CSE 221/ICT221 Analysis and Design of Algorithms Lecture 05: Analysis of time Complexity of Sorting Algorithms

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CSE 221/ICT221 Analysis and Design of
AlgorithmsLecture 05Analysis of time
Complexity of Sorting Algorithms

• Dr.Surasak Mungsing
• E-mail Surasak.mu_at_spu.ac.th

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Sorting Algorithms

• Bin Sort
• Radix Sort
• Insertion Sort
• Shell Sort
• Selection Sort
• Heap Sort
• Bubble Sort
• Quick Sort
• Merge Sort

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Bin Sort
(a) Input chain
(b) Nodes in bins
(c) Sorted chain
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Radix Sort with r10 and d3
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Insertion Sort Concept
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Shell Sort Algorithm
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Shell Sort Algorithm
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Shell Sort Algorithm
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Shell Sort Demohttp//e-learning.mfu.ac.th/mflu/
1302251/demo/Chap03/ShellSort/ShellSort.html
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Selection Sort Concept
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Heap Sort Algorithm
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Bubble Sort Concept
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Quick sort
The fastest known sorting algorithm in practice.
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Quick sort
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Quick sort
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Quick Sort Partitions
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Quick sort
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External Sort A simple merge
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Merge Sort
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Merge Sort
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Sort Algorithm Analysis
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Analysis of Insertion Sort

• for (int i 1 i lt a.length i)
• // insert ai into a0i-1
• int t ai
• int j
• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj
• aj 1 t

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Complexity

• Space/Memory
• Time
• Count a particular operation
• Count number of steps
• Asymptotic complexity

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Comparison Count

• for (int i 1 i lt a.length i)
• // insert ai into a0i-1
• int t ai
• int j
• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj
• aj 1 t

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Comparison Count

• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj
• How many comparisons are made?

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Comparison Count

• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj
• number of compares depends on
• as and t as well as on i

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Comparison Count

• Worst-case count maximum count
• Best-case count minimum count
• Average count

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Worst-Case Comparison Count

• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj

a 1, 2, 3, 4 and t 0 ? 4 compares a
1,2,3,,i and t 0 ? i compares
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Worst-Case Comparison Count

• for (int i 1 i lt n i)
• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj

total compares 1 2 3 (n-1)

(n-1)n/2
O(n2)
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Average-Case Comparison Count

• for (int i 1 i lt n i)
• for (j i - 1 j gt 0 t lt aj j--)
• aj 1 aj

O(n2)
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Analysis of Quick Sort
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MainQuick SortRoutine
private static void quicksort( Comparable a,
int left, int right ) / 1/ if(
left CUTOFF lt 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 ) lt 0 )
/ 6/ while( a --j .compareTo(
pivot ) gt 0 ) / 7/ if( i lt 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 )
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Worst-Case Analysis
??????????????? run quick sort ???????????????????
????? recursive call 2 ????? linear time
???????????????? pivot ????????? basic quick sort
relation ??????? T(n) T(i) T(n-i-1) cn
?????? Worst- case ???? ?????? pivot
??????????????????? ????????????????? recursion
??? T(n) T(n-1) cn ngt1 T(n-1)
T(n-2)c(n-1) T(n-2) T(n-3)c(n-2)
T(2) T(1)c(2) ?????????????? T(n) T(1)
c
O(n2)
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Best-Case Analysis

• The pivot is in the middle T(n) 2 T(n/2) cn
• Divide both sides by n

T(n) c n logn n
O(n log n)
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Average-Case Analysis (1/4)
Total time T(N) T(i) T(N-i-1) c N ..(1)
..(2)
Therefore
..(3)
..(4)
..(5)
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Average-Case Analysis (2/4)
..(6)
NT(N) (N-1) T(N-1) 2T(N-1) 2cN - c
(5) (4)
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Average-Case Analysis (3/4)
Sun equations (8) to (11)
..(12)
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Average-Case Analysis (4/4)
In summary, time complexity of Quick sort
algorithm for Average-Case is T(n) O(n log n)
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