted = a[unsorted] insertInOrder(firstUnsorted, a, first, unsorted-1)} Algorithm insertInOrder(element, a, begin, end)// Inserts element into the sorted array elements a[begin] through a[end].index = endwhile ( (index >= begin) and (element < a[index]) ) { a[index+1] = a[index] // make roomindex - -}// Assertion: a[index+1] is available.a[index+1] = element // insert

13. Iterative Insertion Sort An insertion sort inserts the next unsorted element into its proper location within the sorted portion of an array

14. Iterative Insertion Sort An insertion sort of an array of integers into ascending order

15. Recursive Insertion Sort • Algorithm for recursive insertion sort Algorithm insertionSort(a, first, last) // Sorts the array elements a[first] through a[last] recursively. if (the array contains more than one element){ Sort the array elements a[first] through a[last-1]Insert the last element a[last] into its correct sorted position within the rest of the array }

16. Recursive Insertion Sort public static void insertionSort( Comparable[] a, int first, int last) { If ( first < last) { //sort all but the last element insertionSort( a, first, last -1 ); //insert the last element in sorted order from first through last positions insertInOrder(a[last], a, first, last-1); } } insertInorder( element, a, first, last) If (element >= a[last]) a[last+1] = element; else if (first < last) { a[last+1] = a[last]; insertInOrder(element, a, first, last-1); } else // first == last and element < a[last] { a[last+1] = a[last]; a[last] = element }

17. Recursive Insertion Sort Inserting the first unsorted element into the sorted portion of the array. (a) The element is ≥ last sorted element; (b) The element is < than last sorted element

18. Efficiency of Insertion Sort • Best time efficiency is O(n) • Worst time efficiency is O(n2) • If array is closer to sorted order • Less work the insertion sort does • More efficient the sort is • Insertion sort is acceptable for small array sizes

19. Insertion Sort of Chain of Linked Nodes A chain of integers sorted into ascending order.

20. Insertion Sort of Chain of Linked Nodes During the traversal of a chain to locate the insertion point, save a reference to the node before the current one.

21. Insertion Sort of Chain of Linked Nodes Efficiency of insertion sort of a chain is O(n2) Breaking a chain of nodes into two pieces as the first step in an insertion sort: (a) the original chain; (b) the two pieces

22. 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

23. 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.

24. Shell Sort The subarrays after they are sorted, and the array that contains them.

25. Shell Sort The subarrays by grouping elements whose indices are 3 apart

26. Shell Sort The subarrays after they are sorted, and the array that contains them.

27. 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.

28. Comparing the Algorithms Best Average Worst Case Case Case Selection sort O(n2) O(n2) O(n2) Insertion sort O(n) O(n2) O(n2) Shell sort O(n) O(n1.5) O(n1.5) or O(n2) The time efficiencies of three sorting algorithms, expressed in Big Oh notation.