Lesson no 3 - Algorithm Analysis - II.pptx

Algorithm Analysis – Part II
Algorithm
Analysis
&
Design April 2020
1
S. Hassan Adelyar, Ph.D
Instructor of Computer Science Faculty
Kabul University
December 2020
Lesson III: Algorithm-Analysis Part II

Algorithm
Analysis
&
Design April 2020
2
Learning Outcome
 By the end of this presentation, students will be
able to:
 Count the number of steps for simple operations
 Count the number of steps for loops &
conditional statements.
 Count the number of steps for simple
Algorithms.
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Algorithm
Analysis
&
Design April 2020
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Simple Operation
 Two classes of operations:
 Comparison
 Arithmetic.
 Comparison operations
 Include equal, not equal, less than, greater
than, less than or equal, and greater than or
equal.
 The comparison operators are counted as a
single step.
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Algorithm
Analysis
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Design April 2020
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 Arithmetic operators are counted in three
groups:
 Additive
 Multiplicative
 Logarithms and geometric functions
 Additive operators include:
 addition, subtraction, increment, and
decrement.
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Algorithm
Analysis
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Design April 2020
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 Multiplicative operators include:
 multiplication, division, and modulus.
 Additive & multiplicative are counted separately
because multiplications are considered to take
longer than additions.
 Reduce the number of multiplications.
 Logarithms and geometric functions:
 Consume more time than multiplications.
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Algorithm
Analysis
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Design April 2020
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 A special case is integer multiplication or division
by a power of 2.
 This operation can be reduced to a shift operation,
which is considered as fast as an addition.
 Multiplication or division by 2 is commonly found
in divide and conquer algorithms.
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Algorithm
Analysis
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Design April 2020
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 We define a set of primitive operations such as the
following:
 Assigning a value to a variable.
 Calling a method.
 Performing an arithmetic operation (for
example, adding two numbers).
 Comparing two numbers.
 Indexing into an array.
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Analysis
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Design April 2020
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 Instead of trying to determine the specific
execution time of each primitive operation, we will
simply count how many primitive operations are
executed, and use this number as a measure of the
running-time of the algorithm.
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Algorithm
Analysis
&
Design April 2020
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 String operation
 Linear, but indexing and length are constant.
 For loops
 The total running time of a statement inside a
nested loops is the running time of the statement
multiplied by the product of the loops.
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Algorithm
Analysis
&
Design April 2020
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 If/else
 For the fragment if (condition) S1 else S2, the
running time of an if/else statement is never more
than the running time of the test plus the larger of
the running times of S1 and S2.
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Algorithm
Analysis
&
Design April 2020
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Examples
 Examples
 1) x = 0;
 2) for(i=0; i<n; i++)
 3) x++;
 The number of steps can be calculate as follow:
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Algorithm
Analysis
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Design April 2020
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 Line Time Number
 1 c1 1
 2 c2 n+1
 3 c3 n
 Therefore, T(n) is:
 T(n) = c1+c2(n+1)+c3n
 Now considering c as the larger of c1, c2 & c3:
 T(n)= c(2n+2)
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Algorithm
Analysis
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Design April 2020
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 Example 2:
 1) x=0;
 2) for(i = 0; i<n; i++)
 3) for(j=0; j< n; j++)
 4) x++;
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Algorithm
Analysis
&
Design April 2020
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 1 c1 1
 2 c2 n+1
 3 c3 n(n+1)
 4 c4 n x n
 Therefore, T(n) is:
 T(n) = c1 + c2(n+1) + c3n(n+1)+c4n2
 T(n) = C(2n2
+2n +2)
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Algorithm
Analysis
&
Design April 2020
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 Example 3:
 1) int Factorial(int n) {
 2) int fact = 1;
 3) for(int i=1; i<n; i++)
 4) fact *=i;
 5) return fact;
 }
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Algorithm
Analysis
&
Design April 2020
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 2 c1 1
 3 c2 n
 4 c3 n-1
 5 c4 1
 T(n) = c1+c2n+c3(n-1)+c4
 T(n) = C(2n+1)
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Analysis
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Design April 2020
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Summary
 In this presentation we study Algorithm analysis by
counting the number of steps for simple operations.
 In addition to counting the number of steps for
simple operation we studied how to count the
number of steps for loops & conditional statements.
 Furthermore, we studied the ways for counting the
number of steps for simple Algorithms.
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Algorithm
Analysis
&
Design April 2020
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:‫نمائید‬ ‫دریافت‬ ‫را‬ ‫ذیل‬ ‫کود‬ ‫زمانی‬ ‫تابع‬ :‫تمرین‬
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for (i=0; i<10; i++){
a = a + i;
b = b * i;
}

Algorithm
Analysis
&
Design April 2020
19
Exercises
1) Count the number of steps for the following code (5
score):
int a=0;
for(i = 0; i<n; i++)
for(j=0; j< n; j++)
a++;
2) Why we measure the time of Algorithm by counting the
number of steps instead of measuring the time by
seconds?
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Lesson no 3 - Algorithm Analysis - II.pptx

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