Algorithms

Program

By :- Eman El-ma’asarawi

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What is Program? How to make program? How to think about the program? Steps for making a good project.

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DS Algorithm+ Progra

m =

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list of instruction to achieve specific task.

Program

O/P instruction

Processing instruction

I/P instruction

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H/W S/W

Os Compiler Text editor

Idea ware Algorithm DS Programming Methodologies

Structure Design Oop

S/W Concepts Information hiding Data encapsulation Data abstraction

Programming Toolboxes

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HW

OS

Compiler

Application (text editor)

Algorithm DS Methodologie

s

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Data :- factual information Structure :- Arrangement or relationship of

elements as particles

Data structure :-A means of storing a collection of data

DS (Data Structure)

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A data structure can be viewed as consisting of a set of algorithms for performing operations on the data it stores.

DS cont…

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Variable:-data storage location that has a

value that can change during program execution.

Constant:-fixed value that can’t change.

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Applications of Data Structure

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Linear

DS

Non-Linear

Sequential

Linked

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linke

dlis

tsstac

k

Link

edst

acks

Link

edqu

eues

Tree

Gra

phs

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1-Array2-Linked List3-Stack4-Queue5-Tree

Is a data structure where all elements are stored in contiguous memory

1-Array

Individual elements Array

n

12

0

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Random access Fixed size Less efficient used on data stored in array

Array properties

Array size

Element size

1-in bytes2-the number of element

Data type of array

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Exambel

1 2 34 56 2 78 0 10

0 1 2 3 4 5 6 7

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2-linked list

List head /Dummy Node

Data

Null

Is a data structure in which the first element is (head) stored on its own

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The size not fixed grow/shrink The extra space is needed to store the

pointer of each element More time Implement linked list use array but not

efficient Complex to code and manage

Linked list properties

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Appending Nodeadd Node to the end of the list.

Traversingcheck the linked list.

Inserting Nodeadd Node in the middle of a list.

Deleting Node from memory Modified links

Destroy

Operation on linked list

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Which is fast array or linked list?Why?

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Is a data structure consisting of data nodes connected to each other with pointers.

Each node connected to 2 or more other nodes.

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5-Trees

The order of the tree:-is the max number of nodes to which any

single node connected. binary tree:-

is the simplest tree is of order 2.each child have two pointers.

Root node:-the topmost node in the tree.

leaf node:-node with no children.

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Tree properties

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Data rightleft

Data rightleftData righ

tleft

Create binary tree Inserting Node Traversing the tree recursion Searching the tree Deleting the tree

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Binary search tree op…

Logical sequence of steps describe complete solution to solve problem in finite amount of time

May involve alternation, iteration or recursion

More than one algorithm possible for the same task

Algorithms

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Sorting Searching-Insertion sort -linear search-Merge sort -Binary search-Heap sort-Quick sort

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Algorithmes

What resources are required to accomplish the task

How one algorithm compares with other algorithms

How much time or space is required Measured in terms of common basic

operations

Efficiency of Algorithms

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Worst-case: (usually) T(n) = maximum time of algorithm on

any input of size n. Average-case: (sometimes) T(n) = expected time of algorithm over

all inputs of size n. Need assumption of statistical distribution of

inputs. Best-case: (bogus) Cheat with a slow algorithm that works

fast on some input.

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Kinds of analyses

Running time: On a particular input, it is the number

of primitive operations (steps) executed. One line may take a different amount of

time than another, but each execution of line i takes

the same amount of time c . Running time of the algorithm is

Ʃ (cost of statement)*(number of times statement is

executed)

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How efficiency varies with the size of the task

E.g. if the size of the task increases linearly, do the efficiency measures increase linearly, quadratically, exponentially,...?

Expressed in terms of standard functions of n

Complexity

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notations

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Big o

Big ƟBig Ω

F(n)<=g(n) F(n)>=g(n) F(n)=g(n)C1g(n)<=f(n)<=c2g(n)

Data shapeEx - insertion sort

best case

Data methodEx – insertion sort

worst case

PseudoCode conventions

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Indentation indicates block structure. While, for, repeat , if , then, and else.

indicates comment. indicates assignment. Variables are local by default. Array elements accessed as in A[i].

Calculate x^n, where n is an integer greater than 0.

Algorithm

1. Set r = 1 2. Set i = 1 3. Repeat 3.1 If i=n then terminate loop 3.2 Multiply r by x 3.3 Add 1 to i 4. Exit with result r

ExampleSimple Power Algorithm

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O(n) - the algorithm requires n multiplications.

Time complexity

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Formulate a Concise Specification of the Problem

Design a high-level strategy for a solution Refine steps 1 and 2 if necessary Complete the details of the design Implement in the chosen language

How to Approach Recursive Algorithms

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Calculate x^n, where n is an integer greater than 0.

Recursive Algorithm

1. If n=0 then 1.1 set r = 1 2. Else 2.1 set p = n/2 (dropping any remainder) 2.2 set r = (xp)2 2.3 if n is odd then 2.3.1 multiply r by x 3. Exit with result r

Recursive Power Algorithm

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static int power (int x, int n) int r; if (n == 0) r = 1; else r = power(x,n/2); r = r*r; if (n%2 != 0) r *= x; return r;

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implementation

O(log n) - the algorithm requires approximately log n multiplications.

2

Time complexity

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Quiz

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Towers of Hanoi

1 2 3

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1 2 3

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1 2 3

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1 2 3

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1 2 3

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1 2 3

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1 2 3

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1 2 3

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Solution' ≡ shortest path Recursive Solution:

1. Identify biggest discrepancy (=disk N)

2. If moveable to goal peg Then move Else

3. Subgoal: set-up (N−1)-disk tower on non-goal peg.

4. Go to 1. ...

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SDLC (s/w Development life cycle)

Problem analysis

Requ

irem

ent d

efini

tions

Desig

nIm

plem

enta

tion

Testing

Delivery

Usi

ng p

rogr

am

Maintenance

The End

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