1. Advanced Algorithms Analysis and DesignBy
Dr. Nazir Ahmad Zafar
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

2. Lecture No 2 Mathematical Tools for Design and Analysis of Algorithms (Fundamentals of Algorithms)
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

3. Review of Lecture No 1 Introduction to Algorithms Designing Techniques
Algorithm is a Technology
Model of Computation
Even we have supercomputers, it requires algorithm
Algorithms makes difference in users and modeler
Some of the applications areas of algorithms
Management and manipulation of data
Electronic commerce
Manufacturing and other commercial settings
Shortest paths etc.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

4. A Sequence of Mathematical Tools Sets Sequences Order pairs
Today Covered
A Sequence of Mathematical Tools
Sets
Sequences
Order pairs
Cross Product
Relation
Functions
Operators over above structures
Conclusion
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

5. Set A set is well defined collection of objects, which are unordered
distinct
same type
with common properties
Notation:
Elements of set are listed between a pair of curly braces S1 = {R, R, R, B, G} = {R, B, G} = {B, G, R}
Empty Set
S3 = { } = , has not elements, called empty set
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

6. Representation of Sets
Three ways to represent a set
Descriptive Form
Tabular form
Set Builder Form (Set Comprehension)
Example
S = set of all prime numbers
{2, 3, 5, . . .}
Set Builder Form
{x : N| ( i  {2, 3, . . ., x-1}   (i / x))  x}
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

7. Set Comprehension Some More Examples
{x : s | p  x} = {x : s | p} = all x in s that satisfy p
{x : Z | x2 = x  x} = {0, 1}
{x : N | x  0 mod 2  x} = {0, 2, 4, }
{x : N | x  1 mod 2  x} = {1, 3, 5, }
{x : Z | x ≥ 0  x ≤ 6  x} = {0, 1, 2, 3, 4, 5, 6}
{x : Z | x ≥ 0  x ≤ 6  x2} = {0, 1, 4, , 25, 36}
{x : N | x  1 mod 2  x3} = {1, 27, 125, }
Dr. Nazir A. Zafar Formal Methods

8. All collections are not sets
The prime numbers
Primes == {2, 3, 5, 7, }
The four oceans of the world
Oceans == {Atlantic, Arctic, Indian, Pacific}
The passwords that may be generated using eight lower-case letters, when repetition is allowed
Hard working students in MSCS class session at Virtual University
Intelligent students in your class
Kind teachers at VU
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

9. Operators Over Sets Membership Operator
If an element e is a member of set S then it is denoted as e  S and read e is in S. Let S is sub-collection of X
X = a set
S  X
Now
 : X x P X Bool
 (x, S) = = 1 if x is in S
0 if x is not in S
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

10. Operators Over Sets Subset:
If each element of A is also in B, then A is said to be a subset of B, A  B and B is superset of A, B  A.
Let X = a universal set, A  X, B  X, Now
 : X x X  Bool
 (A, B) = = 1, if  x : X, x  A  x  B
0 if  x : X, x  A  x  B
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

11. Operators Over Sets Intersection  : X x X  X
 (A, B) = = {x : X | x  A and x  B  x}
Union
 : X x X  X
 (A, B) = = {x : X | x  A or x  B  x}
Set Difference
\ : X x X  X
\ (A, B) = = {x : X | x  A but x  B  x}
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

12. Cardinality and Power Set of a given Set
A set, S, is finite if there is an integer n such that the elements of S can be placed in a one-to-one correspondence with {1, 2, 3, …, n}, and we say that cardinality is n. We write as: |S| = n
Power Set
How many distinct subsets does a finite set on n elements have? There are 2n subsets.
How many distinct subsets of cardinality k does a finite set of n elements have?
There are
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

13. Partitioning of a set
A partition of a set A is a set of non-empty subsets of A such that every element x in A exactly belong to one of these subsets.
Equivalently, set P of subsets of A, is a partition of A
If no element of P is empty
The union of the elements of P is equal to A. We say the elements of P cover A.
The intersection of any two elements of P is empty. That means the elements of P are pair wise disjoints
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

14. Mathematical Way of Defining Partitioning
A partition P of a set A is a collection {A1, A2, . . ., An} such that following are satisfied
 Ai  P, Ai  ,
Ai  Aj = ,  i, j  {1, 2, . . ., n} and i  j
A = A1  A2  . . . An
Example: Partitioning of Z Modulo 4
{x : Z | x  0 mod 4} = { , -4, 0, 4, 8, } = [0]
{x : Z | x  1 mod 4} = { , -3, 1, 5, 9, } = [1]
{x : Z | x  2 mod 4} = { , -2, 2, 6, 10, } = [ 2]
{x : Z | x  3 mod 4} = { , -1, 3, 7, 11, } = [3]
{x : Z | x  4 mod 4} = {. . .,-8, -4, 0, 4, 8, } = [4]
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

15. Sequence
It is sometimes necessary to record the order in which objects are arranged, e.g.,
Data may be indexed by an ordered collection of keys
Messages may be stored in order of arrival
Tasks may be performed in order of importance.
Names can be sorted in order of alphabets etc.
Definition
A group of elements in a specified order is called a sequence.
A sequence can have repeated elements.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

16. Sequence
Notation: Sequence is defined by listing elements in order, enclosed in parentheses. e.g.
S = (a, b, c), T = (b, c, a), U = (a, a, b, c)
Sequence is a set
S = {(1, a), (2, b), (3, c)} = {(3, c), (2, b), (1, a)}
Permutation: If all elements of a finite sequence are distinct, that sequence is said to be a permutation of the finite set consisting of the same elements.
No. of Permutations: If a set has n elements, then there are n! distinct permutations over it.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

17. Operators over Sequences
Operators are for manipulations
Concatenation
(a, b, c )  ( d, a ) = ( a, b, c, d, a )
Other operators
Extraction of information: ( a, b, c, d, a )(2) = b
Filter Operator: {a, d}  ( a, b, c, d, a ) = (a, d, a)
Note:
We can think how resulting theory of sequences falls within our existing theory of sets
And how operators in set theory can be used in case of sequences
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

18. Tuples and Cross Product
A tuple is a finite sequence.
Ordered pair (x, y), triple (x, y, z), quintuple
A k-tuple is a tuple of k elements.
Construction to ordered pairs
The cross product of two sets, say A and B, is A  B = {(x, y) | x  A, y  B}
| A  B | = |A| |B|
Some times, A and B are of same set, e.g., Z  Z, where Z denotes set of Integers
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

19. Binary Relations Definition: If X and Y are two non-empty sets, then
X  Y = {(x, y) | x X and y  Y}
Example: If X = {a, b}, Y = {0,1} Then
X  Y = {(a, 0), (a, 1), (b, 0), (b, 1)}
Definition: A subset of X  Y is a relation over X  Y
Example: Compute all relations over X  Y, where
X = {a, b}, Y = {0,1}
R1 = , R2 = {(a, 0)}, R3 = {(a, 1)}
R4 = {(b, 0)}, R5 = {(b, 1)}, R6 = {(a, 0), (b, 0)}, . . .
There will be 24 = 16 number of relations
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

20. Binary Relations Example: If X = {a, b}, Y = {0,1} Then
Definition: XY == (X  Y)
Example: If X = {a, b}, Y = {0,1} Then
X  Y = {(a, 0), (a, 1), (b, 0), (b, 1)} and
X  Y = {R1, R2, R3, R4, R5, R6, R7, R8, R9, R10, R11, R12, R13, R14, R15, R16}
Lemma 1: if #X = m, #Y = n then
# (X  Y) = 2m  n
Algorithm: All binary relations, cardinality of it
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

21. Equivalence Relation A relation R  X  X, is Reflexive:
(x, x)  R,  x  X
Symmetric:
(x, y)  R  (y, x)  R,  x, y  X
Transitive:
(x, y)  R  (y, z)  R  (x, z)  R
Equivalence:
If reflexive, symmetric and transitive
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

22. Applications : Order Pairs in Terms of String
Definition
A relation R over X, Y, Z is some subset of X x Y x Z and so on
Example 1
If  = {0, 1}, then construct set of all strings of length 2 and 3
Set of length 2 =  x  = {0,1} x {0,1} = {(0,0), (0,1), (1,0), (1,1)} = {00, 01, 10, 11}
Set of length 3 =  x  x  = {0, 1} x {0, 1} x {0,1}
= {(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)}
= {000, 010, 100, 110, 001, 011, 101, 111}
Example 2
If  = {0, 1}, then construct set of all strings of length ≤ 3
Construction = {}     x    x  x 
Similarly we can construct collection of all sets of length n
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

23. Partial Function
Definition: a partial function is a relation that maps each element of X to at most one element of Y. XY denotes the set of all partial functions.
X  Y == { f | f  X  Y and
(x, y1)  f  (x, y2)  f  y1 = y2 ,  x  X, y1, y2  Y} b c d 1 2 3 X Y f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

24. Function
Definition: if each element of X is related to unique element of Y then partial function is a total function denoted by X  Y.
X  Y == { f | f  XY and dom f = X} b c d 1 2 3 X Y f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

25. Is Algorithm a Function?
Not good algorithms
Input1
Input2
Input3
. . .
Inputn
output1
output2
output3
outputn X Y f
Input1
Input2
. . .
Inputn
output1
output2
outputn X Y f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

26. Conclusion We started with sets and built other structures e.g.
Sequences
Relations
Functions, etc.
We discussed operators over above structures
All these tools are fundaments of mathematics
Sets play key role in algorithms design and analysis
Finally proving correctness of algorithms, we required logic.
Our next lecture will be on proving techniques
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design