1. Sorting

2. RHS – SOC 2 Sorting Searching in sorted data is much faster (O(log(n)), than searching in unsorted data (O(n)). Being able to sort data efficiently is thus a quite important ability But how fast can be sort data…?

3. RHS – SOC 3 Selection sort A very simple algorithm for sorting an array of n integers works like this: –Search the array from element 0 to element (n-1), to find the smallest element –If the smallest element is element i, then swap element 0 and element i –Now repeat the process from element 1 to element (n-1) –…and so on…

4. RHS – SOC 4 Selection sort 1056264827634186040

5. RHS – SOC 5 Selection sort 1056264827634186040

6. RHS – SOC 6 Selection sort 1056263482764186040

7. RHS – SOC 7 Selection sort 1056263482764186040

8. RHS – SOC 8 Selection sort 1056263482764186040

9. RHS – SOC 9 Selection sort 1056263482764186040

10. RHS – SOC 10 Selection sort 1056263482764186040

11. RHS – SOC 11 Selection sort 1018263440564607682

12. RHS – SOC 12 Selection sort How fast is selection sort? We scan for the smallest element n times –In scan 1, we examine n element –In scan 2, we examine (n-1) element –…and so on A total of n + (n -1) + (n – 2) +…+ 2 + 1 examinations The sum is n(n + 1)/2

13. RHS – SOC 13 Selection sort The total number of examinations is equal to n(n + 1)/2 = (n 2 + n)/2 The run-time complexity of selection sort is therefore O(n 2 ) O(n 2 ) grows fairly fast…

14. RHS – SOC 14 Selection sort nn2n2 24 525 20400 502500 20040000

15. RHS – SOC 15 Exercise 1 Download the project selectionSortInJava from the PSL website Examine the code – see how selection sort is implemented in Java The project contains two helper classes ArrayUtil (generates a random array of integers), and StopWatch (can measure the time needed to execute some code). Using these classes, the program measures how long it takes to sort an array using selection sort Try to run the program with various array sizes. For each run, write down the array size and the elapsed time. Make sure to try some array sizes that take several seconds to complete Enter the data into an Excel spreadsheet, plot a curve from the data, and see how the run time behaves when the array size increases

16. RHS – SOC 16 Merge sort Selection sort is conceptually very simple, but not very efficient… A different algorithm for sorting is merge sort Merge sort is an example of a divide-and- conquer algorithm It is also a recursive algorithm

17. RHS – SOC 17 Merge sort The principle in merge sort is to merge two already sorted arrays: 1026345618404607682

18. RHS – SOC 18 Merge sort 1026345618404607682

19. RHS – SOC 19 Merge sort Merging two sorted arrays is pretty simple, but how did the arrays get sorted…? Recursion to the rescue! Sort the two arrays simply by appying merge sort to them… If the array has length 1 (or 0), it is sorted

20. RHS – SOC 20 Merge sort public void sort() // Sort the array a { if (a.length <= 1) return; // Base case int[] a1 = new int[a.length/2];// Create two new int[] a2 = new int[a.length – a1.length];// arrays to sort System.arraycopy(a,0,a1,0,a1.length);// Copy data to System.arraycopy(a,a1.length,a2,0,a2.length);// the new arrays MergeSorter ms1 = new MergeSorter(a1);// Create two new MergeSorter ms2 = new MergeSorter(a2);// sorter objects ms1.sort();// Sort the two ms2.sort();// new arrays merge(a1,a2);// Merge the arrays }

21. RHS – SOC 21 Merge sort All that is left is the method for merging two arrays A little bit tedious, but as such trivial… Time needed to merge two arrays to the total length of the arrays, i.e to n We can now analyse the run-time com- plexity for merge sort

22. RHS – SOC 22 Merge sort Merge sort of an array of length n requires –Two merge sorts of arrays of length n/2 –Merging two arrays of length n/2 The running time T(n) then becomes: T(n) = 2×T(n/2) + n

23. RHS – SOC 23 Merge sort If we re-insert the expression for T(n) into itself m times, we get T(n) = 2 m ×T(n/2 m ) + mn If we choose m such that n = 2 m, we get T(n) = n×T(1) + mn = n + n×log(n)

24. RHS – SOC 24 Merge sort The run-time complexity of merge sort is therefore O(n log(n)) Many other sorting algorithms have this run-time complexity This is the fastest we can sort, except under very special circumstances Much better than O(n 2 )…

25. RHS – SOC 25 Merge sort nn log(n)n2n2 224 51225 2086400 502822500 200152940000

26. RHS – SOC 26 Exercise 2 Download the project mergeSortInJava from the PSL website Examine the code – see how merge sort is implemented in Java (the project contains the same helper classes as the selectionSortInJava project – ArrayUtil and StopWatch) Try to run the program with various array sizes. For each run, write down the array size and the elapsed time. Make sure to try some array sizes that take several seconds to complete Enter the data into an Excel spreadsheet, plot a curve from the data, and see how the run time behaves when the array size increases Compare the results with the results obtained for selection sort – when do the curves for run time cross each other (if at all)?

27. RHS – SOC 27 Sorting in practice It does matter which sorting algorithm you use… …but do I have to code sorting algorithms myself? No! You can – and should – use sorting algorithms found in the Java library

28. RHS – SOC 28 Sorting in practice Sorting an array: Car[] cars = new Car[n]; … Arrays.sort(cars);

29. RHS – SOC 29 Sorting in practice Sorting an arraylist: ArrayList cars = new ArrayList (); … Collections.sort(cars);

30. RHS – SOC 30 Sorting in practice Why not code my own sorting algorithms? Sorting algorithms in Java library are better than anything you can produce… –Carefully debugged –Highly optimised –Used by thousands You cannot beat them

31. RHS – SOC 31 Sorting in practice In order to sort an array of data, we need to be able to compare the elements ”Larger than” should make sense for the elements in the array Easy for numeric types (>) What about types we define ourselves…?

32. RHS – SOC 32 Sorting in practice If a class T implements the Comparable interface, objects of type T can be compared: public interface Comparable { int compareTo(T other); }

33. RHS – SOC 33 Sorting in practice In the interface definition, T is a type parameter It is used the same way as we use an arraylist ArrayList : an arraylist holding elements of type Car

34. RHS – SOC 34 Sorting in practice In order for the sorting algorithms to work properly, an implementation of the interface must obey these rules: The call a.compareTo(b) must return: –A negative number if a < b –Zero if a = b –A positive number if a > b

35. RHS – SOC 35 Sorting in practice The implementation of compareTo must define a so-called total ordering: –Antisymmetric: If a.compareTo(b) ≤ 0, then b.compareTo(a) ≥ 0 –Reflexive: a.compareTo(a) = 0 –Transitive: If a.compareTo(b) ≤ 0 and b.compareTo(c) ≤ 0, then a.compareTo(c) ≤ 0

36. RHS – SOC 36 Sorting in practice public class Car implements Comparable {... // Here using weight as ordering criterion // public int compareTo(Car other) { if (getWeight() < other.getWeight()) return -1; if (getWeight() == other.getWeight()) return 0; return 1; }... }

37. RHS – SOC 37 Exercises Programming P14.12, P.14.14 For P14.14, read about the Comparator interface in Advanced Topic 14.5, page 657-658