The document discusses algorithms and their complexity analysis. It defines an algorithm as a finite set of steps to solve a problem. Algorithms have inputs, outputs, definiteness, finiteness, and effectiveness. Their complexity depends on time and space requirements. Time complexity measures how long an algorithm takes to run, while space complexity measures the memory needed. Complexity is often expressed using Big O notation and algorithms can have constant, logarithmic, linear, polynomial or exponential time complexities in the worst, best or average cases.

1. ALGORITHM
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2. Introduction to Algorithm
● An algorithm is finite set of steps defining the
solution of a particular problem.
● It can be expressed in English like language,
called pseudocode, in programming language
or in the form of flowchart.
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3. Characteristics
● Input – There are zero or more values which are
externally supplied.
● Output – At least one value is produced
● Definiteness – Each step must be clear and
unambiguous
● Finiteness – If we trace the steps of an
algorithm, then for all cases, the algorithm must
terminate after a finite number of steps
● Effectiveness – Each step must be sufficiently
basic that it can in principal be carried out by a
person using only paper and pencil
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4. Algorithm Complexity
● There are basically two aspects of computer
programming.
○ Data Organization (i.e Data & structure)
○ Algorithm
● The choice of a particular algorithm depends
on the following consideration :
○ Time Complexity (Performance Requirements)
○ Space Complexity (Memory Requirements)
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5. Space Complexity
● It means the amount of memory, algorithm or
program needs to run to completion.
● Some of the reasons for studying space
complexity are
○ If a multi user system
○ Sufficient memory is available to run the
program
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6. Space Complexity (Cont.)
● The space needed by a program consists of
the following components.
○ Instruction Space : Space needed to store the
executable version of the program
○ Data Space : Space needed to store all
constants, variable values and has further
following components
■ Constants, variables, structured variables like
array and structure
■ Dynamically allocated space
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7. Space Complexity (Cont.)
● Environmental Stack Space : Space needed
to store the information needed to resume the
suspended functions.
○ Return Address
○ Values of all local variables and the values of
formal parameters in the function being
involved.
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8. Time Complexity
● It means the amount of time it needs to run to
completion
● Reasons for studying time complexity
○ Interested to know in advance that whether a
program will provide a satisfactory real time
response
○ There may be several possible solutions with
different time requirements
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9. Expressing Space and Time
Complexity
● The space and/or time complexity is usually
expressed in form of the function f(n)
○ Where n is the input size for a given instance of the
problem being solved.
● Expressing space and/ or time complexity as a
function has following reasons
○ We may be interested to predict the rate of growth
of complexity as the size of problem increases.
○ To Compare the complexities of two or more
algorithm
○ Since in modern computers, the memory is not a
sence constraint, therefore, our analysis of
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algorithms will be on the basis of time complexity.

10. Big ‘O’ Notation
● Big O is a characterization scheme that
allows to measure properties of algorithms
such time and space complexity.
● It is useful to set the prerequisites of
algorithms and to develop and design efficient
algorithms in terms of time and space
complexity.
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11. Categories of Algorithm
● Based on Big O notation, the algorithm can
be categorized as follows
○ Constant time O(1)
○ Logarithmic time O(log n)
○ Linear time O(n)
○ Polynomial time O(nk) (for k > 1)
○ Exponentail time O(kn) for k > 1
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12. Algorithm Analysis
● Different ways or algorithms to solve a
problem.
● Hence, some of the algorithm may be more
efficient than the others
● There are different types of time complexities
which can be analyzed for an algorithm
○ Best Case Time Analysis
○ Average Case Time Analysis
○ Worst Case Time Analysis
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13. Best Case Time Complexity
● It is a measure of the minimum time that the
algorithm will require for an input of size ‘n’.
● The running time of many algorithms varies
not only for the inputs of different sizes but
also for the different inputs of same size
○ Example : Sorting or Searching
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14. Worst Case Time Complexity
● The worst case time complexity of an
algorithm is a measure of the maximum time
that the algorithm will require for an input of
size ‘n’.
● Example : If ‘n’ input data items are supplied in
reverse order for any sorting algorithm, the
algorithm will require n2 operations to perform the
sort.
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15. Average Case Time Complexity
● The time that an algorithm will require to
execute a typical input data of size ‘n’ is
known as average case time complexity.
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