Data Structures - 09. Introduction to the Asymptotic Analysis, I

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Introduction to theAsymptotic Analysis I

Lecture 12

1107186 – Estrutura de Dados – Turma 02

Prof. Christian Azambuja PagotCI / UFPB

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Algorithm Efficiency

● Can be measured in terms of:– Space.

– Bandwidth.

– Time !!!

– Etc.

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Algorithm Efficiency

● Is influenced by – Input (Problem Instance):

● Sorting: number of elements in a list.● Multiplication: number of bits of each operand. ● Graph traversal: number of nodes and edges.● Etc.

– Number of instructions executed.● Depends on the computer organization.

– Thus, we must define some assumptions regarding “our machine”...

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Algorithm Efficiency

● Assumptions regarding the architecture of our virtual computer:– Sequential processor.

– Simple (regarding microinstructions) instructions.

– No concurrency.

– RAM memory.

– No memory hierarchy.

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Example: Insertion Sort

● The algorithm:

void InsertionSort(int* v, int size){int i, j, key;for (i=1; i<size; i++){

key = v[i];j = i­1;while ((j>=0) && (v[j]>key)){

v[j+1] = v[j];j = j­1;

}v[j+1] = key;

}}

C code excerpt:

v = [5, 2, 4, 6, 1, 3]

Input vector:

v = [1, 2, 3, 4, 5, 6]

Output vector:

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Example: Insertion Sort

● The cost of each line:

void InsertionSort(int* v, int size){int i, j, key;for (i=1; i<size; i++){

key = v[i];j = i­1;while ((j>=0) && (v[j]>key)){

v[j+1] = v[j];j = j­1;

}v[j+1] = key;

}}

C code excerpt:

Cost(μs) # steps

c1

c2

c3

c4

c5

c6

c7

nn−1n−1∑i=1

n−1ti

∑i=1

n−1(t i−1)

∑i=1

n−1(t i−1)

n−1

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Example: Insertion Sort

● Total cost T(n):Cost(μs) # steps

c1

c2

c3

c4

c5

c6

c7

nn−1n−1∑i=1

n−1ti

∑i=1

n−1(t i−1)

∑i=1

n−1(t i−1)

n−1

T (n)=c1n+c2(n−1)+c3(n−1)+c 4∑i=1

n−1t i+c5∑i=1

n−1(ti−1)+c6∑i=1

n−1(t i−1)+c7(n−1)

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Example: Insertion Sort

● Best T(n):

void InsertionSort(int* v, int size){int i, j, key;for (i=1; i<size; i++){

key = v[i];j = i­1;while ((j>=0) && (v[j]>key)){

v[j+1] = v[j];j = j­1;

}v[j+1] = key;

}}

C code excerpt:

Cost(μs) # steps

c1

c2

c3

c4

c5

c6

c7

nn−1n−1

n−1

n−100

ti = 1

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Example: Insertion Sort

● Best T(n) (ti=1):

Cost(μs)

c1

c2

c3

c4

c5

c6

c7

T (n)=c1n+c2(n−1)+c3(n−1)+c 4∑i=1

n−1t i+c5∑i=1

n−1(ti−1)+c6∑i=1

n−1(t i−1)+c7(n−1)

nn−1n−1

n−1

n−100

# steps

T (n)=c1n+c2(n−1)+c3(n−1)+c 4(n−1)+c7(n−1)Total Cost:(for comparision)

Best case:

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Example: Insertion Sort

● Best T(n) (ti=1):

a=c1+c2+c3+c4+c7

b=−(c2+c3+c 4+c7)

n

T(n)

T (n)=c1n+c2(n−1)+c3(n−1)+c 4(n−1)+c7(n−1)1

T (n)=(c1+c2+c3+c4+c7)n−(c2+c3+c4+c7)2

T (n)=an+b3

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Example: Insertion Sort

● Worst T(n):

void InsertionSort(int* v, int size){int i, j, key;for (i=1; i<size; i++){

key = v[i];j = i­1;while ((j>=0) && (v[j]>key)){

v[j+1] = v[j];j = j­1;

}v[j+1] = key;

}}

C code excerpt:

Cost(μs) # steps

c1

c2

c3

c4

c5

c6

c7

nn−1n−1

n−1

∑i=1

n−1(i+1)

∑i=1

n−1i

∑i=1

n−1i

ti = i+1

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Example: Insertion Sort

● Worst T(n) (ti=i+1):

Cost(μs)

c1

c2

c3

c4

c5

c6

c7

T (n)=c1n+c2(n−1)+c3(n−1)+c 4∑i=1

n−1t i+c5∑i=1

n−1(ti−1)+c6∑i=1

n−1(t i−1)+c7(n−1)

# steps

Total Cost (for comparision) :

Worst case:

nn−1n−1

n−1

∑i=1

n−1(i+1)

∑i=1

n−1i

∑i=1

n−1i

T (n)=c1n+c2(n−1)+c3(n−1)+c 4∑i=1

n−1( i+1)+c5∑i=1

n−1i+c6∑i=1

n−1i+c7(n−1)

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c4 ( n(n+1)−22 )+c5(n(n−1)

2 )+T (n)=c1n+c2(n−1)+c3(n−1)+

c6( n (n−1)

2 )+c7 (n−1)

2

∑i=1

n−1(i+1)=

n(n+1)−22

T (n)=c1n+c2(n−1)+c3(n−1)+

c4∑i=1

n−1(i+1)+c5∑i=1

n−1i+

c6∑i=1

n−1i+c7 (n−1)

1

Example: Insertion Sort

● Worst T(n) (ti=i+1):

a=c4

2+

c5

2+

c6

2

b=c1+c2+c3+c4

2−

c5

2−

c6

2+c7

c=−(c2+c3+c4+c7)

∑i=1

n−1i=

n(n−1)

2

n

T(n)

T (n)=( c4

2+

c5

2+

c6

2 )n2+(c1+c2+c3+

c4

2−

c5

2−

c6

2+c7)n−(c2+c3+c4+c7 )3

T (n)=an2+bn+c

4

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Example: Insertion Sort

● Comparison:– Best case:

– Worst case:

T (n)=an−b

T (n)=an2+bn+c

n

T(n)

n

T(n)Running time is dominated by n.

Running time is dominated by n2.

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Discussion

● Sometimes we can determine the exact running time of an algorithm. However:– Evaluation of long expressions are time consuming.

– Most of the terms contribute too little to the description of the algorithm behavior.

– Obtained running times (in terms of time units) may vary from machine to machine.

– Small inputs are efficiently processed. We are concerned about very large inputs.