02 Introduction to Data Structures & Algorithms.pptx

INTRODUCTION TO DATA
STRUCTURES &
ALGORITHMS

OUTLINE
• Introduction
• Data structures
• Computational model
• Algorithms
• Time complexity
• Asymptotic analysis
• Recursion
• Divide –and- conquer strategy
• Computational problem

OBJECTIVES
• Understand computational models
• Understand asymptotic notations
• Write recurrence relation
• Understand recursion and how to trace the recursive function
• Discuss strategies for solving computational problems

DATA STRUCTURES
• Studies the way of organizing data in a computer so that it
can be used effectively
• Organizing
• Manipulation
• Effective/ efficient manner
• Types: Linear and Non Linear
• Examples: Stacks, Queues,Trees, Dictionaries, Graphs

COMPUTATIONAL MODEL
• A computational model is a mathematical model in computational science
• It requires extensive computational resources to study the behavior of a
complex system by computer simulation
• It is using mathematical model (representations) to make better sense of
data or complex systems
• Programming language
• Computer architecture

ALGORITHMS
• It a step-by-step procedure for solving a computational problem
• They are important in studying complex systems design (computational
model)
• They can be studied in a language and machine independent way
• Techniques to compare algorithms efficiency without implementing them:
• RAM model of computation
• Asymptotic analysis of worst-case complexity

RAM MODEL
• Random Access Machine suggests that each simple operation (+, -,/,*,=, if…) takes
exactly 1 sept
• Each memory access takes 1 step
• Loops and methods aren’t simple operations rather they are dependent on the size of
data and content of method
• The run time of an algorithm is measured by counting the number of steps
• The RAM model: a fixed program, unbounded memory, read only input, write only
output, memory holds arbitrary integer.

SPACE COMPLEXITY
• The amount of memory required by an algorithm to run to completion
• Fixed part:The size required to store certain data/variables, that is
independent of the size of the problem: Such as int a (2 bytes, float b (4 bytes)
etc
• Variable part: Space needed by variables, whose size is dependent on the size of
the problem: Dynamic array a[ ].

RUNNING TIME OF ALGORITHMS
• Depends on input size n.
• the input vertices and edges in a graph gives the running time
of an algorithm (bit representations)
• It is based upon the number of primitive operations performed
• primitive operation is the unit of operation that can be identified in the
pseudo-code

DETERMINING TIME COMPLEXITY
• Step-1. Determine how you will measure input size.
• Step-2. Choose the type of operation (or perhaps two operations) Comparisons,
Swaps, Copies, Additions.
• Step-3. Decide whether you wish to count operations in the
• Best case? - the fewest possible operations.
• Worst case? - the most possible operations.
• Average case? - This is harder as it is not always clear what is meant by an "average case".
NB: these calculation requires some level of probability theory
• Step-4. For the algorithm and the chosen case (best, worst, average), express the
count as a function of the input size of the problem.

RASP MACHINE OR STORED PROGRAM
MODEL
• It is same as RAM but allow program to change itself
as it is now stored in the memory
• memory of modern computers hosts both a program
and its corresponding data

CHARACTERISTICS OF ALGORITHM
• Input – It may take 0 or more inputs.
• Output – It must generate at least 1 output.
• Definiteness – Every statement must have one single and exact
meaning.
• Finiteness – Algorithm must have finite set of statements. It must
terminate at some point.
• Effectiveness – Statements must serve some purpose.

ALGORITHM WRITING
• A simple algorithm of swapping can be written as:
Algorithm swap (a, b)
{
temp = a;
a = b;
b = temp;
}

ANALYSIS OF ALGORITHMS
• There are few criteria for analysis:
• Analysis of time taken:
• every simple statement takes 1 unit of time
• in the algorithm of swapping, the time taken is 3 which is a constant
• So, F(n) = 3
• The constant is represented as O (1)

ANALYSIS OF SPACE TAKEN
• The variables taken in the algorithm are a, b, and temp
• The space taken is 3 words which is a constant
• Constant is represented as O (1)
• The analysis can be done by using the frequency count
method.

ANALYSIS OF SPACE
• The variables taken in the algorithm are a, b, and temp
• The space taken is 3 words which is a constant
• Constant is represented as O (1)
• The analysis can be done by using the frequency count
method

SPACE COMPLEXITY
• F (n) = 2n +3, so degree of polynomial = 1.
• Then the time complexity = O (n)
• Space complexity:
• Variables used = A, n, S, I
• A:n words, n:1 word, S:1 word, i:1 word
• S(n) = n + 3 words,
• so space complexity = O(n)

SPACE COMPLEXITY (2)
• F(n) = 2n2
+ 2n +1, here the highest degree is 2.
• So, the time complexity is O(n2
).
• Variables: A, B, C, n, i, j.
• A n2
, B n2
, C n2
, n 1, i 1 and j 1 word.
• Space complexity S(n) is 3 n2
+ 3.
• Degree is O(n2
)

TIME COMPLEXITY
We need to check for how many times the statement inside the loop will
run. So, the degree will depend upon that

TYPES OF TIME FUNCTIONS
• O (1) Constant
🡺
• O(log n) Logarithmic
🡺
• O(n) Linear
🡺
• O(n2
) Quadratic
🡺
• O(n3
) Cubic
🡺
• O(2n
) , O(3n
), O(nn
) Exponential
🡺

COMPARISON OF CLASSES OF FUNCTIONS
• 1 < log n < (n)1/2< n < n log n < n2< n3< --- < 2n< 3n ---< nn
• to see which alg is taking less time:
• implement both the algorithms and run the two programs with different
inputs
• There are problems with this approach for analysis of algorithms
• What problems? NB: Input and machine dependent

HOW TO COMPARE ALGORITHMS
• To compare algorithms, let us define a few objective measures:
• Execution times?
• Not a good measure (specific computer).
• Number of statements executed?
• Not a good measure (programming language)
• Ideal solution?
• algorithm as a function of the input size n (i.e., f(n))

TYPES OF ANALYSIS
• know which inputs the algorithm takes less time or long time
• First case: best case
• Second case: worse case
• We need some kind of syntax = asymptotic analysis or notation
• Worse case
• Average case
• Best case

ASYMPTOTIC ANALYSIS
• evaluate the performance of an algorithm in terms of input size
• calculate, how the time (or space) taken by an algorithm increases with the input size
• Asymptotic notations are mathematical tools to represent the time complexity of
algorithms
• The notations used are:
• O (Big – Oh notation):Works as an upper bound of a function.
• (Big – Omega notation): Works as a lower bound of a function.
Ω
• (Theta notation): Works as an average bound of a function.
Θ

BIG-0 NOTATION
• It gives tight upper bound of a given function f(n)

BIG OH NOTATION
• The function f(n) = O(g(n)) iff +ve constants C and n
∃ 0. such that f(n) ≤ C * g(n) n
∀ 0
• Understanding the Statement:
📌 The function f(n) = O(g(n)) This means that f(n) grows at most as fast as g(n) when
n becomes large.
In other words, for sufficiently large values of n, f(n) is always less than or equal to
some multiple of g(n).
📌 iff +ve constants C and n
∃ ₀ This means that there exist some positive constants
C and n₀ such that the condition holds.
📌 such that f(n) ≤ C * g(n) n ≥ n
∀ ₀ This means that for all values of n greater than
or equal to n₀, the function f(n) will always be less than or equal to C times g(n).

02 Introduction to Data Structures & Algorithms.pptx

02 Introduction to Data Structures & Algorithms.pptx

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