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Dr.Surasak Mungsing E-mail: Surasak.mu@spu.ac.th

CSE 221/ICT221 Analysis and Design of Algorithms Lecture 05: Analysis of time Complexity of Sorting Algorithms. Dr.Surasak Mungsing E-mail: Surasak.mu@spu.ac.th. Sorting Algorithms. Bin Sort Radix Sort Insertion Sort Shell Sort Selection Sort Heap Sort Bubble Sort Quick Sort

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Dr.Surasak Mungsing E-mail: Surasak.mu@spu.ac.th

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  1. CSE 221/ICT221 Analysis and Design of AlgorithmsLecture 05:Analysis of time Complexity of Sorting Algorithms Dr.SurasakMungsing E-mail: Surasak.mu@spu.ac.th

  2. Sorting Algorithms • Bin Sort • Radix Sort • Insertion Sort • Shell Sort • Selection Sort • Heap Sort • Bubble Sort • Quick Sort • Merge Sort CSE221/ICT221 Analysis and Design of Algorithms

  3. CSE221/ICT221 Analysis and Design of Algorithms

  4. D H C J B C I B D E A G F E F G A H J I 4 0 4 2 3 3 4 4 4 4 4 5 5 3 0 2 3 4 3 3 I J G H D C F A E B Bin 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin Sort (a) Input chain (b) Nodes in bins (c) Sorted chain CSE221/ICT221 Analysis and Design of Algorithms

  5. 216 521 425 116 91 515 124 34 96 24 24 521 515 216 91 34 116 124 91 34 96 521 124 24 116 425 124 24 425 216 515 216 34 425 116 91 515 521 96 96 (a) Input chain (b) Chain after sorting on least significant digit (c) Chain after sorting on second-least significant digit (d) Chain after sorting on most significant digit Radix Sort with r=10 and d=3

  6. Insertion Sort Concept

  7. CSE221/ICT221 Analysis and Design of Algorithms

  8. Shell Sort Algorithm CSE221/ICT221 Analysis and Design of Algorithms

  9. Shell Sort Algorithm CSE221/ICT221 Analysis and Design of Algorithms

  10. Shell Sort Algorithm CSE221/ICT221 Analysis and Design of Algorithms

  11. CSE221/ICT221 Analysis and Design of Algorithms

  12. Shell Sort Demohttp://e-learning.mfu.ac.th/mflu/1302251/demo/Chap03/ShellSort/ShellSort.html

  13. Selection Sort Concept CSE221/ICT221 Analysis and Design of Algorithms

  14. Heap Sort Algorithm CSE221/ICT221 Analysis and Design of Algorithms

  15. CSE221/ICT221 Analysis and Design of Algorithms

  16. Bubble Sort Concept

  17. 31 75 81 57 43 0 13 26 65 92 Select pivot 31 75 81 57 43 0 13 26 65 92 Partition 31 75 92 57 13 65 43 81 0 26 Quick sort The fastest known sorting algorithm in practice. CSE221/ICT221 Analysis and Design of Algorithms

  18. 31 75 92 57 13 65 43 81 0 26 Quick sort small Quick sort large 13 43 26 31 0 57 75 65 81 92 26 31 13 0 43 75 57 65 81 92 Quick sort CSE221/ICT221 Analysis and Design of Algorithms

  19. Quick sort

  20. Quick Sort Partitions CSE221/ICT221 Analysis and Design of Algorithms

  21. Quick sort CSE221/ICT221 Analysis and Design of Algorithms

  22. External Sort: A simple merge CSE221/ICT221 Analysis and Design of Algorithms

  23. Merge Sort CSE221/ICT221 Analysis and Design of Algorithms

  24. Merge Sort CSE221/ICT221 Analysis and Design of Algorithms

  25. CSE221/ICT221 Analysis and Design of Algorithms

  26. Sort Algorithm Analysis

  27. Analysis of Insertion Sort for (int i = 1; i < a.length; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; } CSE221/ICT221 Analysis and Design of Algorithms

  28. Complexity • Space/Memory • Time • Count a particular operation • Count number of steps • Asymptotic complexity CSE221/ICT221 Analysis and Design of Algorithms

  29. Comparison Count for (int i = 1; i < a.length; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; } CSE221/ICT221 Analysis and Design of Algorithms

  30. Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; How many comparisons are made? CSE221/ICT221 Analysis and Design of Algorithms

  31. Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; number of compares depends on a[]s and t as well as on i CSE221/ICT221 Analysis and Design of Algorithms

  32. Comparison Count • Worst-case count = maximum count • Best-case count = minimum count • Average count CSE221/ICT221 Analysis and Design of Algorithms

  33. Worst-Case Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a = [1, 2, 3, 4] and t = 0  4 compares a = [1,2,3,…,i] and t = 0  i compares CSE221/ICT221 Analysis and Design of Algorithms

  34. n  T(n) (i-1) = i=2 Worst-Case Comparison Count for (int i = 1; i < n; i++) for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; total compares = 1 + 2 + 3 + … + (n-1) = (n-1)n/2 = O(n2) CSE221/ICT221 Analysis and Design of Algorithms

  35. i-1 2 n  = T i=2 Average-Case Comparison Count for (int i = 1; i < n; i++) for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; = O(n2) CSE221/ICT221 Analysis and Design of Algorithms

  36. 31 75 81 57 43 0 13 26 65 92 Select pivot 31 75 81 57 43 0 13 26 65 92 Partition 31 75 92 57 13 65 43 81 0 26 Analysis of Quick Sort CSE221/ICT221 Analysis and Design of Algorithms

  37. private static void quicksort( Comparable [ ] a, int left, int right ) { /* 1*/ if( left + CUTOFF <= right ) { /* 2*/ Comparable pivot = median3( a, left, right ); // Begin partitioning /* 3*/ int i = left, j = right - 1; /* 4*/ for( ; ; ) { /* 5*/ while( a[ ++i ].compareTo( pivot ) < 0 ) { } /* 6*/ while( a[ --j ].compareTo( pivot ) > 0 ) { } /* 7*/ if( i < j ) /* 8*/ swapReferences( a, i, j ); else /* 9*/ break; } /*10*/ swapReferences( a, i, right - 1 ); // Restore pivot /*11*/ quicksort( a, left, i - 1 ); // Sort small elements /*12*/ quicksort( a, i + 1, right ); // Sort large elements } else // Do an insertion sort on the subarray /*13*/ insertionSort( a, left, right ); } MainQuick SortRoutine CSE221/ICT221 Analysis and Design of Algorithms

  38. n  i i=2 Worst-Case Analysis เวลาที่ใช้ในการ runquick sortเท่ากับเวลาที่ใช้ในการทำrecursive call2 ครั้ง + linear timeที่ใช้ในการเลือกpivot ซึ่งทำให้basic quicksort relation เท่ากับT(n) = T(i) + T(n-i-1) + cn ในกรณีWorst- caseเช่น การที่pivotมีค่าน้อยที่สุดเสมอ เวลาที่ใช้ในการทำrecursion คือT(n) = T(n-1) + cn n>1 T(n-1)= T(n-2)+c(n-1) T(n-2)= T(n-3)+c(n-2) … T(2) = T(1)+c(2) รวมเวลาทั้งหมดT(n) = T(1) + c = O(n2) CSE221/ICT221 Analysis and Design of Algorithms

  39. = + c = + c T(n/2) T(1) T(n/4) T(n/4) T(n/8) T(2) T(n) T(n/2) T(n) T(1) = + c n/2 n/4 n/4 n/8 n/2 1 2 n n 1 + c log n + c = = Best-Case Analysis The pivot is in the middle; T(n) = 2 T(n/2) + cn Divide both sides by n; Add all equations; … T(n) = c n logn + n = O(n log n) CSE221/ICT221 Analysis and Design of Algorithms

  40. + cN T(N) = Average timeofT(i)andT(N-i-1)is N -1 N -1 N -1    T( j ) T( j ) T( j ) j=0 j=0 j=0 + cN2 2 NT(N) = 2 (N-1) T(N-1) = + c(N – 1)2 1 2 T( j ) N N N -2  j=0 Average-Case Analysis (1/4) Total time; T(N) = T(i) + T(N-i-1) + c N…………..(1) …………..(2) Therefore …………..(3) …………..(4) …………..(5) CSE221/ICT221 Analysis and Design of Algorithms

  41. Rearrange terms in equation and ignore c onthe right-hand side; NT(N) = (N+1) T(N-1) +2cN …………..(7) …………..(8) (7) divides by N(N+1); T(N-1) 2c T(N) + = N N+1 N+1 Average-Case Analysis(2/4) …………..(6) NT(N) – (N-1) T(N-1) = 2T(N-1) +2cN - c (5)– (4); CSE221/ICT221 Analysis and Design of Algorithms

  42. …………..(8) …………..(9) …………..(10) . . . …………..(11) T(N-1) T(N-2) T(N-3) T(N-1) T(1) 2c 2c 2c 2c T(N-2) T(N) T(2) T(1) T(N) + + + + = = = = + = 2c N-1 N-2 N 2 N+1 N-1 N 3 N+1 N-1 N 3 2 N+1 N +1  i=3 Average-Case Analysis(3/4) Sun equations (8)to(11); …………..(12) CSE221/ICT221 Analysis and Design of Algorithms

  43. …………..(12) Sum in equation (12)ia approximately logC(N+1)+  - 3/2, whichisEuler’s constant  0.577 …………..(13) therefore T(1) T(N) T(N) + = 2c and =O(Nlog N) T(N) …………..(14) =O(log N) 2 N+1 N+1 N +1  i=3 Average-Case Analysis(4/4) In summary, time complexity ofQuick sort algorithmforAverage-Case is T(n) = O(n log n) CSE221/ICT221 Analysis and Design of Algorithms

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