sorting - cs.montana.edu fileto learn how to implement the following sorting algorithms: selection...
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SortingSorting
Chapter 10Chapter 10
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Chapter ObjectivesChapter Objectives
To learn how to use the standard sorting methods in the Java APITo learn how to implement the following sorting algorithms: selection sort, bubble sort, insertion sort, Shell sort, merge sort, heapsort, and quicksortTo 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
To learn how to use the standard sorting methods in the Java APITo learn how to implement the following sorting algorithms: selection sort, bubble sort, insertion sort, Shell sort, merge sort, heapsort, and quicksortTo 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
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Using Java Sorting MethodsUsing Java Sorting Methods
Java API provides a class Arrays with several overloaded sort methods for different array typesThe Collections class provides similar sorting methodsSorting methods for arrays of primitive types are based on quicksort algorithmMethod of sorting for arrays of objects and Lists based on mergesort
Java API provides a class Arrays with several overloaded sort methods for different array typesThe Collections class provides similar sorting methodsSorting methods for arrays of primitive types are based on quicksort algorithmMethod of sorting for arrays of objects and Lists based on mergesort
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Using Java Sorting Methods (continued)
Using Java Sorting Methods (continued)
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Declaring a Generic MethodDeclaring a Generic Method
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Selection SortSelection Sort
Selection sort is a relatively easy to understand algorithmSorts 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 arrayEfficiency is O(n*n)
Selection sort is a relatively easy to understand algorithmSorts 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 arrayEfficiency is O(n*n)
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Selection Sort (continued)Selection Sort (continued)
Selection sort is called a quadratic sortNumber of comparisons is O(n*n)Number of exchanges is O(n)
Selection sort is called a quadratic sortNumber of comparisons is O(n*n)Number of exchanges is O(n)
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Selection Sort (continued)Selection Sort (continued)
Basic rule: on each pass select the smallest remaining item and place it in its proper locationBasic rule: on each pass select the smallest remaining item and place it in its proper location
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Bubble SortBubble Sort
Compares adjacent array elements and exchanges their values if they are out of orderSmaller values bubble up to the top of the array and larger values sink to the bottom
Compares adjacent array elements and exchanges their values if they are out of orderSmaller values bubble up to the top of the array and larger values sink to the bottom
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Analysis of Bubble SortAnalysis of Bubble Sort
Provides excellent performance in some cases and very poor performances in other casesWorks best when array is nearly sorted to begin withWorst 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) comparisonsO(1) exchanges
Provides excellent performance in some cases and very poor performances in other casesWorks best when array is nearly sorted to begin withWorst 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) comparisonsO(1) exchanges
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Insertion SortInsertion 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 orderWhen the player picks up a new card, he makes room for the new card and then inserts it in its proper place
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 orderWhen the player picks up a new card, he makes room for the new card and then inserts it in its proper place
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Insertion Sort AlgorithmInsertion 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
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
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Analysis of Insertion SortAnalysis of Insertion Sort
Maximum number of comparisons is O(n*n)In the best case, 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 comparisonsA 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
Maximum number of comparisons is O(n*n)In the best case, 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 comparisonsA 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
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Comparison of Quadratic SortsComparison of Quadratic Sorts
None of the algorithms are particularly good for large arraysNone of the algorithms are particularly good for large arrays
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Shell Sort: A Better Insertion Sort
Shell Sort: A Better Insertion Sort
Shell sort is a type of insertion sort but with O(n^(3/2)) or better performanceNamed after its discoverer, Donald ShellDivide and conquer approach to insertion sortInstead of sorting the entire array, sort many smaller subarrays using insertion sort before sorting the entire array
Shell sort is a type of insertion sort but with O(n^(3/2)) or better performanceNamed after its discoverer, Donald ShellDivide and conquer approach to insertion sortInstead of sorting the entire array, sort many smaller subarrays using insertion sort before sorting the entire array
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Analysis of Shell SortAnalysis of Shell Sort
A general analysis of Shell sort is an open research problem in computer sciencePerformance depends on how the decreasing sequence of values for gap is chosenIf 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))
A general analysis of Shell sort is an open research problem in computer sciencePerformance depends on how the decreasing sequence of values for gap is chosenIf 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))
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Merge SortMerge 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 methodThe objects in both sequences are ordered in accordance with this compareTo method
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 methodThe objects in both sequences are ordered in accordance with this compareTo method
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Merge AlgorithmMerge Algorithm
Merge AlgorithmAccess the first item from both sequencesWhile 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 sequenceCopy any remaining items from the second sequence to the output sequence
Merge AlgorithmAccess the first item from both sequencesWhile 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 sequenceCopy any remaining items from the second sequence to the output sequence
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Analysis of MergeAnalysis 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 placeAdditional space usage is O(n)
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 placeAdditional space usage is O(n)
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Algorithm and Trace of Merge SortAlgorithm and Trace of Merge Sort
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Algorithm and Trace of Merge Sort (continued)
Algorithm and Trace of Merge Sort (continued)
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HeapsortHeapsort
Merge sort time is O(n log n) but still requires, temporarily, n extra storage itemsHeapsort does not require any additional storage
Merge sort time is O(n log n) but still requires, temporarily, n extra storage itemsHeapsort does not require any additional storage
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Algorithm for In-Place HeapsortAlgorithm for In-Place Heapsort
Build a heap by arranging the elements in an unsorted arrayWhile the heap is not empty
Remove the first item from the heap by swapping it with the last item and restoring the heap property
Build a heap by arranging the elements in an unsorted arrayWhile the heap is not empty
Remove the first item from the heap by swapping it with the last item and restoring the heap property
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QuicksortQuicksort
Developed in 1962Quicksort 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 pivotQuicksort ensures that the elements in the right subarray are larger than the pivotAverage case for Quicksort is O(n log n)
Developed in 1962Quicksort 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 pivotQuicksort ensures that the elements in the right subarray are larger than the pivotAverage case for Quicksort is O(n log n)
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Quicksort (continued)Quicksort (continued)
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Algorithm for PartitioningAlgorithm for Partitioning
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Revised Partition AlgorithmRevised Partition Algorithm
Quicksort is O(n*n) when each split yields one empty subarray, which is the case when the array is presortedBest solution is to pick the pivot value in a way that is less likely to lead to a bad split
Requires three markersFirst, middle, last
Select the median of the these items as the pivot
Quicksort is O(n*n) when each split yields one empty subarray, which is the case when the array is presortedBest solution is to pick the pivot value in a way that is less likely to lead to a bad split
Requires three markersFirst, middle, last
Select the median of the these items as the pivot
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Testing the Sort AlgorithmsTesting the Sort Algorithms
Need to use a variety of test casesSmall and large arraysArrays in random orderArrays that are already sortedArrays with duplicate values
Compare performance on each type of array
Need to use a variety of test casesSmall and large arraysArrays in random orderArrays that are already sortedArrays with duplicate values
Compare performance on each type of array
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The Dutch National Flag ProblemThe Dutch National Flag Problem
A variety of partitioning algorithms for quicksort have been publishedA partitioning algorithm for partitioning an array into three segments was introduced by Edsger W. DijkstraProblem is to partition a disordered three-color flag into the appropriate three segments
A variety of partitioning algorithms for quicksort have been publishedA partitioning algorithm for partitioning an array into three segments was introduced by Edsger W. DijkstraProblem is to partition a disordered three-color flag into the appropriate three segments
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The Dutch National Flag ProblemThe Dutch National Flag Problem
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Chapter ReviewChapter Review
Comparison of several sorting algorithms were madeThree quadratic sorting algorithms are selection sort, bubble sort, and insertion sortShell sort gives satisfactory performance for arrays up to 5000 elementsQuicksort 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
Comparison of several sorting algorithms were madeThree quadratic sorting algorithms are selection sort, bubble sort, and insertion sortShell sort gives satisfactory performance for arrays up to 5000 elementsQuicksort 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
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Chapter Review (continued)Chapter Review (continued)
The Java API contains “industrial strength” sort algorithms in the classes java.util.Arrays and java.util.Collections
The Java API contains “industrial strength” sort algorithms in the classes java.util.Arrays and java.util.Collections