consolidated dsad

8/12/2019 Consolidated DSAD

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1st

Feb 2014:

Data structures: conceptual and concrete ways to organize data for efficient storage andefficient manipulation

Employment of this data structures in the design of efficient algorithms

Topics Covered:

Introduction to Data Structure Abstract Data Type (ADT) Introduction to Algorithm Analysis and Design

Why to study Data Structure:

Any organization has a collection of records that can be searched, processed in any order, or modified.

o Data structures organize data: Good choice: more efficient programs Bad choice: poor program performance

The choice can make a difference between the program running in afew seconds or many day

o Characteristics of a problems solution efficient: if it solves problem within resource constraints

time space

Cost: amount of resources a solution consumesCosts & Benefits :

A data structure requires a certain amount of:

space for each data item it stores time to perform a single basic operation programming effort.


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Selecting a data Structure :

Select a data structure as follows:

1. Analyze the problem to determine the resource constraints a solution must meet.2. Determine the basic operations that must be supported. Quantify the resource constraints

for each operation.

3. Select the data structure that best meets these requirements.Abstract Data Type :

o A logical view of the data objects together with specifications of the operations required tocreate and manipulate them.

Describe an algorithmpseudo-code Describe a data structureADT

o A data structure is the physical implementation of an ADT Each ADT operation is implemented by one or more subroutines. Data structures are used to organize data in main memory

Abstract Data Type (ADT) is Simple or structured data type whose implementation details arehidden.

An abstract data type is not a part of a program, because a program written in a programinglanguage requires the definition of a data structure, not only the operations on data structure.

An object oriented language (OOL) such as C++ has direct link to abstract data types byimplementing them as a class.

Data Type :

Set of values

Operations that can be performed on those values

Ex: short int

can take values (-32768 to 32767)

Operations are , -, , /


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Data Type Classification:

ADT :

Both an interface and an implementation

Interface and implementation are independent

Interface defines

The type of the data stored

Operations that are performed on the data

parameters of each operation

Implementation defines

data organization

developing efficient algorithm for each operation

Primitive Non-

Sint,

float,short

UserReferen

Lang.

Arrays,strings,

structurePoint

classAD

Data Types


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Basics Operations of ADT :

insert(S, x)

delete(S, x)

search(S, x)

findMin(S)

findMax(S)

findSuccessor(S, x)

findPredecessor(S, x)

Classification of ADT :

Linear

Arrays Linked list Circular list Doubly linked list Stack Queue

Circular queue Priority queue

Non-linear

Trees

Binary Trees and Types

Binary Search Trees and Variants

Threaded Binary Trees

Heaps

Graphs


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Undirected

Directed

Hash Tables

ADT Summary :

Standard data collection organizations (Data Structures) with desired operations

Described by an interface

Many implementations are possible

Facilitate reuse and easy extensibility

Design issues are Time and Space Complexity

Problems, Algorithms and Programs

Programmers deal with:

problems, algorithms and computer programs.

Problem: a task to be performed.

Best thought of as inputs and matching outputs. Problem definition should include constraints on the resources that may be

consumed by any acceptable solution.

Problemsmathematical functions

A function is a matching between inputs (the domain) and outputs (the range). An input to a function may be single number, or a collection of information. The values making up an input are called the parameters of the function. A particular input must always result in the same output every time the function is

computed.

Algorithm: a method or aprocess followed to solve a problem.

A recipe: The algorithm gives us a recipe for solving the problem by performing aseries of steps, where each step is completely understood and can be implemented.


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An algorithm takes the input to a problem (function) and transforms it to the output.

A mapping of input to output.A problem can be solved by many algorithms.

For example, the problem of sorting can be solved by the

Following algorithms:

Insertion sort Bubble sort Selection sort Shellsort Mergesort

An algorithm possesses the following properties:

It must be correct. It must be composed of a series of concrete steps. There can be no ambiguity as to which step will be performed next. It must be composed of a finite number of steps. It must terminate.

A computer program is an instance, or concrete representation, for an algorithm in some programming

language.

Algorithm Design Techniques

The design of algorithms is also an important focus.

Types of algorithms:

Greedy algorithms Divide and Conquer Dynamic programming Randomized algorithms Backtracking


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Algorithm Analysis:

Predict the amount of resources required:

Memory: how much space is needed? Computational time: how fast the algorithm runs?

FACT: running time grows with the size of the input

Input size (number of elements in the input)

Size of an array, polynomial degree, # of elements in a matrix, # of bits in the binaryrepresentation of the input, vertices and edges in a graph

Def: Running time = the number of primitive operations (steps) executed before termination

Running time is expressed as T(n) for some function T on input size n.

Two approaches to obtaining running time:

Measuring under standard benchmark conditions. Estimating the algorithms performance

Estimation is based on:

The size of the input The number of basic operations

The time to complete a basic operation does not depend on the value of its operands.

Lists of ADT :

List is an ordered sequence of elements List has the property length(count of elements) The elements are arranged consecutively. Can be implemented as static(Array implementation) or dynamic (Linked List implementation)

Array

Fundamental data structure

Homogeneous collection of values store values sequentially in memory


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associate INDEX with each value use array name and index to quickly access the value. efficient method for working with large collection of data. An array can be

Single-dimensional Multi-dimensional

Array Memory Layout:

The index in a one-dimensional array directly defines the relative positions of the element inactual memory.

Two-dimensional array is stored in memory using row-major or column-major storageOperations on Array:

The common operations on arrays are searching, insertion, deletion and traversal. An array is more suitable when the number of deletions and insertions is small, but a lot of

searching and retrieval activities are expected.

Pros and cons:

Advantages

Simple and easy to use Faster Access to elements (Constant time random access)

Disadvantages

Fixed size Inefficient insertions and deletions

Ques:We have stored the two-dimensional array students in memory. The array is 100 4 (100 rows

and 4 columns). Show the address of the element students [5][3] assuming that the element student[1][1] is stored in the memory location with address 1000 and each element occupies only one memory

location. The computer uses row-major storage.

Solutions:We can use the following formula to find the location of an element, assuming each element

occupies one memory location.

Y=x+ {cols * (i-1)}+(j-1)


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Advantages

Not so simple Sequential access

Disadvantages

Dynamic size Efficient insertions and deletions

Linked List Application:

It is a dynamic data structure in which the list can start with no nodes and then grow as newnodes are needed

It is a suitable structure if a large number of insertions and deletions are needed, but searching alinked list is slower that searching an array.

It is a very efficient data structure for sorted list that will go through many insertions anddeletions

Linked List Operations:

Comparisons of linked list and Array


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Variations of linked list :

Singly linked list:It has only head part and corresponding references to the next nodes.

Doubly linked list: A linked list which has both head and tail parts, thus allowing the traversal in

bi-directional fashion. Except the first node, the head node refers to the previous node.

Circular linked list:A linked list whose last node has reference to the first node.

Try:

How many pointers are contained as data members in the nodes of a circular, doubly linked listof integers with five nodes?

If the address of A [1][1] and A[2][1] are 1000 and 1010 respectively and each element occupies2 bytes then the array has been stored in _________ order.

The operation of processing each element in the list is known as _____________8thFeb 2014:

Topics:

Arrays Linked Lists Stacks Queues

Stacks:

A stack is a restricted linear list in which all additions and deletions are made at one end, thetop. (LIFO)

Operations on stack ADT:

No search No adding in arbitrary positions No sorting No access to anything beyond the top element. Stack --- stack(stackName) Push ---push(stackName,dataItem)


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Pop ---- pop(stackName,dataItem) Empty--- empty(stackName)

Stack ADT implementation:

Stack ADTs can be implemented using either AS an array or a linked list.Stack array Implementation:

createStack(S): Define an array S for some fixed sixe N

top -1

push(x,S): if top = N-1 then error

else top top + 1

S*top+ x

StackEmpty(S): return (top < 0)

pop(S): if isStackEmpty() then error

else item S*top+

top top 1

return(item)

Application of stack

Expression Evaluation Function calls Memory Management (Run time Environment) Backtracking Parenthesis Matching

Expression Evaluation:The three Notations of Expressions are: Infix a+b Postfix(RPN) ab+


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Prefix(PN) +abConversion from Infix to postfix:

1. Print operands as they arrive. 2. If the stack is empty or contains a left parenthesis on top, push the incoming

operator onto the stack.

3. If the incoming symbol is a left parenthesis, push it on the stack. 4. If the incoming symbol is a right parenthesis, pop the stack and print the

operators until you see a left parenthesis. Discard the pair of parenthesis.

5. If the incoming symbol has higher precedence than the top of the stack, push iton the stack.

6. If the incoming symbol has equal precedence with the top of the stack, useassociation. If the association is left to right, pop and print the top of the stack and

then push the incoming operator. If the association is right to left, push the

incoming operator.

7. If the incoming symbol has lower precedence than the symbol on the top of thestack, pop the stack and print the top operator. Then test the incoming operator

against the new top of stack.

8. At the end of the expression, pop and print all operators on the stack. (Noparentheses should remain.)

Ex:

Infix arithmetic expression a + b * cd.

Input: a + b * cd Output: a opStack: empty

Input: a + b * cd Output: a opStack: +

Input: a + b* cd Output: a b opStack: +

Input: a + b*cd Output: a b opStack: + *

Input: a + b * cd Output: a b c opStack: + *

Input: a + b * cd Output: a b c opStack: + *

Input: a + b * cd Output: a b c * opStack: +

Input: a + b * cd Output: a b c * + opStack: empty


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Input: a + b * cd Output: a b c * + opStack:

Input: a + b * cd Output: a b c * +d opStack:

Input: a + b * cd Output: a b c * + d opStack: empty

Ex:

a/b^c-d abc^/d- a-b+c ab-c+ a*(b+c) abc+c* a * (b + c * d) + e a b c d * + * e +

Queue ADT:

A queue is a linear list in which data can only be inserted at one end, called the rear, and deletedfrom the other end, called thefront.(FIFO).

Operations on Queue ADT:

Queue Enqueue Dequeue Empty

Queue Implementation:

A queue ADT can be implemented using either as an array or a linked listApplication of Queue ADT:

For implementing any "natural" FIFO service, like telephone enquiries, reservation requests,traffic flow, etc.

For implementing any "computational" FIFO service, for instance, to access some resources.Examples: printer queues, disk queues, etc.

For searching in special data structures (breadth-first search in graphs and trees). For handling scheduling of processes in a multitasking operating system.


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Try :

How many pointers are contained as data members in the nodes of a circular, doubly linked listof integers with five nodes?

If the address of A[1][1] and A[2][1] are 1000 and 1010 respectively and each element occupies2 bytes then the array has been stored in _________ order.

The operation of processing each element in the list is known as _____________22 feb 2014:

O-Notation :Intuitively: O(g(n)) = the set of functions with a smaller or same order of growthas g(n)

Examples :

3n + 2 = O(n) ; 3n + 2 = 2

3n + 3 = O(n) ; 3n + 3 = 3

100n + 6 = O(n) ; 100n + 6 = 6

= O(n2)

10 n2+ 4n + 2 < = 11 n

2for n >= 5


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- notationIntuitively: (g(n)) = the set of functions with a larger or same order of growth asg(n)

3n + 2 = ? 3n + 2 >= 3n for all n >= 1

3n + 3 = ? 3n + 3 >= 3n for all n >= 1

100n + 6 = ? 100n + 6 >= 100n for all n >= 1

3n + 3 = ?

3n + 3 =1,c2 = 6

3n+3 >= 3n for all n>=1, c1=3

3n


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Sorting:

Iterative methods: Insertion sort Bubble sort Selection sort

Divide and conquer

Merge sort Quicksort

Counting sort

Radix sort Bucket sort

Insertion Sort

Alg.: INSERTION-SORT(A)

forj 2 to n

do key A* j +

Insert A[ j ] into the sorted sequence A[1 . . j -1]

i j - 1

while i > 0 and A[i] > key

do A*i 1+ A*i+

i i 1

A*i 1+ key

Insertion sortsorts the elements in place


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Analysis of Insertion ADT:

INSERTION-SORT(A)

forj 2 to n

do key A* j +

Insert A[ j ] into the sorted sequence A[1 . . j -1]

i j - 1

while i > 0 and A[i] > key

do A*i 1+ A*i+

i i 1

A[i + 1+ key

Best Case Analysis :

The array is already sorted

A[i] key upon the first time the while loop test is run (when i =j -1) tj= 1

T(n) = c1n + c2(n -1) + c4(n -1) + c5(n -1) + c8(n-1) = (c1+ c2+ c4+ c5+ c8)n + (c2

+ c4+ c5+ c8)

= an + b = (n)

Worst Case Analysis :

The array is in reverse sorted order

Always A[i] > key in whileloop test Have to compare keywith all elements to the left of the j-th position compare

with j-1 elements tj= j

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Alg.: SELECTION-SORT(A)

n length*A+

forj 1 to n - 1

do smallest j

for i j 1 to n

do if A[i] < A[smallest]

then smallest i

exchange A*j+ A*smallest+

Dividethe problem into a number of sub-problems

Similar sub-problems of smaller sizeConquerthe sub-problems

Solve the sub-problems recursively Sub-problem size small enoughsolve the problems in straightforward manner

Combinethe solutions to the sub-problems

Obtain the solution for the original problemMerger and Sort APPROACH :

TO SORT AN ARRAY A[P . . R]:

DIVIDE

DIVIDE THE N-ELEMENT SEQUENCE TO BE SORTED INTO TWO SUBSEQUENCES OF N/2ELEMENTS EACH

CONQUER

SORT THE SUBSEQUENCES RECURSIVELY USING MERGE SORT WHEN THE SIZE OF THE SEQUENCES IS 1 THERE IS NOTHING MORE TO DO

COMBINE

MERGE THE TWO SORTED SUBSEQUENCES


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MERGE SORT :

ALG.: MERGE-SORT(A, P, R) IF P < R CHECK FOR BASE CASE THEN Q (P + R)/2 DIVIDE MERGE-SORT(A, P, Q) CONQUER MERGE-SORT(A, Q + 1, R) CONQUER MERGE(A, P, Q, R) COMBINE INITIAL CALL:MERGE-SORT(A, 1, N)

MERGING:

INPUT: ARRAY AAND INDICES P, Q, RSUCH THAT P Q


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COMBINE:

MERGE ON AN N-ELEMENT SUBARRAY TAKES(N) TIMEC(N) = (N)(1) IF N =1

T(N) = 2T(N/2) + (N) IF N > 1

SOLVE THE RESCURRSION:

T(N) = C IF N = 1

2T(N/2) + CN IF N > 1

USE MASTERS THEOREM:

COMPARE N WITH F(N) = CN

CASE 2: T(N) = (NLGN)

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