An Introduction to Sorting Chapter 11

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 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 Fig Before and after exchanging shortest book and the first book.

5 Selection Sort Fig 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 among a[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 among a[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, inner loop executes n – 2 times (n – 1) + (n – 2) + …+ 1 = n(n – 1)/2 = O(n 2 ) Recursive selection sort performs same operations Also O(n 2 )

9 Insertion Sort If first two books are out of order Remove second book Slide first book to right Insert removed book into first slot Then look at third book, if it is out of order Remove that book Slide 2 nd book to right Insert removed book into 2 nd slot Recheck first two books again Etc.

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

11 Insertion Sort Fig An insertion sort of books

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 = end while ( (index >= begin) and (element < a[index]) ) {a[index+1] = a[index] // make room index - - } // Assertion: a[index+1] is available. a[index+1] = element // insert

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

14 Iterative Insertion Sort Fig 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 Fig 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

17 Efficiency of Insertion Sort Best time efficiency is O(n) Worst time efficiency is O(n 2 ) 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

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

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

20 Insertion Sort of Chain of Linked Nodes Fig Breaking a chain of nodes into two pieces as the first step in an insertion sort: (a) the original chain; (b) the two pieces Efficiency of insertion sort of a chain is O(n 2 )

21 Shell Sort A variation of the insertion sort But faster than O(n 2 ) 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

22 Shell Sort Fig An array and the subarrays formed by grouping elements whose indices are 6 apart.

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

24 Shell Sort Fig The subarrays of the array in Fig formed by grouping elements whose indices are 3 apart

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

26 Efficiency of Shell Sort Efficiency is O(n 2 ) for worst case If n is a power of 2 Average-case behavior is O(n 1.5 ) Any time the variable space (Java code, section 11.22) is even, add 1 This also results in O(n 1.5 )

27 Comparing the Algorithms BestAverageWorst Case CaseCase Selection sort O(n 2 ) O(n 2 ) O(n 2 ) Insertion sort O(n) O(n 2 ) O(n 2 ) Shell sort O(n) O(n 1.5 ) O(n 1.5 ) Fig The time efficiencies of three sorting algorithms, expressed in Big Oh notation.