1 - frequency count

8/3/2019 1 - Frequency Count

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College of Computer Studies, De La Salle University Manila2401 Taft Avenue, Metro Manila 1004 Philippines

Data Structures and

Algorithms

Algorithm Analysis


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Outline

What is an Algorithm?

Algorithm Formulation

Algorithm Analysis

Frequency Count

Operation Count


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An organized sequence or list of clearsteps or operations needed to solve a

given programming problem 3 components:

Inputs

Steps or instructions

Output


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Any well-defined computational procedurethat takes some value(s) as input and

produces some value(s) as output Sequence of computational steps that

transform the input to output


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A means to solve a problem

A problem can be solved in many different ways:

Many algorithms can be used to solve the sameproblem

We choose between different algorithms basedon their efficiency

A good metric of efficiency is the computationalcomplexity of an algorithm


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Algorithms and Computer Science

Can a particular task be accomplished by acomputing device?

What is the minimum number of stepsoroperations for any algorithm to perform acertain function?

Concerns: effectiveness, efficiency


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Steps:

1. Understand the problem

Determine the given inputs, the desiredoutput, and the steps to reach the output

2. Design the algorithm

Write the algorithm that will solve theproblem given the inputs


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3. Analyze the algorithm

Determine the resources that the algorithm

requires memory, bandwidth, hardware,computational time

4. Refine the algorithm

5. Verification


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Algorithm Analysis

Study the behavior of the algorithm todetermine its pattern and performance

Measured in terms ofexecution time (speed)and amount of memory (space) consumed

Designing efficient algorithms


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How to analyze programs?

2 phases:

1. Priori estimates

Obtain a function which bounds thealgorithms complexitytime

The amount of time a single execution will

take, or the number of times a statement isexecuted

Space consumed may also be estimated


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How to analyze programs

It is impossible to know the exact amountof time to execute any command unless

the following are known: The machine for execution

machine instruction set

time required by each machine instruction

translation of the compiler from source tomachine language


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How to analyze programs?

2. Posteriori estimates

Study of exact time & space required for

execution


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Frequency Count

Number of statements or steps neededby the algorithm to finish

Simple statements:

a 10 Count: 1

b a * 2 Count: 1


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Frequency Count

Conditional Statements

if ()

;else

;

Sum of the following:

1 + Max{ Count(), Count() }


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Frequency Count - Example

if x > 1

y 10;

else

{ y 20;

z 30;}

Total = 3

1

1

2

2


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Loop Statements

for i

to

Frequency Count

Example

for i 1 to nx x + 1

ub - lb + 2

ub - lb + 1

n - 1 + 2 = n+1n

Total = 2n+1


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if x < 1

y 10else if x < 2

{y 20; z 30}

else{ for i 1 to x

print(i)}

Frequency Count - Example1

1

Total = 2x+3

1

2

x

x+12x+1


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Frequency Count - Exercise

1. k 500;for i 1 to k-1

z z + 1

2. for k 0 to n

{ print (k);print(n-k);

}


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Nested Statements

for i 1 to nfor j 1 to n

x x + 1

Frequency Count

n - 1 + 2 = n+1(n+1)(n)

Total = n+1 + n2+n + n2= 2n2 + 2n + 1

(n)(n)

(n) =


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for i 2 to n-1

for j

1 to nx x + 1

Frequency Count - Examplen-1 - 2 + 2 = n-1

(n+1)(n)

Total = n-1 + n2-2n+n-2 + n2-2n

= 2n2 - 2n - 3

(n-2)(n-2)

= (n-2)


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for i 1 to n{ x x + 1

for j

3 to n+1{ y y + 1z z + 1

}

}

Frequency Count - Example

Total = n+1 + n + n2 + (n-1)n + (n-1)n

= 2n+1 + n2 + n2-n + n2-n

= 3n2 + 1

= (n)

= (n-1)

n+1n

(n)n(n)(n)

(n-1)(n-1)


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for i 1 to n

for j 1 to nfor k 1 to n

z z + 1


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Loop Statementsdo

while

Frequency Count

Ex.x 1 1

dox x + 1 n-x=n-1

while x < n n-x=n-1Total = 2n-1

What if while x


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Loop Statementswhile

Frequency Count

What if while x


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1. x 1while x


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1. for i 1 to n

for j

1 to 2nx x + 1

2. for k 2 to n+1

for j 3 to n-3x x + 1


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Frequency Count - Exercise3. for i 2 to n+1

for j 3 to n-3for k 4 to n-4

x x + 1

4. for i 1 to n{ j 2

while j


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5. for i 1 to nfor j 1 to i

x x + 1

6. for i 1 to nfor j 2i downto 1

x x + 1

7. for i 1 to nfor j i to n

x x + 1


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8. for i 1 to n-1for j 1 to ix x + 1

9. for i 4 to nfor j 1 to ix x + 1

10. for i 1 to nfor j 1 to i

for k 1 to jx x + 1


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College of Computer Studies, De La Salle University Manila2401 Taft Avenue Metro Manila 1004 Philippines


11. while (i < n) 13. while (i > n){ k = k+1; { k = k+1;

i = i+1; i = i-1;

} }

12. while (i>=n) 14. while (b != n-10){ k= k+1; b= b+1;

x = x+1;i = i-1; 15. do { h = h-1;

} } while (h >= n);

1 - frequency count

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