Chapter Contents. An Introduction to Sorting. Selection Sort. Selection Sort. Selection Sort. Iterative Selection Sort. Chapter 9

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Joel Theodore Griffith
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An Introduction to Sorting Chapter 9 Chapter Contents Iterative Recursive The Efficiency of Iterative Recursive The Efficiency of of a Chain of Linked Nodes The Java Code The Efficiency of Comparing

Swap with first book Look at remaining books, select shortest Swap with second book Before and after exchanging shortest book and the first book.

1 An Introduction to Sorting Chapter 9 Chapter Contents Iterative Recursive The Efficiency of Iterative Recursive The Efficiency of of a Chain of Linked Nodes The Java Code The Efficiency of Comparing the Algorithms 2 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 Before and after exchanging shortest book and the first book. Repeat 3 4 Iterative Iterative algorithm for selection sort A selection sort of an array of integers into ascending order. 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]

Recursive Recursive algorithm for selection sort Algorithm selectionsort(a, first, last) // Sorts the array elements a[first] through a[last] recursively.

.. a[first] and these are the smallest // of the original array elements. // The remaining array elements begin at a[first+1].

2 Recursive 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) 7 The Efficiency of 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) = n(n 1)/2 = O(n 2 ) It does not depends on the nature of the data in the array. Recursive selection sort performs same operations Also O(n 2 ) 8 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 2 nd book Remove 3 rd book Slide 2 nd book to right Compare with the 1 st book, if is taller than 3 rd, slide 1 st to right, insert the 3 rd book into first slot 9 The placement of the third book during an insertion sort. 10 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 11by one element at each pass Iterative 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 12 2

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

Recursive 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

3 Iterative Iterative An insertion sort inserts the next unsorted element into its proper location within the sorted portion of an array 13 An insertion sort of an array of integers into ascending order 14 Recursive Algorithm for recursive insertion sort Algorithm insertionsort(a, first, last) // Sorts the array elements a[first] through a[last] recursively. Recursive 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); 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 15 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 16 Recursive Efficiency of 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 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

20 of Chain of Linked Nodes 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 Efficiency of of insertion

4 of Chain of Linked Nodes of Chain of Linked Nodes A chain of integers sorted into ascending order. 19 During the traversal of a chain to locate the insertion point, save a reference to the node before the current one. 20 of Chain of Linked Nodes 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 Efficiency of of insertion insertion sort sort of of a chain chain is is O(n O(n 2 2 )) 21 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 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. 23 The subarrays after they are sorted, and the array that contains them. 24 4

5 The subarrays by grouping elements whose indices are 3 apart 25 The subarrays after they are sorted, and the array that contains them. 26 Efficiency of Efficiency is O(n 2 ) for worst case If n is a power of 2 Average-case behavior is O(n 1.5 ) 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. 27 Comparing the Algorithms Best Average Worst Case Case Case 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 ) The time efficiencies of three sorting algorithms, expressed in Big Oh notation. 28 5

CSC 273 Data Structures

CSC 273 Data Structures Lecture 5 - Introduction to Sorting The Interface Comparable Consider the method compareto for class String if s and t are strings, s.compareto(t) is Negative if s comes before