1. Design and Analysis of Algorithms
Lecture # 02
Muhammad Nasir
Department of Computer Science
COMSATS University Islamabad,
Lahore Campus

2. Model of Computation
Our analysis will be independent of the variations in:
Machine hardware
Operating system
Compiler
Programming language etc
Algorithms need to understood by the people not machines

3. Model of Computation
Ideally this model should be a reasonable abstraction of a standard generic single-processor machine
We will call this machine as Random Access Machine

4. Random Access Machine (RAM)
A Random Access Machine (RAM) is an ideal machine with
Infinite large memory
Executes one instruction at a time (no parallelism)
Each instruction performs some basic operations on two values in memory

5. Basic RAM operations Basic operations include
Assigning a value to a variable
Performing basic arithmetic operation (+, -, *, /) on any size of integer values
Accessing an element of an Array
like array[100]

6. An Important Assumption
Each basic operation takes same constant time to execute
There is no difference between performing addition or multiplication on any size of the values

7. Running Time Analysis Criteria
Worst case analysis
Maximum running time over all (legal) inputs of size n
Average case analysis
average running time over all inputs of size n
Best case analysis
Running time over the best case input of size n
We always look for the worst case running time

8. Theoretical versus empirical
The difference is the theoretical analysis, as the name suggests, is based on theory and not actual events. Empirical analysis is based on real testing events. Empirical is what you KNOW, theoretical is what you THINK.

9. Example: 2-Dimension Maxima
Suppose that you want to buy a computer
You want to buy a fastest computer But fastest computer is expensive and you want the cheapest
You are unable to decide which is more important:
Speed or
Price

10. Example: 2-Dimension Maxima
Certainly, we don’t want a computer if there is another which is faster and cheaper
We can say that the fast and cheap dominates the slow and expensive computer
So, we want a list of computers that are not dominated by any other

11. Example: 2-Dimension Maxima
This how we might model this as a formal problem
Let a point p in 2-dimentional space be represented by its integer coordinate s, p = (p.x, p.y)
A point p is said to be dominated by a point q if,
p.x ≤ q.x and
P.y ≤ q.y

12. Example: 2-Dimension Maxima
Given a set of n points, P ={p1, p2, p3, … , pn}, a point p is said to be maximal if it is not dominated by any other point in the set P.

13. Example: 2-Dimension Maxima
The computer selection problem can be modeled this way
For each computer we associate (x, y) pair in such a way that
x is the speed of computer
y is the negation of price
High y value means cheap computer and low y value means expensive computer
Maximal point corresponds to fastest and cheapest computer

14. Example: 2-Dimension Maxima
Input
A set of n points i.e. P = {p1, p2, p3, …, pn} in 2 dimensional space
Output
The set of maximal points of P, the set of those points which are not dominated by the others

15. Example

16. Brute-Force Algorithm for 2-D Maxima Example

17. Analysis of 2-D Maxima
To measure the running time of brute force maxima algorithm we can
Count the number of steps of the pseudo code that are executed
Or, count the number of times an element of set P is accessed
Or, the number of comparisons that are performed

18. Running Time Analysis Depends upon the input size e.g. n
Depends upon the structure of input
For example, the breaking of inner loop in brute force algorithm depends upon the input size.

19. Analysis of 2-D Maxima Analysis of brute-force maxima algorithm
Input size is n
We are going to count the number of times an element of P is accessed

20. Analysis of 2-D Maxima (Cont…)

21. Analysis of 2-D Maxima (Cont…)
The outer loop runs n times
For each iteration of the outer loop, the inner loop runs n times
P is accessed four times in the if statement
P is accessed two times in output statement
In worst case, every point is maximal so every point is output

22. Analysis of 2-D Maxima (Cont…)

23. Analysis of 2-D Maxima (Cont…)
The worst case running time is

24. Some Important Summations

25. Some Important Summations

26. Analysis: A Harder Example