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Transcript of Algorithm analysis examples
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Algorithm AnalysisE lExamples
Muhammad Shoaib Farooq
1M.Shoaib Farooq
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Complexity of Algorithm
- Each instruction executes within const timetime
e.g. Assignment Statementx=a ....................1x a ....................1x= a+b*c/h-u ......1a>b 1a>b ......................1
M.Shoaib Farooq 2
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If StatementComplexity= ConstTime(bool exp) +
max(ThanPart /else Part)max(ThanPart /else Part)
e g if (a>b) than 1e.g. if (a>b) than .........1a=2......................1
elseelsex=x+2*3 .................1
T(n)= 1+1 = 2= O(1) Or O(c)
M.Shoaib Farooq 3
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If StatementIf Statemente.g.g
if (a>b) than……..........1a=2.......................1b b+3*t 1b= b+3*t...............1
elsex=x+2*3 ..............1x x 2 3 ..............1
T(n)= 1+2 = 3= O(1)
M.Shoaib Farooq4
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LOOP
For n complete iterations- while ............n- for loop ..........n+1- repeat..until ....n
M.Shoaib Farooq
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An Example: LOOP
i=0 .............................1while i<=n ..................n
i++ n-1i++ ........................n-1a=i ...........................1
T(n)=1+n+n-1+1( )= 2n+1= O(n)= O(n)
M.Shoaib Farooq 6
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An Example: LOOPi=1 ...........................1while i<=n nwhile i<=n ................n
i++ ........................1a=i ............................1
T (n)= 1+n+1(n-1)+1 = 2n+1 = o(n)( ) ( ) ( )OR
T(n) = 1+ +1∑n
1T(n) = 1+ +1= 1+ n+1
∑=i 1
M.Shoaib Farooq
= O(n)7
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An Example: LOOPpi=1 ...............................1while (i<=n) nwhile (i<=n)....................n
a=2+g........................n-1i i+1 1i=i+1 ............. ..........n-1
if (i<=n)........... .............12 1a=2 ..........................1
elsea=3............................1
T(n) = 1+ n+n-1+n-1+1+1 =3n+1 = O(n)
M.Shoaib Farooq 8
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An Example: LOOPi=1..............................1while (i<=10)................10
i i 1 9i=i+1........................9i=1 ..............................1while (i<=n) nwhile (i<=n)..................n
a=2+g......................n-1i=i+1 .......................n-1
if (i<=n)........................1a=2 ..........................1 T ( n) = ?
elsea=3...........................1
M.Shoaib Farooq9
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Linear SearchLinearSearch(A, Key)
i 1 1
while (A[i] !=key) and (i<=length[A])i 1
nn-1i 1
if ( i<=length[A]) 1
return trueelse
1
return false 1
M.Shoaib Farooq
T(n) = 1 + n + n-1 + 1 +1 = 2n+2
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Linear Search- Best-caseLinear Search Best caseBest case occurs when key element occurs at first location of Array then loop executes only once
T(n) = 1 + n + (n-1) + 1 +1T(n) 1 n (n 1) 1 1= 1+1+0+1+1
O(1)= O(1)
M.Shoaib Farooq 11
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Linear Search- Worst-caseLinear Search Worst caseWorst case occurs when key element is the last element in array or is not there at all loop executes n timesT(n) = 1 + n + (n-1) + 1 +1
= 2n +2 2n 2= O(n)
M.Shoaib Farooq12
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Linear Search- Average-caseLinear Search Average caseKey element is equally likely to occur at any position in the array. The number of comparisons can be any of the numbers 1,2,3…..n and each number occurs with probability p=1/nT(n) = 1.1/n + 2. 1/n +……….+ n.1/n
= (1+2+3+……+n).1/n (1 2 3 …… n).1/n= [n(n+1)/2] 1/n = n+1/2
O( )= O(n) M.Shoaib Farooq 13
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Binary Search BinarySearch(A, key)1. mid (low + high)/22. while (A[mid]!=key) and ( low<=high)3. if (key>A[mid])4 low mid +14. low mid +15. else6 high mid - 16. high mid 17. mid (low + high)/28. if (low<=high) T(n)=1+k+(k-1)+(k-1)+(k-1)+1+1
4K 4 l O(l )( g )
9. return true10. else
= 4K= 4 log2n O(log2n)
n/2 + n/4 + n/8 + n/2k
M.Shoaib Farooq
11. return false n/2 + n/4 + n/8 ……..+ n/2k
2K<=n k=log2n14
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An Example: Insertion SortAn Example: Insertion SortInsertionSort(A, n)( , )1. for i = 2 to n 2. key = A[i] 3. j = i – 14. while (j > 0) and (A[j] > key)
j 1 j5. A[j+1] = A[j]6. j = j – 17 A[j+1] = key7. A[j+1] = key
15M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort30 10 40 20 i = ∅ j = ∅ key = ∅
A[j] = ∅ A[j+1] = ∅
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = ∅ A[j+1] = ∅
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
16M.Shoaib Farooq
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An Example: Insertion Sort30 10 40 20 i = 2 j = 1 key = 10
A[j] = 30 A[j+1] = 10
An Example: Insertion Sort
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 10
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
17M.Shoaib Farooq
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An Example: Insertion Sort30 30 40 20 i = 2 j = 1 key = 10
A[j] = 30 A[j+1] = 30
An Example: Insertion Sort
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
18M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort30 30 40 20 i = 2 j = 1 key = 10
A[j] = 30 A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
19M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort30 30 40 20 i = 2 j = 0 key = 10
A[j] = ∅ A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = ∅ A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
20M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort30 30 40 20 i = 2 j = 0 key = 10
A[j] = ∅ A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = ∅ A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
21M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 2 j = 0 key = 10
A[j] = ∅ A[j+1] = 10
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = ∅ A[j+1] = 10
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
22M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 3 j = 0 key = 10
A[j] = ∅ A[j+1] = 10
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = ∅ A[j+1] = 10
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
23M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 3 j = 2 key = 40
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
24M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 4 j = 2 key = 40
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
25M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
26M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
27M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 4 j = 3 key = 20
A[j] = 40 A[j+1] = 20
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 40 A[j+1] = 20
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
28M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 20 i = 4 j = 3 key = 20
A[j] = 40 A[j+1] = 20
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 40 A[j+1] = 20
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
29M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 40 i = 4 j = 3 key = 20
A[j] = 40 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 40 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
30M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 40 i = 4 j = 3 key = 20
A[j] = 40 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 40 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
31M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 40 i = 4 j = 3 key = 20
A[j] = 40 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 40 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
32M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 40 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
33M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 40 40 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 40
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 40
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
34M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 30 40 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
35M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 30 40 i = 4 j = 2 key = 20
A[j] = 30 A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 30 A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
36M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 30 30 40 i = 4 j = 1 key = 20
A[j] = 10 A[j+1] = 30
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 10 A[j+1] = 30
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
37M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 20 30 40 i = 4 j = 1 key = 20
A[j] = 10 A[j+1] = 20
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 10 A[j+1] = 20
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}}}
38M.Shoaib Farooq
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An Example: Insertion SortAn Example: Insertion Sort10 20 30 40 i = 4 j = 1 key = 20
A[j] = 10 A[j+1] = 20
InsertionSort(A, n) {f i 2 t {
1 2 3 4A[j] = 10 A[j+1] = 20
for i = 2 to n {key = A[i]j = i - 1;while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) {
A[j+1] = A[j]j = j - 1
}}A[j+1] = key
}} Done!} Done!
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Analysis: Insertion SortAnalysis: Insertion Sort
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Worst Case
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Analysis: Insertion SortAnalysis: Insertion Sort
tj = 1
n
∑=
−=n
jn
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T(n)= c1n + c2(n - 1) + c4(n - 1) + c5(n - 1) + c8(n - 1)
= (c1 + c2 + c4 + c5 + c8)n - (c2+ c4 + c5 + c8).
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Best Case
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Selection Sort SelectionSort(A)
f i 1 1 dfor i 1 to n-1 dosmall ifor j i+1 to n do
if (A[small]>A[j])if (A[small]>A[j])small=j
A[i] A[ ll]A[i] A[small]
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Sort (Which one?)Sort (Which one?)
SORT ( A)SORT ( A)for i 1 to n-1 do
f j i 1 t dfor j i+1 to n doif ( A[i]>A[j])
A[i] A[j]
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Bubble Sort BubbleSort (A)for i 1 to n-1 do
for j 1 to n-1 dofor j 1 to n 1 do if ( A[j] > A[j+1])
A[j] A[j+1]A[j] A[j+1]
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An Example: LOOPAnalysis (Bottom Up Approach)
T(n)= ?
M.Shoaib Farooq
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Bottom Up Analysisp y• Bottom while Loop (line 5 and 6)
∑j
while(j) =
• Inner For Loop (line 3 and 4)∑=
+=k
j0
11
for(i) = ∑ ∑= =
+=i
j
i
j
jjwhile2
1
2
1
1)(
i i2 2
= = 2i(2i+1)/2 + 2i∑ ∑= =
+i
j
i
jj
2
1
2
11
= 2i2 + I + 2i 2i2 +3i
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Bottom Up Analysis
Outer For Loop (Line 1 )T( )
n nT(n) = = ∑=i
ifor1
)( ∑=
+n
i
ii1
2 )32(
∑∑nn
2
== 2 [(2n3+3n2+n)/6] + 3 [n(n+1)/2]
∑∑==
+ii
ii11
2 32 Quadratic Series
= 2 [(2n +3n +n)/6] + 3 [n(n+1)/2]= O (n3)
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