The document discusses various mathematical functions essential in algorithm analysis, including floor, ceiling, remainder, integer, and absolute value functions, along with summation and factorial functions. It also covers algorithmic notations, control structures, and the complexity of algorithms, explaining the efficiency measurements through time and space, as well as asymptotic notations like big O, omega, and theta notations. Additionally, it details the types of logic including sequence, selection, and iteration logic in algorithms.

1. Preliminaries Chapter-2
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

2. Mathematical Notations and Functions
• The following mathematical functions appear very
often in the analysis of algorithms and in computer
science.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

3. Floor & Ceiling Functions
• Let x be any real number.
• Then x lies between two integers called the floor and the ceiling of x.
• Լ x┘ called the floor of x, denotes the greatest integer that does not exceed x.
• Floor(3.14)=3
• Floor(-8.5)=-9
• ΓxꞀ called the ceiling of x, denotes the least integer that is not less than x.
• Example
• Ceiling(3.14)=4
• Ceiling(-8.5)=-8
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

4. Remainder Function; Modular Arithmetic
• Let k be any integer and let M be a positive integer, then
• K Mod M
• Will denote the integer remainder when k is divided by M.
• More exactly, k Mod M is the unique integer r such that
• k = Mq + r where 0 ≤ r < M
• Example:
• 25 Mod 7 = 4 (25 = 7*3 + 4)
• 35 Mod 11 = 2 (35 = 11*3 + 2)
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

5. Integer Function
• Let x be any real number.
• The integer value of x, written as INT(x) converts x into an
integer by deleting the fractional part of the number.
• INT(3.14)=3
• INT(-8.5)=-8
• OBSERVE:
• INT(x) = Floor(x)
• Or
• INT(x) = Ceiling(x)
o according to whether x is positive or negative.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

6. Absolute Value Function
• The absolute value of the real number x, written as ABS(x) or |x|
is defined as the greater of x or –x.
• ABS(0) = 0
• |-15| = 15
• |4.44| = 4.44
• Note that |x| = |-x|
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

7. Summation
• The summation symbol is ∑ (sigma).
• Consider the sequence a1,a2,a3,… then the sums
• a1+a2+a3…+an
and
• am+am+1+…+an
• Will be denoted respectively by
• σ𝒋=𝟏
𝒏
𝒂𝒋
• σ𝒋=𝒎
𝒏
𝒂𝒋
• Here j is called as the dummy index or dummy variable.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

8. Factorial Function
• The product of the positive integers from 1 to n, inclusive, is
denoted by n!
• That is
• n! = 1 x 2 x . . . X (n-2) x (n-1) x n
• Example:
• 4 ! = 1 x 2 x 3 x 4 = 24
• 5 ! = 5 x 4! = 5 x 24 = 120
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

9. Permutations
❖A permutation of a set of n elements is an arrangement of
the elements in a given order.
❖For example, the permutations of the set consisting of the
elements a, b and c are as follows:
❖Abc
❖Acb
❖Bac
❖Bca
❖Cab
❖Cba
❖There are n! permutations of a set of n elements.M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

10. Exponents & Logarithms:
• Recall for any integer m,
• am= a . a. . . . a (m times)
• a0 = 1
• a-m =
1
𝑎 𝑚
• Exponents are extended to include all rational numbers by
defining, for any rational number m/n,
• am/n = 𝑛
𝑎 𝑚 = ( 𝑛
𝑎 )m
• Exponents are extended to include all real numbers by
defining for any real number x,
• ax = lim
𝑟→𝑥
𝑎 𝑟
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

11. • Logarithms are related to exponents as follows.
• Let b be a positive number the logarithm of any positive
number x to the base b written as
• log 𝑏 𝑥
• Represents the exponent to which b must be raised to
obtain x. that is
• Y = log 𝑏 𝑥
and
• by = x
• The logarithm of 0 and the logarithm of negative number
are NOT DEFINED.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

12. Algorithmic Notations.
1. Identifying Number
✓Each algorithm is assigned an identifying number.
✓Example Algorithm 4.3 refers to the 3rd algorithm in chapter 4.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

13. Algorithmic Notations.
2. Steps, Control, Exit
✓Each steps of the algorithm are executed one after the other, beginning
with step-1.
✓Control may be transferred to step n of the algorithm by the statement
“Go to Step n”.
✓These “goto” statements may be eliminated by using certain control
structures.
✓If several statements appear in the same step like SET K:=1, LOC:=1,
then they are executed from left to right.
✓The algorithm is completed with the statement EXIT.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

14. Algorithmic Notations.
3. Comments
✓Each step may contain a comment in brackets which indicates the
main purpose of the step.
✓The comment will usually appear at the beginning or at the end of
the step.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

15. Algorithmic Notations.
4. Variable Names
✓Variable names will use capital letters (Eg. MAX, DATA)
✓Single letter names of variables used as counters or subscripts will
also be capitalized in the algorithms.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

16. Algorithmic Notations.
5. Assignment statement
✓Assignment statements will use the dots-equal notation :=
✓Example MAX := DATA[1]
✓Will assign the value in DATA[1] to MAX.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

17. Algorithmic Notations.
6. Input and output
✓Data may be input and assigned to variables by means of Read
statement with the following form.
✓Read: Variable_names
✓Message placed in quotation marks and data in variables may be
output by means of a Write or Print statement with the following
form.
✓Write: Message and/or variable_names
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

18. Algorithmic Notations.
7. Procedures
✓This term is used for an independent algorithmic module which
solves a particular problem.
✓The use of the word Procedure or Module denotes it.
✓It is used to describe a certain type of sub-algorithm.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

19. 2.4 CONTROL STRUCTURES
Three types of logic or flow-of-control are used:
Sequence Logic (or) Sequential Flow
Selection Logic (or) Conditional Flow
Iteration Logic (or) Repratitive Flow
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

20. Sequence Logic
• Unless instructions given to the
contrary, the modules are executed in
the obvious sequence.
• The sequence may be preseted
explicitly, by means of numbered
steps, or implicitly, by the order in
which the modules are written.
• Most processing will generally follow
this elementary flow pattern.
Module A
Module B
Module CM.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

21. Selection Logic
• Employs a number of conditions which lead to a selection of one out of
several alternative modules.
• The structures which implement this logic are called conditional structures
or IF structures.
• Most processing will generally follow this elementary flow pattern.
• These condition structures fall into three categories.
• Single alternative
• Double alternative
• Multiple alternatives
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

22. Selection Logic – Single Alternative.
• If the given condition holds, then Module-A,
which may consist of one or more statements,
is executed;
• Otherwise Module-A is skipped and control
transfers to the next step of the algorithm.
Structure
If condition then:
[ Module-A]
[End of If Structure]
Condition?
Module - A
Yes
No
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

23. Selection Logic – Double Alternative.
• If the given condition holds, then Module-A,
gets executed;
• Otherwise Module-B is is executed.
Structure
If condition then:
[ Module-A]
Else:
[ Module-B]
[End of If Structure]
Condition?
Module - A
Yes
No
Module - B
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

24. Selection Logic – Multiple Alternatives.
• The logic of this structure allows only one of the modules to be executed.
• Either the modulewhich follows the first condition which holds is executed, or the
module which follows the final Else statement is executed.
Structure
If condition-1 then:
[ Module-A1]
Else If condition-2 then :
[ Module-A2]
…
Else:
[ Module-B]
[End of If Structure]
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

25. Iteration Logic (Repetitive Flow)
• Refers to either of 2 types of structures involving loops.
• Each type begins with a Repeat statement and is followed by a
module, called the body of the loop.
• 2 Types:
• Repeat-for loop
• Repeat-while loop.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

26. Iteration Logic – Repeat-for Loop
• Uses an index variable to control the loop (such
as I,j or k).
• Format:
Repeat for K = R to S by T:
[Module]
[End of loop]
• Here R is called the initial value, S the end
value or test value and T the increment.
• The body of the loop is executed first with K=R,
then with K=R+T, then with K=R+2T and so on.
• The cycling ends when K>s.
Is K > S ?
Module
[ body of loop]
No
Yes
K = K + T
K = R
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

27. Iteration Logic – Repeat-while Loop
• Uses a ocndition to control the loop.
• Format:
Repeat while condition:
[Module]
[End of loop]
• The cycling continues until the condition si
false.
• There must be a statement before the structure
that initializes the condition controlling the
loop.
• There must be a statement in the body of the
loop that changes the condition.
Yes
condition?
Module
[ body of loop]
No
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

28. 2.5 COMPLEXITY OF ALGORITHMS
• Inorder to compare algorithms, we must have some criteria to
measure the efficiency of out algorithm.
• Suppose
• M is an algorithm, and
• n is he size of input data
• The time & space used by the algorithm M are the two main measures for
the efficiency of M.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

29. 2.5 COMPLEXITY OF ALGORITHMS
• The complexity of an algorithm M is the function f(x) which gives
the running time and/or storage space requirement of the
algorithm in terms of the size n of the input data.
• The storage space required by an algorithm is simply a multiple of
the data size n.
• So mostly the term “complexity” refer to the running time of the
algorithm.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

30. 2.5 COMPLEXITY OF ALGORITHMS
• Finding the complexity function f(x) deals with 3 cases in
complexity theory.
• Worst case: the maximum value of f(n) for any possible input.
• Average case: the expected value of f(n)
• Best case: the minimum value of f(n).
Example – Searching an element in Linear search fashion:
• Worst case: The ITEM is at the LAST position or NOT PRESENT in the list.
• Average case: The ITEM appears in the list.
• Best case: The ITEM appears at the FIRST position in the list.M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

31. Rate of Growth; Big O Notation
• Suppose M is an algorithm with the size of input n, the complexity
f(n) of M increases as n increases.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

32. Rate of Growth; Big O Notation
• The rate of increase of f(n) is done by comparing f(n) with some
standard function such as log2 n, n log2 n, n2 , n3 , 2n
• The rate of growth of these standard functions are indicated in
the below table
G(n)
Log n N N log n n2 n3 2n
N
5 3 5 15 25 125 32
10 4 10 40 100 103 103
3100 7 100 700 104 106 1030
1000 10 103 104 106 109 10300M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

33. Rate of Growth; Big O Notation
• Supposef(n) and g(n) are functions with property that f(n) is
bounded by some multiple of g(n) for almost all n,
• Then we may write as
• F(n) = O(g(n))
• This is called as the “big O” notation.
• Example complexity functions of well-known searching and sorting
algorithms:
• Linear search : O(n)
• Binary search : O(log n)
• Bubble sort : O(n2)
• Merge sort : O(n log n)M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

34. 2.6 OTHER ASYMPTOTIC NOTATIONS FOR COMPLEXITY
OF ALGORITHMS Ω , 𝜃 , 𝜊
Omega Notation (Ω)
• The omega notation is used when the function g(n) defines a lower
bound for the function f(n).
• F(n) = Ω (g(n) ), iff there exists a positive integer n0 and a positive
integer M such that |f(n)| >= M|g(n)|, for all n>= n0
• For f(n)=18n+9, f(n)>18n for all n, hence f(n)= Ω(n)
• For f(n)=90n2+18n+6, f(n)>90n2 for n2=0 and therefore f(n)= Ω(n2)
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

35. 2.6 OTHER ASYMPTOTIC NOTATIONS FOR COMPLEXITY
OF ALGORITHMS Ω , 𝜃 , 𝜊
Theta Notation (𝜃)
• The theta notation is used when the function f(n) is bounded both
from the above and below by the function g(n).
• It implies that the function g(n) is both an upper bound and a
lower bound for the function f(n) for all values of n, n>= n0.
• That is f(n) is such that f(n) = O(g(n)) and f(n) = Ω(g(n))
• F(n) = 𝜃 (g(n) ), iff there exists two positive constants c1 and c2
and a positive integer n0 such that c1|g(n)| <= c2 |g(n)|, for all
n>= n0.
• For f(n)=18n+9, f(n)>18n for all n, hence f(n)= Ω(n)
• For f(n)=90n2+18n+6, f(n)>90n2 for n2=0 and therefore f(n)= Ω(n2)
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

36. 2.6 OTHER ASYMPTOTIC NOTATIONS FOR COMPLEXITY
OF ALGORITHMS Ω , 𝜃 , 𝜊
Little Oh Notation (o)
• F(n) = 𝑜(g(n) ), iff f(n)=O(g(n)) and f(n) ≠ Ω(g(n)).
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

37. 2.7 SUBALGORITHMS
• A subalgorithm is a complete and independently defined
algorithmic module which is used by some main module or by
some other subalgorithm.
• A subalgorithm
• receives values called arguments, from an originating algorithm,
• performs calculations, and then
• sends back the result to the calling algorithm.
• The subalgorithm is defined independently so that it may be
called by many different algorithms or called at different times in
the same algorithm.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

38. 2.7 SUBALGORITHMS
• The subalgorithm will have a RETURN statement instead of EXIT
statement. It implies that the control is transferred back to the
calling program when execution of the subalgorithm is completed.
• Subalgorithms fall into 2 basic categories:
• Function subalgorithms – return only a single value to the calling algorithm.
• Procedure subalgorithms – can send back more than one value to the calling
algorithm.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

39. 2.8 VARIABLES, DATA TYPES
• Each variable in any of our algorithms has a data type which
determines the code that is used for storing its value. Four such
data types are:
• Character
• Real (floating point)
• Integer (fixed point)
• Logical
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

40. Local & Global Variables
• The organization of a computer program into a main program and
various subprograms has led to the notion of local and global
variables.
• Each program module contains its own lis of variables called local
variables, which can be accessed only by the particular module.
• Variables that can be accessed by all program modules are called
global variables.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.

41. Local & Global Variables
• There are 2 basic ways for modules to communicate with each
other:
• Directly, by means of well-defined parameters.
• Indirectly, by means of non-local and global variables.
M.Priyavani,MCA,DCHN,M.Phil, V.V.V.College for Women, Virudhunagar.