1. Design and Analysis of Algorithms
Text Book: Horowitz, S. Sahni, Fundamentals of Computer Algorithms
Reference Book: Introduction to algorithms
 By Thomas H. Cormen

2. Notion: Algorithms
An algorithm is a sequence of unambiguous instructions for solving a computational problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.
“computer”
problem
algorithm
input
output
Algorithm is thus a sequence of computational steps that transform the input into the output.

3. More precisely, an algorithm is a method or process to solve a problem satisfying the following properties:
Finiteness
terminates after a finite number of steps
Definiteness
Each step must be rigorously and unambiguously specified.
Input
Valid inputs must be clearly specified.
Output
can be proved to produce the correct output given a valid can be proved to produce the correct output given a valid input.
Effectiveness
Steps must be sufficiently simple and basic.
Can be carried out with pen and paper
These are also called characteristics of an algorithm

4. Examples Is the following a legitimate algorithm? i 1
While (i <= 10) do
a  i + 1
Print the value of a
End of loop
Stop

5. Four key terms:
Natural Language
Algorithm
Program : A program is the expression of an algorithm in a programming language
Psuedocode-mix of algorithm and some programming language.
Key Points:
Each step of an algorithm must be unambiguous.
The same algorithm can be represented in several different ways.
There might exists more than one algorithm for a certain problem.
Algorithms for the same problem can be based on very different ideas and can solve the problem with dramatically different speeds.

6. Two main issues related to algorithms
How to design algorithms
How to analyze algorithm efficiency

7. Analysis of Algorithms
How good is the algorithm? (Determined by the complexity)
time efficiency
space efficiency
Does there exist a better algorithm?
lower bounds
optimality

8. Importance of Analyze Algorithm
Need to recognize limitations of various algorithms for solving a problem
Need to understand relationship between problem size and running time
When is a running program not good enough?
Need to learn how to analyze an algorithm's running time without coding it
Need to learn techniques for writing more efficient code

9. What do we analyze about them?
Correctness
Does the input/output relation match algorithm requirement?
Amount of work done (aka complexity)
Basic operations to do task
Amount of space used
Memory used
Simplicity, clarity
Verification and implementation.
Optimality
Is it impossible to do better?

10. Models of Computation Reference: The design and analysis of computer algorithms by Aho Ullman

11. Three models of computation
Random Access Machines
RASP Machines(Stored program model)
Turing Machines

12. Random Access Machine (RAM)

13. Each register holds an integer Program can’t modify itself
RAM Assumptions
Each register holds an integer
Program can’t modify itself
Memory instructions involve simple arithmetic
Addition, subtraction
Multiplication, division and control states (got, if-then, etc.)

14. RASP Machine stored program model
Same as RAM but allow program to change itself as it is now stored in the memory
Same power as RAM
Example Von Neumann architecture
Let RAM use memory registers to store modifiable program of RASP

15. Turing Machine A Turing machine includes
A (conceptual) tape that extends infinitely in both directions
Holds the input to the Turing machine
Serves as memory
Is divided into cells
A unit that reads one cell of the tape at a time and writes a symbol in that cell
It is controlled by finite automaton that has finite number of states.

16. Example of a finite automaton containing two states is a light
switch.
In this example, the two states are on and off. If the state of the lightswitch is on, pushing the switch is the input that will cause a transition to the off state.
Each cell contains one symbol
Symbols must come from a finite set of symbols called the alphabet
Alphabet for a given Turing machine
Contains a special symbol b (for “blank”)
Usually contains the symbols 0 and 1
Sometimes contains additional symbols

17. Each operation involves
Writing a symbol in the cell (replacing the symbol already there)
Going into a new state (could be same state)
Moving one cell left or right

18. Each instruction says something like
if (you are in state i) and (you are reading symbol j) then
write symbol k onto the tape
go into state s
move in direction d