1. Algorithms
Prof. Dr. Péter Iványi

2. One of the oldest algorithms
bc. IV century, Alexandria, Euclides
Largest common denominator of two integers
Lets assume a and b positive integer number and the largest common denominator is ( a , b )

3. Largest common denominator
If a = b * q + r , then ( a , b ) = ( b , r ) , thus the problem is reduced to a problem with smaller numbers. If we continue the procedure, then the last remainder that is different from zero is the largest common denominator

4. Largest common denominator
( 360, 225 ) = ?
According to the Eucledian algorithm:
       a    b     r    360 = 225*    225 = 135*1 +  90    135 =  90*1 +  45     90 =  45*2 +   0        
Therefore ( 360, 225 ) = 45.

5. Programming Two basic concepts:
Description of relations between quantities and information
x+y
Evaluation of these relations, where values are substituted into the expression
x = 2
y = 3
2+3 = 5

6. Operations We can specify simple operations on every data For example:
Numbers: addition, subtraction, multiplication, ...
Characters: Concatenate, sort,...
A programmer creates a program such simple operations

7. Information Information is describd as data
There are different types of data:
Simple, (atomic, not divisible), like numbers
Complex, like number series
Although data represents the information, but the interpretation is up to us
For example can represent
temperature
time
distance

8. Programming, practical approach
To divide up the problem to simple operations (steps) that a computer can understand
The program: It is a sequence of operations that can be understood by a computer and that solves a problem.

9. Elements of a program Input Output Operations, instructions
(variables)
Input
Operations
Output

10. Input and output Input Output Keyboard Mouse File Serial port, etc
Screen

11. Variables We store data in variables Name Value Assigning
Denotes a storage place
Value
The stored data
Assigning
We put a data into a storage place

12. Data, simple What kind of data can we store? Numbers (bináris szám)
Integer
Real
Complex, etb
Characters
Binary number is converted to character
The same data can be stored in different ways
1, ‘1’, “1”

13. Derived, complex data Character series, string, Vectors Matrices Lists
Etc.
Depends on the programming language!

14. Operations Manipulating data Arithmetic: Relational: Logical
+, -, *, /
Relational:
>, <, =, <=, >=
Logical
NOT, AND, OR

15. Order of operations a=6+12*5 Evaluation order,
How to, in what order to execute the operations?
Generally from left to right
Precedence
Priority of operations
Brackets
a=(6+12)*5
a=6+(12*5)

16. Program, summarised Program, consists of several programs
Properties of algorithms:
Can be executed (consits of simple steps)
Every step is well defined
After a finite number of steps it stops
It is valid for a defined set of input
It produces the appropriate output
It solves a specific problem

17. Algorithms, design of programs

18. Programming methodology
Until 1960 monolithic programming
Properties:
One programmer – one program
There is no structure in the program

19. Programming methodology
The most important criteria of a good program
It has a visibly good structure
It is well documented
It can be „proven” that it does what it should

20. Modular programming The basis of design: decomposition
The complicated problem is divided in pieces
We create a solution for the pieces
We assemble the pieces

21. Guidelines Divide and conquer Divide the problem to modules
Modules do not „get into” each other’s data
Modules operate independently
Easier to detec mistakes
Easier to overview the task
Easier to modify

22. Guidelines Hiding data Delaying decisions
Modules work only with their own data
Delaying decisions
Decision is only made when it is necessary
If there is not enough information, delay the decision
An early decision may have to be modified later
Once a decision has been made, declare it
Write it down, or record it
So later no contradictory decision is made

23. Modular programming Top-Down decomposition Bottom-Up composition
The task is divided into subtasks, which are also divided into further subtasks, until their size is small enough that they can be handler.
Bottom-Up composition
We use already existing building blocks
We do not know how they will connect together, which will connect to which other one
These building blocks must already exist

24. Modular programming Advantages: Subprograms can be easily overviewed
Easy to create them
Easy to test them
Several modules can be developed at the same time (parallel development)
Easy to correct mistakes
Modules can be standardised
It is possible to create module libraries
Reusable modules

25. Structured programming
Go To Statement Considered Harmful.
DOI: /
Dijkstra:
Based on the Top-Down dekomposition
The original problem is divided into subproblems. This is an abstract program which works on an abstract computer. It can be proven that it works according to specification.
Refinement, which reduces the abstraction, e.g. a subproblem is described in more detail, which runs on another abstract machine. Still possible to prove that is satisfies the specification.
Further refinement, until we reach the level of a real computer, with its instructions.
2015 Szept 23

26. Structured programming
Outcome:
The resulting is provably correct.
Problem:
The required amount of work is large.
Solution:
We miss certain steps, and therefore an approximately correct program is created, which can be checked by tests

27. Boehm, Jackopini 1964
Every algorithm can be described by three control structure
Selection
Iteration
Sequence

28. Mills 1968
Proved the theory of Boehm and Jackopini with certain conditions:
The basic structure of a program is a sequence
Every sequence has one entry and one exit point. An exit point of a module is an entry point of another module.
Every sequence can structured internally as required.

29. Graphic representation of algorithms
Pseudo code
Almost like sentences
Flowchart
Structogram
Nassi–Shneiderman diagram
Starting point is one rectangle, which can be divided.
...

30. Flowchart, basic elements
Start:
Input data
Output data
Action, statement

31. Flowchart, basic elements
Conditional action, branching
Label
End:

32. Structogram
Module name, beginning or algorithm
Input data
Output data

33. Structogram Action, statement Conditional action, statement, branching
Iteration

34. Program execution The computer executes one step after another one.
Control structure: deviation from the sequential order

35. Control structure Jump Conditional execution
Multiple conditional execution
Count controlled cycle, iteration, loop
Pre-testing cycle, iteration, loop
Post-testing cycle, iteration, loop
(Subroutines)

36. Jump The program continues at a new given point, not at the next step.
Usually results in structure that is difficult to understand
Not used in this semester!!!

37. Subprogram Repeated task Subtasks to separate Names: Subroutine
Procedure
Function

38. Control structure, flowchart
Sequence
Selection with else
Selection

39. Control structure, flowchart
Multiple selection

40. Control structure, flowchart
Pretesting iteration
Posttesting iteration

41. Control structure, flowchart
Count controlled iteration

42. Control structure, structogram
Sequence
Selection, branching
Selection with else

43. Control structure, structogram
Multi branching
Pretesting iteration
Posttesting iteration
Count controlled iteration

44. Pseudo code
Function Something BEGIN actions END IF cond THEN action1 ELSE action2 END IF

45. Pseudo code
WHILE cond DO actions END WHILE DO WHILE cond

46. Pseudo code
FOR var=a…b DO actions END FOR

47. Sequence

48. Selection, branching

49. Selection, branching

50. Selection, branching

51. Pre-testing iteration
We do pushup as long as we can.

52. Post-testing iteration
The previous example.

53. Count controlled iteration
Lets do 10 push ups.

54. Making tea

55. The problem
A farmer (Joe) would like to buy cattle, goat, chicken, in total 100 pieces of animal for 100 dollar. The price of 1 cattle is 3,5 dollar, the price of 1 goat is 1,33 dollar and the price of 1 chicken is 0,5 dollar. How many animals can Joe buy?

56. Variables and conditions
s : number of cattles
k : number of goats
j : number of chicken
Conditions:
1 <= s <= 100
1 <= k <= 100
1 <= j <= 100
Number: s + k + j = 100
Price: 7/2*s + 4/3*k + 1/2*j = 100

57. 1. Algorithm

58. 1. Algorithm, analysis Number of executions
Can we create a better algorithm?
If Joe buys only one type of animal, then:
cattle: 7/2 * s = 100 thus ‘s’ can be: 28
goat: 4/3 * k = 100 thus k = 75
chicken: 1/2 * j = 100 thus j = 200
But Joe can buy only 100 animals, so j = 100

59. 2. Algorithm

60. 2. Algorithm, analysis 28 * 75 * 100 = 210 000
Can we create a better algorithm?
If we already know the number of animals for two types of animal, then the third can be calculated
j = 100 – s – k
Consequence:
The internal iteration can be removed
The value of j can be determined, however j cannot be negative, therefore the condition will change a little bit

61. 3. Algorithm

62. 3. Algorithm, analysis 28 * 75 = 2100 O(n2)
Can we create a better algorithm? If
j = 100 – s – k
then substitute this into
7/2*s + 4/3*k + 1/2*j = 100
we get a new result
k = /5 * s
Condition will also change: ‘k’ must be positive, integer

63. 4. Algorithm

64. 4. Algorithm, analysis
28 steps to be execute!!! O(n)
Can we create a better algorithm?
We know that ‘k’ must be positive and integer, which is only possible if ‘s’ is divisible by 5:
s = 5, 10, 15, 20, 25
However if:
s = 25 then k = -30 AND
s = 20 then k = -12
Therefore these number are not good for our purpose

65. 5. Algorithm
The problem can be solved in three steps
O(1)

66. Programs S T R U C T O R I Z E R: Flowgorithm:
Flowgorithm: