Complexity analysis in Algorithms

Course:CSE221 (Algorithms)
Course Teacher: Md. Al-Amin Hossain ( Lecturer )
Section: P
Depertment: CSE(43 Batch)
Group Members:
01. Md. Ashaf Uddaula (161-15-7473)
02. Alamin Hossain (161-15-7483)
03. Md. Khasrur Rahman (161-15-7214)
04. Md. Eram Talukder(161-15-7485)
05. Ijaz Ahmed Utsa (161-15-7180)
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GOING TO TELL ABOUT……….
Motivations for Complexity Analysis.
Machine independence.
Best, Average, and Worst case complexities.
Simple Complexity Analysis Rules.
Simple Complexity Analysis Examples.
Asymptotic Notations.
Determining complexity of code structures.
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MOTIVATIONS FOR COMPLEXITY
ANALYSIS
• There are often many different algorithms which can be used to solve the same
problem. Thus, it makes sense to develop techniques that allow us to:
o compare different algorithms with respect to their “efficiency”
o choose the most efficient algorithm for the problem
• The efficiency of any algorithmic solution to a problem is a measure of the:
o Time efficiency: the time it takes to execute.
o Space efficiency: the space (primary or secondary memory) it uses.
• We will focus on an algorithm’s efficiency with respect to time.
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MACHINE INDEPENDENCE
• The evaluation of efficiency should be as machine independent as
possible.
• It is not useful to measure how fast the algorithm runs as this
depends on which particular computer, OS, programming
language, compiler, and kind of inputs are used in testing
• Instead,
o we count the number of basic operations the algorithm performs.
o we calculate how this number depends on the size of the input.
• A basic operation is an operation which takes a constant amount of
time to execute.
• Hence, the efficiency of an algorithm is the number of basic
operations it performs. This number is a function of the input size
n.
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BEST, AVERAGE, AND WORST CASE
COMPLEXITIES
• We are usually interested in the worst case
complexity: what are the most operations that
might be performed for a given problem size.
We will not discuss the other cases -- best
and average case
• Best case depends on the input
• Average case is difficult to compute
• So we usually focus on worst case analysis
• Easier to compute
• Usually close to the actual running time
• Crucial to real-time systems (e.g. air-traffic control)
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BEST, AVERAGE, AND WORST CASE
COMPLEXITIES
• Example: Linear Search Complexity
• Best Case : Item found at the beginning: One comparison
• Worst Case : Item found at the end: n comparisons
• Average Case :Item may be found at index 0, or 1, or 2, . . . or n - 1
-Average number of comparisons is: (1 + 2 + . . . + n) / n = (n+1) / 2
• Worst and Average complexities of common sorting algorithms
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SIMPLE COMPLEXITY ANALYSIS:
LOOPS
• We start by considering how to count operations in for-loops.
• We use integer division throughout.
• First of all, we should know the number of iterations of the
loop; say it is x.
• Then the loop condition is executed x + 1 times.
• Each of the statements in the loop body is executed x times.
• The loop-index update statement is executed x times.
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SIMPLE COMPLEXITY ANALYSIS: LOOPS (WITH
<)
• In the following for-loop:
The number of iterations is: (n – k ) / m
• The initialization statement, i = k, is executed one time.
• The condition, i < n, is executed (n – k) / m + 1 times.
• The update statement, i = i + m, is executed (n – k) / m times.
• Each of statement1 and statement2 is executed (n – k) / m times.
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for (int i = k; i < n; i = i + m){
statement1;
statement2;
}

SIMPLE COMPLEXITY ANALYSIS : LOOPS (WITH
<=)
• In the following for-loop:
• The number of iterations is: (n – k) / m + 1
• The initialization statement, i = k, is executed one time.
• The condition, i <= n, is executed (n – k) / m + 2 times.
• The update statement, i = i + m, is executed (n – k) / m + 1 times.
• Each of statement1 and statement2 is executed (n – k) / m + 1 times.
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SIMPLE COMPLEXITY ANALYSIS: LOOP
EXAMPLE
• Find the exact number of basic operations in the following program fragment:
• There are 2 assignments outside the loop => 2 operations.
• The for loop actually comprises
• an assignment (i = 0) => 1 operation
• a test (i < n) => n + 1 operations
• an increment (i++) => 2 n operations
• the loop body that has three assignments, two multiplications, and an addition => 6 n operations
Thus the total number of basic operations is 6 * n + 2 * n + (n + 1) + 3
= 9n + 4
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double x, y;
x = 2.5 ; y = 3.0;
for(int i = 0; i < n; i++){
a[i] = x * y;
x = 2.5 * x;
y = y + a[i];
}

SIMPLE COMPLEXITY ANALYSIS:
LOOPS WITH LOGARITHMIC ITERATIONS
• In the following for-loop: (with <)
-The number of iterations is: (Logm (n / k) )
• In the following for-loop: (with <=)
-The number of iterations is:  (Logm (n / k) + 1) 
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ASYMPTOTIC NOTATIONS
Following are the commonly used asymptotic notations to calculate the
running time complexity of an algorithm.
• Ο Notation
• Ω Notation
• θ Notation
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DETERMINING COMPLEXITY OF CODE STRUCTURES
Loops:
Complexity is determined by the number of iterations in the loop times the complexity of the body
of the loop.
Examples:
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Nested independent loops:
Complexity of inner loop * complexity of outer loop.
Examples:
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DETERMINING COMPLEXITY OF CODE
STRUCTURES
Nested dependent loops: Examples:
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Number of repetitions of the inner loop is: 1 + 2 + 3 + . . . + n = n(n+2)/2
Hence the segment is O(n2)
Number of repetitions of the inner loop is: 0
The outer loop iterates n times.
Hence the segment is O(n)
An important question to consider in complexity analysis is
whether the problem size is a variable or a constant.

STRUCTURES
If Statement:
O(max(O(condition1), O(condition2), . . ,
O(branch1), O(branch2), . . ., O(branchN))
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STRUCTURES
O(if-else) = Max[O(Condition), O(if), O(else)]
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Switch: Take the complexity of the most expensive case including the default case
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o(n)
o(n2)
Overall Complexity: o(n2)

THE END
Success comes from Experience
&
Experience comes from Bad Experience
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Complexity analysis in Algorithms

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