2. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 DCO 20105 Data structures and algorithms  Lecture 7: Big-O analysis Sorting Algorithms  Big-O analysis on different ways of multiplication  Sorting Algorithms: selection sort, bubble sort, insertion sort, radix sort, partition sort, merge sort  Comparison of different sorting algorithms -- By Rossella Lau

3. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance re-visit  For multiplication, we can use (at least) three different ways:  The one we used to use in primary school  bFunction(m, n) in slide 9 of Lecture 6  funny(a, b) in slide 10 of Lecture 6 int funny(int a, int b) { if ( a == 1 ) return b; return a & 1 ? funny ( a>>1, b<<1) + b : funny ( a>>1, b<<1); } int bFunction( int n, int m) { if (!n) return 0; return bFunction (n-1, m) + m; }

4. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance analysis The traditional waybFunction(n,m)funny(a, b) Memorizing time table is required (memory) Memorizing time table is not required Memorizing is not required for binary system (in a computer) Number of operations: Multiplications: #(n) * #(m) – memory recall Additions: #(n) #(n) is the number of digits of n Number of operations: number of n additions Number of operations: Shift operations: 2 * #b(a) Additions: worst case, #b(a) #b(a) is the number of digits of a in binary

5. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Execution time vs memory  The traditional multiplication has the least operations but it requires the most memory at O (log10 n)  bFunction() does not require additional memory but it spends a terrible amount of time getting the result : O(n)  funny() does not require additional memory and it has a bit more operations at O (log2 n)  The traditional way may have less operations but hard to say if it really outperforms funny() since memory load may not be faster than shift operation

6. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Ordering of data  In order to search a record efficiently, records are stored in the order of key values  A key is a field or some fields of a record that can uniquely identify the record in a file  Usually, only the key values are stored in memory and the corresponding record is loaded into the memory only when it is necessary  The key values, therefore, usually are sorted in a special order to allow efficient searching

7. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Classification of sorting methods  Comparison-Based Methods  Insertion Sorts  Selection Sorts  Heapsort (tree sorting) – in future lesson  Exchange sorts Bubble sort Quick sort  Merge sorts  Distribution Methods: Radix sorting

8. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Selection sort  Selection: choose the smaller element from a list and place it in the 1 st position.  The process is from the first element to the second to last element on a list and for each element to apply the “selection” on the sub-list starting from the element being processed.  Ford’ text book slides 2-9 in Chapter 3

9. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Bubble sort  To pass through the array n-1 times, where n is the number of data in the array  For each pass:  compare each element in the array with its successor  interchange the two elements if they are not in order  The algorithm bubble (int x[], int n) { for (i=0; i<n-1; i++) for (j=0; j<n-1; j++) if (x[j] > x[j+1]) SWAP (x[j], x[j+1]) }

10. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 An example trace of bubble sort Given data sequence: 25 57 48 37 12 92 86 33 The first pass: 25 57 48 37 12 92 86 33 25 48 57 37 12 92 86 33 25 48 37 57 12 92 86 33 25 48 37 12 57 92 86 33 25 48 37 12 57 86 92 33 25 48 37 12 57 86 33 92 Subsequent passes: Pass2: 25 37 12 48 57 33 86 92 Pass3: 25 12 37 48 33 57 86 92 Pass4: 12 25 37 33 48 57 86 92 Pass5: 12 25 33 37 48 57 86 92 Pass6: 12 25 33 37 48 57 86 92 Pass7: 12 25 33 37 48 57 86 92

11. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Improvement can be made  At pass i, the last i elements should be in proper positions since, at the first pass the largest element should be placed at the end of the array. At the second pass, the second large element should be placed before the last element, and so on.  The comparison only requires from x[0] to x[n-i-1]  The array has already been sorted at the fifth iteration and the sixth and seventh are redundant  Therefore, once no exchange is required in an iteration, the array is already sorted and the subsequent iterations are redundant

12. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The improved algorithm for bubble sort void bubble1 (int x[], int n) { exchange = TRUE; for (i=0; i<n-1 && exchange; i++) { exchange = FALSE; for (j=0; i<n-i-1; j++) if (x[j] > x[j+1]) { exchange = TRUE; SWAP(x[j], x[j+1]); }/*end if */ }/* end for i */ }

13. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance considerations of bubble sort  For the first version, it requires (n-1) comparisons in (n-1) passes  the total number of comparisons is n 2 - 2n +1, i.e., O(n 2 )  For the improved version, it requires (n-1) + (n-2) +... + (n-k) for k (<n) passes  the total number of comparisons is (2kn-k 2 -k)/2. However, the average k is O(n) yielding the overall complexity as O(n 2 ) and the overhead (set and check exchange) introduced should also be considered  It only requires little additional space

14. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Insertion sort  Insert an item into a previous sorted order one by one for each of the data.  It is similar to repeatedly picking up playing cards and inserting them into the proper position in a partial hand of cards

15. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 An example trace of insertion sort 25 57 48 37 12 92 86 33 25 57 37 12 92 86 33 25 48 57 37 12 92 86 33 25 48 57 12 92 86 33 25 37 48 57 12 92 86 33 25 37 48 57 92 86 33 12 25 37 48 57 92 86 33 12 25 37 48 57 86 92 33 …… 12 25 33 37 48 57 86 92

16. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The algorithm of insertion sort void insertsort(x,n) int x[], n) {for (k=1; k<n; k++) { y = x[k]; for (i = k-1; i >=0 && y<x[i]; i--) x[i+1] = x[i]; x[i+1] = y; } /* end for k */ }  The checking of i>=0 is time consuming. Setting a sentinel in the beginning of the array will prevent y from going beyond the array void insertsort(int x[], int m) /* m is n+1, data from x[1] */ X[0] = MAXNEGINT {for (k=2; k < m; k++) { y = x[k]; for (i = k-1; y<x[i]; i--) x[i+1] = x[i]; x[i+1] = y; } /* end for k */ }

17. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance analysis of insertion sort  If the original sequence is already in order, only one comparison is made on each pass ==> O(n)  If the original sequence is in a reversed order, it requires n comparison in each pass ==> O(n 2 )  The complexity is from O(n) to O(n 2 )  It requires little additional space

18. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Quick sort  It is also called partition exchange sort  In each step, the original sequence is partitioned into 3 parts: a. all the items less than the partitioning element b. the partitioning element in its final position c. all the items greater than the partitioning element  The partitioning process continues in the left and right partitions

19. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The partitioning in each step of quicksort  To pick one of the elements as the partitioning element, p, usually the first element of the sequence  To find the proper position for p while partitioning the sequence into 3 parts a) it employs two indexes, down and up b) down goes from left to right to find elements greater than p c) up goes from right to left to find elements less than p d) elements found by up and down are exchanged e) process until up and down are matched or passed each other f) the position of p should be pointed by up g) exchange p with the element pointed by up

20. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 An example trace of quicksort 25 57 48 37 12 92 86 33 25 12 48 37 57 92 86 33 (12) 25 (48 37 57 92 86 33) Subsequent processes: 12 25 (48 37 57 92 86 33) 12 25 (48 37 33 92 86 57) 12 25 (33 37) 48 (92 86 57) 12 25()33 (37) 48 (57 86) 92() 12 25 33 37 48 (57 86) 92 12 25 33 37 48()57(86) 92 12 25 33 37 48 57 86 92 _ down, _ up

21. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The algorithm for quicksort void quickSort(int x[], int left, int right) { int down, up, partition; down=left; up=right+1; partition=x[left]; while (down<up) { while (x[++down] <= partition); while (x[--up] > partition); if (down<up) SWAP(x[down], x[up]) } x[left] = x[up]; x[up] = partition; if (left < up - 1) quickSort(x, left, up-1); if (down < right) quickSort(x, down, right); }

22. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance considerations of quicksort  Quciksort got its name because it quickly puts an element into its proper position by employing two indexes to speed up the partioning process and to minimize the exchange  Each pass reduces the comparisons about a half  total number of comparisons is about O(nlog 2 n)  It requires spaces for the recursive process or stacks for an iterative process, it is about O(log 2 n)

23. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Merge  Merge means to combine two or more sorted sequences into another sorted sequence  The merging of two sequences, for example, are as follows 32 45 78 90 92 | 25 30 52 88 98 | 32 45 78 90 92 | 25 30 52 88 98 |25 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 52 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 52 78 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 52 78 88 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 52 78 88 90 32 45 78 90 92 | 25 30 52 88 98 |25 30 32 45 52 78 88 90 92 32 45 78 90 92_| 25 30 52 88 98_|25 30 32 45 52 78 88 90 92 98

24. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Merge sort  It employs the merging technique in the following way: 1. Divide the sequence into n parts 2. Merge adjacent parts yielding the sequence n/2 parts 3. Merge adjacent parts again yielding the sequence n/4 parts...... Process goes on until the sequence becomes 1 part

25. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 An example of merge sort 8 parts 25 57 48 37 12 92 86 33 merge 25 57 37 48 12 92 33 86 4 parts 25 57 37 48 12 92 33 86 merge 25 37 48 57 12 33 86 92 2 parts 25 37 48 57 12 33 86 92 merge 12 25 33 37 48 57 86 92

26. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance considerations of merge sort  There are only log 2 n passes yielding a complexity of O(nlogn)  It never requires n* log 2 n comparison while quicksort may require O(n 2 ) at the worst case  However, it requires about double of assignment statements as quicksort  It also requires more additional spaces, about O(n), than quicksort's O(log 2 n)

27. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Radix Sort  It is based on the values of the actual digits of its octal position  Starting from the least significant digit to the most significant digit  define 10 vectors for each digit and number the vectors from v0 to v9 for digit 0 to 9 respectively  scan the data sequence once and add x i into the significant digit's respective vector  new data sequence is as follows: remove elements from each vector from the beginning one by one until it is empty from q0 to q9  After the above actions, the new data sequence is the sorted sequence!

28. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 An example of radix sort 25 57 48 37 12 92 86 33 12 92 33 25 86 57 37 48 2525 5757 4848 3737 12129292 8686 3 1212 9292 3 2525 8686 5757 3737 4848 12 25 33 37 48 57 86 92

29. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Performance considerations of radix sort  It does not require any comparison between data  It requires number of digits, log10 m, passes   O(n*log10 m)  O(n), treating log10 m a constant  It requires 10 times of the memory for numbers  It seems that radix sort has the “best” performance; however, it is not popularly used because  It consumes a terrible amount of memory  Log10 m depends on the digit (length) of a key and may not be treated as a small constant when the key length is long

30. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The real life sort for vector based data  Although quick sort is known to be the fastest in many cases, the library will not usually directly use quick sort as the sort method  Usually, a carefully designed library will implement its sort method with quick sort and insertion sort  Quick sort divides partitions until a partition is about the size from 8 to 16, insertion is applied to the partition since the partitions usually are near being sorted

31. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 The real life sort for non vector data  Quick sort requires a container with random access  A container such as a linked list does not support random access and cannot apply quick sort  Merge sort is preferred to be applied

32. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Sample timing of sort methods  Ford’s prg15_2.cpp & d_sort.h Timing for some sample runs : timeSort.out

33. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Summary  Bubble sort and insertion sort have complexity of O(n 2 ) but insertion sort is still preferred for short data stream  Partition sort, merge sort have a less complexity at O(n logn)  Radix sort seemed at O(n) complexity but it consumes more memory and may depend on the key length  Many times, the trade off is space

34. Rossella Lau Lecture 7, DCO20105, Semester A,2005-6 Reference  Ford: 3.1, 4.4, 8.3 15.1  Data Structures using C and C++ by Yedidyah Langsam, Moshe J. Augenstein & Aaron M. Tenenbaum: Chapter 6  Example programs: Ford: prg15_2.cpp, d_sort.h, -- END --