Discuss seven functions, Analysis of algorithms- Experimental Studies/Primitive operations/Asymptotic notation- Big Oh/Big-Omega/Big-Theta
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2. Session 4 is about
Seven Functions
Analysis of algorithms
3. 1. Constant Function
Simplest function:
f(n) = c (some fixed constant)
Value of n doesn’t matter.
Ex: adding two numbers, assigning variable,
comparing two numbers
4. 2. Logarithm Function
f (n) = logb n
for some constant b > 1
This function is defined:
x = logbn if and only if bx = n
Value b is known as the base of the logarithm.
log n = log2 n.
5. Rules for algorithms
Given real numbers a , c > 0, and b , d > 1, we have
If b = 2, we either use log n, or lg n.
6. 3. Linear function
f(n) = c n
Function arises when a single basic
operation is required for each of n
elements.
7. 4. N- log- N function
f(n) = n logn
Grows faster than the linear function.
Slower than the quadratic function.
8. 5. Quadratic function
f(n) = n2
For nested loops, where combined operations of
inner and outer loop gives n*n = n2.
9. 6. Cubic Function & Other Polynomials
f(n) = n3
Comparatively less frequent
10. 7. Exponential Function
f(n) = bn
b is a positive constant, called as the base.
n is the argument and called as the exponent.
Also called as exponent function.
14. Experimental Studies
Running time of implemented algorithm is calculated
by:
Executing it on various test inputs.
Recording the actual time spent in each
execution.
Using the System.curent Time Millis () method.
15. Experimental Studies (cont.)
Steps :
1. Write a program implementing the algorithm.
2. Run the program with inputs of varying size and
composition.
3. Use a system call to get running time measure.
4. Plot the results.
5. Perform a statistical analysis.
17. Limitations of Experimental analysis
Need limited set of test inputs.
Need same hardware and software
environments.
have to fully implement and execute an
algorithm.
18. Primitive operations
Set of primitive operations:
1. Assigning a value to a variable.
2. Calling a method.
3. Performing an arithmetic operation (for example, adding two numbers).
4. Comparing two numbers.
5. Indexing into an array.
6. Following an object reference.
7. Returning from a method.
19. Example
Algorithm arrayMax(A, n)
# operations
currentMax ← A[0] 2
for i ← 1 to n − 1 do 2n
if A[i] > currentMax then 2(n − 1)
currentMax ← A[i] 2(n − 1)
{ increment counter i } 2(n − 1)
return currentMax 1
Total 8n − 2
20. Estimating Running Time
Algorithm arrayMax executes 8n − 2 primitive
operations in the worst case.
Define:
a = Time taken by the fastest primitive operation
b = Time taken by the slowest primitive operation
Let T(n) be worst-case time of arrayMax. Then
a (8n − 2) ≤ T(n) ≤ b(8n − 2)
Hence, the running time T(n) is bounded by two
linear functions.
21. Asymptotic Notation
Uses mathematical notation for functions.
Disregards constant factors.
n refer to a chosen measure of the input “size”.
Focus attention on the primary "big-picture"
aspects in a running time function.
23. Big – Oh Notation
Given functions are f(n) and g(n)
f(n) is O(g(n)) if there are positive constants
c>0 and n0 ≥1 such that
f(n) ≤ cg(n) for n ≥ n0
24. Big – Oh Notation (cont.)
Example: 2n + 10 is O(n)
2n + 10 ≤ cn
(c − 2) n ≥ 10
n ≥ 10/(c − 2)
Pick c = 3 and n0 = 10
25. Big – Oh Notation (cont.)
Example: the function n2 is not O(n)
n2 ≤ cn
n ≤ c
The above inequality cannot be satisfied since c
must be a constant
26.
27. Big – Omega Notation
Given functions are f(n) and g(n)
f(n) is Ω(g(n)) if there are positive constants
c>0 and n0 ≥1 such that
f(n) ≥ cg(n) for n ≥ n0
28. Big – Omega Notation(cont.)
Example : the function 5n2 is Ω(n2)
5n2 ≥ c n2
5n ≥ c n
c = 5 and n0 = 1
29. Big – Theta Notation
Given functions are f(n) and g(n)
f(n) is Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n)) ,
that is, there are real constants
c’> 0 and c’’ >0 and n0 ≥1 such that
c’ g(n) ≤ f(n) ≤ c’’ g(n) for n ≥ n0
30. Big – Theta Notation(cont.)
Example : 3nlog n + 4n + 5logn is Θ(n log
n).
Justification:
3nlogn ≤ 3nlogn + 4n + 5logn ≤
(3+4+5)nlog n
for n ≥ 2
