Unit I- Data structures Introduction, Evaluation of Algorithms, Arrays, Sparse Matrix
1. Data Structures
Introduction
Evaluation of Algorithms
Arrays
Sparse Matrix
Dr. R. Khanchana
Assistant Professor
Department of Computer Science
Sri Ramakrishna College of Arts and Science for Women
2. Syllabus
• UNIT I Introduction: Introduction of Algorithms, Analysing Algorithms. Arrays: Sparse Matrices -
Representation of Arrays. Stacks and Queues. Fundamentals - Evaluation of Expression Infix to
Postfix Conversion - Multiple Stacks and Queues
• UNIT II Linked List: Singly Linked List - Linked Stacks and Queues - Polynomial Addition - More on
Linked Lists - Sparse Matrices - Doubly Linked List and Dynamic - Storage Management - Garbage
Collection and Compaction.
• UNIT III Trees: Basic Terminology - Binary Trees - Binary Tree Representations - Binary Trees -
Traversal - More on Binary Trees - Threaded Binary Trees - Binary Tree Representation of Trees -
Counting Binary Trees. Graphs: Terminology and Representations - Traversals, Connected
Components and Spanning Trees, Shortest Paths and Transitive Closure
• UNIT IV External Sorting: Storage Devices -Sorting with Disks: K-Way Merging - Sorting with Tapes
Symbol Tables: Static Tree Tables - Dynamic Tree Tables - Hash Tables: Hashing Functions - Overflow
Handling.
• UNIT V Internal Sorting: Insertion Sort - Quick Sort - 2 Way Merge Sort - Heap Sort - Shell Sort -
Sorting on Several Keys. Files: Files, Queries and Sequential organizations - Index Techniques -File
Organizations.
4. What is a Data structure?
• Data are just a collection of facts and figures
• Data Structure is not a Programming Language
• Data structure is a set of Algorithms that we
are implementing by using any programming
language to solve some Problems.
5. Chapter 1 - Introduction
• Data Structure - Store and organize data
• Data Structure means to structure the information while
storing
• Data deals two things
– Data Storing
– Data Processing
• Data structure deal with storage of the information, how
effectively to store information or data.
• To structure the data n number of Algorithms are proposed.
• Algorithm - all these Algorithms are called Abstract Data
Types(ADT)
• Abstract data types are set of rules.
9. Algorithms - Overview
• An algorithm is a procedure having well
defined steps for solving a particular problem.
• Definition: Algorithm is finite set of logic or
instructions, written in order for accomplish
the certain predefined task.
• It is not the complete program or code, it is
just a solution (logic) of a problem
– Can be represented either as a Flowchart or
Pseudo code.
10. Study of Algorithms
(i) machines for executing algorithms
– From the smallest pocket calculator to the largest general
purpose digital computer.
(ii) languages for describing algorithms
– language design and translation.
(iii) foundations of algorithms
– Particular task accomplishable by a computing device
– What is the minimum number of operations necessary for
any algorithm which performs a certain function?
(iv) analysis of algorithms
Performance is measured in terms of the computing time
and space that are consumed while the algorithm is processing
11. Characteristics of an Algorithm
• Input
– There are zero or more quantities which are externally supplied
• Output
– At least one quantity is produced
• Definiteness
– Each instruction must be clear and unambiguous
• Finiteness
– If trace out the instructions of an algorithm, then for all cases
the algorithm will terminate after a finite number of steps
• Effectiveness
– Time & Space
– Every instruction must be sufficiently basic and also be feasible.
12. Flowchart
• Raw data is input and algorithms are used to
transform it into refined data. So, instead of
saying that computer science is the study of
algorithms, alternatively, it is the study of data:
• (i) machines that hold data;
• (ii) languages for describing data manipulation;
• (iii) foundations which describe what kinds of
refined data can be produced from raw data;
• (iv) structures for representing data.
14. Data Type & Data Object
• A data type is a term which refers to the kinds
of data that variables may "hold" in a
programming language.
• Data object is a term referring to a set of
elements, say D.
– Data object integers refers to D = {0, 1, 2, ...}. The
data object alphabetic character strings of length
less than thirty one implies D = {",'A','B', ...,'Z','AA',
...}. Thus, D may be finite or infinite
21. How to Analyze
Programs/Algorithms?
There are many criteria upon which judge a
program, for instance:
• (i) Does it do what we want it to do?
• (ii) Does it work correctly according to the
original specifications of the task?
• (iii) Is there documentation which describes how
to use it and how it works?
• (iv) Are subroutines created in such a way that
they perform logical sub-functions?
• (v) Is the code readable?
23. Performance evaluation
• Performance evaluation can be divided into 2
major phases:
– (a) a priori estimates and
– (b) a posteriori testing. Both of these are equally
important.
• First consider a priori estimation. Suppose
that somewhere in one of your programs is
the statement
• x = x + 1
24. Performance evaluation
Time Complexity
– first is the amount of time a single execution will
take
– second is the number of times it is executed.
• The product of these numbers will be the total
time taken by this statement.
• The second statistic is called the frequency
count, and this may vary from data set to data
set.
25. Posteriori Testing
• It is impossible to determine exactly how
much time it takes to execute any command
unless we have the following information:
• (i) the machine we are executing on:
• (ii) its machine language instruction set;
• (iii) the time required by each machine
instruction;
• (iv) the translation a compiler will make from
the source to the machine language.
28. Fibonacci Series - Example
• To clarify some of these ideas, let us
look at a simple program for
computing the n-th Fibonacci
number. The Fibonacci sequence
starts as
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
• Each new term is obtained by taking
the sum of the two previous terms.
If we call the first term of the
sequence F0 then F0 = 0, F1 = 1 and in
general
• Fn = Fn-1 + Fn-2, n 2.
• The program on the following page
takes any non-negative
integer n and prints the value Fn.
30. Execution Count
• A complete set
would include four
cases: n < 0, n =
0, n = 1 and n > 1.
Below is a table
which summarizes
the frequency
counts for the first
three cases.
31. Time Computation
• For example n might be the number of inputs
or the number of outputs or their sum or the
magnitude of one of them. For the Fibonacci
program n represents the magnitude of the
input and the time for this program is written
as T(FIBONACCI) = O(n).
35. Chapter 2 - Arrays
AXIOMATIZATION
• Array is a consecutive set of
memory locations.
• Element − Each item stored in an
array is called an element.
• Index − Each location of an
element in an array has a
numerical index, which is used to
identify the element.
• Dimensions
37. Array Dimensions
– One dimensional Array
– Two dimensional Array
– Three dimensional Array
– Multi Dimensional Array
1-D
2-D
38. Definition of Row & Column Major
• In row-major order, the consecutive elements
of a row reside next to each other, whereas
the same holds true for consecutive elements
of a column in column-major order.
39. Row & Column Major
• Row-major order
• Column-major order
41. Array Length
If an array is declared A(l1 :u1,l2:u2, ...,ln:un), then it is easy to see that the
number of elements is
Example :
If we have the declaration A(4:5, 2:4, 1:2, 3:4) then we have a total of
2 * 3 * 2 * 2 = 24 elements.
42. Array Length -Example
• 1-D A[5..12]
• Formula - Ui-li+1
• Elements are
• U,B,F,D,A,E,C
• 7-0+1
• 6
• 2-D A[0..2]B[0..3]
• Formula - (U1-l1+1) * (U2-l2+1)
• Array Size
• =(2-0+1)*(3-0+1) = 3 *4 = 12
SIZE
44. Array Representation
• The 2-D array A(1:u1,1:u2) may be interpreted as u1 rows:
row 1,row 2, ...,row u1, each row consisting of u2 elements. In a
row major representation, these rows would be represented
in memory as in figure 2.4.
If is the address of A(1), then the
address of an arbitrary element A(i) is
just + (i - 1).
array element: A(l), A(2), A(3), ..., A(i), ..., A(u1)
address: , + 1, + 2, ..., + i - 1, ..., + u1 - 1
total number of elements = u1
46. REPRESENTATION OF ARRAYS
• Multidimensional arrays are provided as a
standard data object in which they are
represented in memory. Recall that memory
may be regarded as one dimensional with
words numbered from 1 to m.
– One dimensional array - A(i1,i2, ...,in),
– Two dimensional array A(l1 :u1,l2:u2, ...,ln: un)
– The number of elements is
51. SPARSE MATRICES
• A matrix is a mathematical object which arises in
many physical problems.
• A general matrix consists of m rows and n columns of
numbers as in figure 2.1.
• Sparse Matrix has many zero entries.
52. Sparse Matrix Representation
• To store a matrix in a 2-D
array, say A(1:m, 1:n)
• Each element of a matrix is
uniquely characterized by its
row and column position,
say i , j. S
• Store a matrix as a list of 3-
tuples of the form (i,j,value)
• 3-tuples of any row be
stored so that the columns
are increasing.
• Store the second matrix of
figure 2.1 in the
array A(0:t,1:3) where t = 8
is the number of nonzero
terms.
56. Transpose -Algorithm
Example :
The difficulty is in not knowing where to put the
element (j,i,val) until all other elements which
precede it have been processed.
– (1,1,15) which becomes (1,1,15)
– (1,4,22) which becomes (4,1,22)
– (1,6, - 15) which becomes (6,1, - 15)
