1. Theory of Computer Science - Algorithms

2. Theory of Computer Science - Algorithms
Algorithm is a sequence of elementary operations that give solution to any problem from a specific class of problems in a finite number of steps.
elementary operations - one command precisely defined
all elementary operations should be precisely defined:
x<y precise definition
x<<y without additional comment it is not precise definition
any problem from a specific class of problems - not a specific, one problem but specific class of problems
finite numbers of steps - algorithm must end and give a result

3. Theory of Computer Science - Algorithms
Features of a good algorithms
finiteness - executed algorithm must have the end
specificness - all operations and their order should be precisely defined
efficiency - algorithm must give a result in a shortest way
generality - algorithm must give solution to any problem from a specific class of problems
correctness - must give a correct solution

4. Theory of Computer Science - Algorithms
Basic way in creating algorithms:
problem definition
problem analysis
preparation of rules
create concise procedure for solving problem

5. Theory of Computer Science - Algorithms
Ways of describing algorithms:
Recursive - mathematical function
Natural language
Flow chart – block diagram
Formal languages

6. Theory of Computer Science - Algorithms
Recursive function - example: factorial n
Problem is defined by itself:
n! = n * (n-1)! if n = 1 then n!=1 and 0!=1
4!=4 * (4-1)! = 4 * 3!
3!= 3 * 2!
2!= 2 * 1!= 2 * 1

7. Theory of Computer Science - Algorithms
Ways of describing algorithms:
Recursive - mathematical function
n! = n * (n-1)!
advantage: high precision
disadvantage: can describe only few kinds of problems

8. Theory of Computer Science - Algorithms
Natural language - example
Give an algorithm of finding the value:
y = Max{ xi }, where 1 ≤ i ≤ n
1. i ← 1, go to 2
2. y ← xi, go to 3
3. whether i = n ? If yes – the end, if no – go to 4
4. i ← i + 1, go to 5
5. whether xi > y ? If yes – go to 2, if no– go to 3

9. Theory of Computer Science - Algorithms
Natural language – example: y = Max{ xi }, where 1 ≤ i ≤ n
sequence x:
quantity of x elements: n = 4
index of processing element: i=?
present maximum: y=?
1. i ← 1, go to 2
2. y ← xi, go to 3
3. whether i = n ? If yes – the end, if no – go to 4
4. i ← i + 1, go to 5
5. whether xi > y ? If yes – go to 2, if no– go to 3

10. Theory of Computer Science - Algorithms
Ways of describing algorithms:
Natural language
advantage: easy to understand, simplicity, no need to use special conventions, wide vocabulary
disadvantage: lack of precision, possibility of misunderstanding

11. Theory of Computer Science - Algorithms
Flow chart -block diagram
computational transaction, process
decision - testing condition
begining and the end of algorithm
input/output operations
connecting element, direction

12. Theory of Computer Science - Algorithms
Flow chart– example: y = Max{ xi }, where 1 ≤ i ≤ n

13. Theory of Computer Science - Algorithms
Ways of describing algorithms:
Flow chart formal way of describing algorithm
advantage: precise, can be used at every level of software development, flexible
disadvantage: technical difficulty of representing large algorithms

14. Theory of Computer Science - Algorithms
Formal languages – example: y = Max{ xi }, where 1 ≤ i ≤ n
The same algorithm presenting with Pascal
function Max( x : array of integer ) : integer;
var
i, y : integer;
begin
y := x[ 1 ];
for i := 2 to n do
if x[ i ] > y then
y := x[ i ];
Max := y;
end;

15. Theory of Computer Science - Algorithms
Formal languages – example: y = Max{ xi }, where 1 ≤ i ≤ n
The same algorithm presenting with C
int Max( int *x ) {
int i;
int y = x[0];
for( i = 1; i < n; i++ )
if( x[i] > y )
y = x[i];
return y; }

16. Theory of Computer Science - Algorithms
Ways of describing algorithms:
Formal languages – programming languages
advantage: precision, executive
disadvantage: you have to know the language – difficulty of learning the language

17. Theory of Computer Science - Algorithms
Sequence algorithm
The previous algorithm was a sequence algorithm.
instructions are executed one after one
n + 1 – amount of algorithm’s operations
Oi – operation number i
t( x ) – moment of time when is executed operation x
This is always true that: moment of time, when is the end of previous operation, is in the some time or earlier, then the beginning of next operation.

18. Theory of Computer Science - Algorithms
Parallel algorithm
At least 2 operations can be executed in the some time
n + 1 – amount of algorithm’s operations
Oi – operation number i
t( x ) – moment of time when is executed operation x
There is moment of time, when the end of previous operation, is later, then the beginning of next operation.

19. Theory of Computer Science - Algorithms
Way of comparing algorithms:
Time complexity How much time is required to execute algorithm (usually we don’t measure time but steps)
Space/memory complexity How much memory space is required, to execute the algorithm

20. Theory of Computer Science - Algorithms
Conclusions
In most cases the increase in space complexity, causes the decrease in time complexity and vice versa.
Processed data, have implication on execution time and needed memory.
Pessimistic and average efficiency, have implication on algorithm’s application.
In parallel algorithms, amount of processors, have implication on time execution. The more processors, the shorter time. But there is limited amount of processors, for each algorithm, where if we use more processors above this limit, it doesn’t increase the time.