Spring 2010CS 2251 Sorting Chapter 8

Spring 2010CS 2252 Chapter Objectives To learn how to use the standard sorting methods in the Java API To learn how to implement the following sorting algorithms: selection sort, bubble sort, insertion sort, Shell sort, merge sort, heapsort, and quicksort To understand the difference in performance of these algorithms, and which to use for small arrays, which to use for medium arrays, and which to use for large arrays

Spring 2010CS 2253 Declaring a Generic Method We have seen generic classes throughout the semester. You can also declare generic methods. methodModifiers returnType methodName( methodParameters) List the generic type(s) between the modifiers and the return type; use them in the parameter list, return type(?) and body of method as needed. public static > int binarySearch( T[] items, T target)

Spring 2010CS 2254 Using Java Sorting Methods Java API provides a class Arrays with several overloaded sort methods for different array types The Collections class provides similar sorting methods for Lists Sorting methods for arrays of primitive types are based on quicksort algorithm Method of sorting for arrays of objects and Lists based on mergesort

Spring 2010CS 2255 java.util.Arrays Two versions of sort for each primitive type public void sort( int [] values) public void sort( int [] values, int from, int to) Four sort methods for objects public static void sort( Object [] items) public static void sort( Object [] items, int from, int to) public static void sort( [] items, Comparator comp) public static void sort( [] items, Comparator comp, int from, int to)

Spring 2010CS 2256 java.util.Collections Two methods for sorting collections that implement the List interface public static > void sort( List list) public static void sort( List list, Comparator comp)

Spring 2010CS 2257 Selection Sort Selection sort is a relatively easy to understand algorithm Sorts an array by making several passes through the array, selecting the next smallest item in the array each time and placing it where it belongs in the array Efficiency is O(n*n)

Spring 2010CS 2258 Selection Sort Algorithm for fill = 0 to n-2 do posMin = fill for next = fill + 1 to n-1 if item at next < item at posMin posMin = next swap elements at fill and posMin

Spring 2010CS 2259 Selection Sort Algorithm Basic rule: on each pass select the smallest remaining item and place it in its proper location

Spring 2010CS Selection Sort Efficiency Selection sort is called a quadratic sort –Number of comparisons is O(n*n) –Number of exchanges is O(n) In best case (already sorted) –Number of comparisons is still O(n*n) –Number of exchanges is O(1)

Spring 2010CS Bubble Sort Compares adjacent array elements and exchanges their values if they are out of order Smaller values bubble up to the top of the array and larger values sink to the bottom

Spring 2010CS Analysis of Bubble Sort Provides excellent performance in some cases and very poor performances in other cases Works best when array is nearly sorted to begin with Worst case number of comparisons is O(n*n) Worst case number of exchanges is O(n*n) Best case occurs when the array is already sorted –O(n) comparisons –O(1) exchanges

Spring 2010CS Insertion Sort Based on the technique used by card players to arrange a hand of cards –Player keeps the cards that have been picked up so far in sorted order –When the player picks up a new card, he makes room for the new card and then inserts it in its proper place

Spring 2010CS Insertion Sort Algorithm For each array element from the second to the last (nextPos = 1) –Insert the element at nextPos where it belongs in the array, increasing the length of the sorted subarray by 1

Spring 2010CS Analysis of Insertion Sort Maximum number of comparisons is O(n*n) In the best case (already sorted), number of comparisons is O(n) The number of shifts performed during an insertion is one less than the number of comparisons or, when the new value is the smallest so far, the same as the number of comparisons A shift in an insertion sort requires the movement of only one item whereas in a bubble or selection sort an exchange involves a temporary item and requires the movement of three items

Spring 2010CS Comparison of Quadratic Sorts None of the algorithms are particularly good for large arrays ComparisonsExchanges SortBestWorstBestWorst SelectionO(n 2 ) O(n) BubbleO(n)O(n 2 )O(1)O(n 2 ) InsertionO(n)O(n 2 )O(n)O(n 2 )

Spring 2010CS Shell Sort Shell sort is a type of insertion sort but with O(n (3/2) ) or better performance Named after its discoverer, Donald Shell 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

Spring 2010CS Shell Sort

Spring 2010CS Analysis of Shell Sort A general analysis of Shell sort is an open research problem in computer science Performance depends on how the decreasing sequence of values for gap is chosen If successive powers of two are used for gap, performance is O(n*n) If Hibbard’s sequence is used, performance is O(n 3/2 )

Spring 2010CS 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

Spring 2010CS 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

Spring 2010CS Merge Example

Spring 2010CS Analysis of Merge For two input sequences that contain a total of n elements, we need to move each element’s input sequence to its output sequence –Merge time is O(n) We need to be able to store both initial sequences and the output sequence –The array cannot be merged in place –Additional space usage is O(n)

Spring 2010CS Algorithm and Trace of Merge Sort if tablesize >1 halfsize = tablesize / 2 copy left half to separate leftTable recursively sort leftTable copy right half to separate rightTable recursively sort rightTable merge leftTable and rightTable

Spring 2010CS Trace of Merge Sort

Spring 2010CS Heapsort Merge sort time is O(n log n) but still requires, temporarily, n extra storage items Heapsort does not require any additional storage –build a max-heap from the array –swap first element with last to put largest element at end –build a heap from the unsorted part of the array

Spring 2010CS Heapsort

Spring 2010CS Heapsort

Spring 2010CS Quicksort Developed in 1962 Quicksort rearranges an array into two parts so that all the elements in the left subarray are less than or equal to a specified value, called the pivot Quicksort ensures that the elements in the right subarray are larger than the pivot Average case for Quicksort is O(n log n)

Spring 2010CS Quicksort

Spring 2010CS Algorithm for Partitioning pivot = table[first] up = first down = last while table[up]<=pivot and up<down increment up while table[down]>pivot and down>up decrement down swap table[down] and table[first] pivotIndex = down

Spring 2010CS Algorithm for Partitioning

Spring 2010CS Revised Partition Algorithm Quicksort is O(n*n) when each split yields one empty subarray, which is the case when the array is presorted Best solution is to pick the pivot value in a way that is less likely to lead to a bad split –Requires three markers First, middle, last –Select the median of the these items as the pivot

Spring 2010CS Testing the Sort Algorithms Need to use a variety of test cases –Small and large arrays –Arrays in random order –Arrays that are already sorted –Arrays with duplicate values Compare performance on each type of array

Spring 2010CS Comparing Sorts SortBestAverageWorst SelectionO(n 2 ) BubbleO(n)O(n 2 ) InsertionO(n)O(n 2 ) ShellO(n 7/6 )O(n 5/4 )O(n 2 ) MergeO(n log n) HeapO(n log n) QuickO(n log n) O(n 2 )

Spring 2010CS The Dutch National Flag Problem A variety of partitioning algorithms for quicksort have been published A partitioning algorithm for partitioning an array into three segments was introduced by Edsger W. Dijkstra Problem is to partition a disordered three-color flag into the appropriate three segments

Spring 2010CS The Dutch National Flag Problem

Spring 2010CS Algorithm for 0<=i<red, color is red for red<=i<=white, color is unknown for white<i<=blue, color is blue for blue<i<=height, color is blue red = 0 blue = white = height-1 while red<white if threads[white] is white: white-- red: swap elements at red and white red++ blue: swap elements at blue and white blue-- white--

Spring 2010CS Chapter Review Comparison of several sorting algorithms were made Three quadratic sorting algorithms are selection sort, bubble sort, and insertion sort Shell sort gives satisfactory performance for arrays up to 5000 elements Quicksort has an average-case performance of O(n log n), but if the pivot is picked poorly, the worst case performance is O(n*n) Merge sort and heapsort have O(n log n) performance

Spring 2010CS Chapter Review The Java API contains “industrial strength” sort algorithms in the classes java.util.Arrays and java.util.Collections