Introducction to Algorithm

Nov 15, 2019Download as pptx, pdf0 likes27 views

This document introduces algorithms and asymptotic notation used to analyze algorithm performance. It defines common asymptotic notations like O, Ω, and Θ notation, and gives examples of standard functions like logarithms and factorials. It notes that accurately measuring algorithm performance requires specifying the programming language, working program, computer used, and compiler options.

INTRODUCTION TO ALGORITHM
Prepared by
R. Gomthijayam
Asst. Prof. of CA

Algorithms
• The word algorithm comes from the name of a Persian
mathematician Abu Ja’far Mohammed ibn-i Musa al
Khowarizmi.
• In computer science, this word refers to a special method
useable by a computer for solution of a problem. The
statement of the problem specifies in general terms the
desired input/output relationship.
• For example, sorting a given sequence of numbers into
nondecreasing order provides fertile ground for introducing
many standard design techniques and analysis tools.

Asymptotic Notation
The notation we use to describe the asymptotic running time
of an algorithm are defined in terms of functions whose
domains are the set of natural numbers
 ...,2,1,0N

O-notation
• For a given function , we denote by the set of
functions
• We use O-notation to give an asymptotic upper bound of a
function, to within a constant factor.
• means that there existes some constant c s.t.
is always for large enough n.
)(ng ))(( ngO








0
0
allfor)()(0
s.t.andconstantspositiveexistthere:)(
))((
nnncgnf
ncnf
ngO
))(()( ngOnf 
)(ncg)(nf

Ω-Omega notation
• For a given function , we denote by the set of
functions
• We use Ω-notation to give an asymptotic lower bound on a
function, to within a constant factor.
• means that there exists some constant c s.t.
is always for large enough n.
)(ng ))(( ng








0
0
allfor)()(0
s.t.andconstantspositiveexistthere:)(
))((
nnnfncg
ncnf
ng
))(()( ngnf 
)(nf )(ncg

-Theta notation
• For a given function , we denote by the set of
functions
• A function belongs to the set if there exist
positive constants and such that it can be “sand- wiched”
between and or sufficienly large n.
• means that there exists some constant c1 and c2
s.t. for large enough n.
)(ng ))(( ng








021
021
allfor)()()(c0
s.t.and,,constantspositiveexistthere:)(
))((
nnngcnfng
nccnf
ng
)(nf ))(( ng
1c 2c
)(1 ngc )(2 ngc
Θ
))(()( ngnf 
)()()( 21 ngcnfngc 

Standard notations and common functions
• Logarithms:
)lg(lglglg
)(loglog
logln
loglg 2
nn
nn
nn
nn
kk
e





Standard notations and common functions
• Factorials
For the Stirling approximation:



















ne
n
nn
n
1
12!
0n
)lg()!lg(
)2(!
)(!
nnn
n
non
n
n




Performance Measurement Needs
 programming language
 working program
 computer
 compiler and options to use
javac -o

This document discusses asymptotic notation which is used to describe the running time of algorithms. It introduces the Big O, Big Omega, and Theta notations. Big O notation represents the upper bound or worst case running time. Big Omega notation represents the lower bound or best case running time. Theta notation represents the running time between the upper and lower bounds. The document provides mathematical definitions and examples of how to determine if a function is Big O, Big Omega, or Theta notation.

Asymptotic notationDr Shashikant Athawale

This document discusses asymptotic notations that are used to characterize an algorithm's efficiency. It introduces Big-Oh, Big-Omega, and Theta notations. Big-Oh gives an upper bound on running time. Big-Omega gives a lower bound. Theta gives both upper and lower bounds, representing an algorithm's exact running time. Examples are provided for each notation. Time complexity is also introduced, which describes how the time to solve a problem changes with problem size. Worst-case time analysis provides an upper bound on runtime.

Asymptoptic notationsAlisha Jindal

This document defines and explains asymptotic notations used to analyze the time complexity of algorithms. It discusses Big-O, Big-Omega, Theta, Little-o and Little-Omega notations. Big-O notation expresses the upper bound of an algorithm's running time and measures worst case time complexity. Big-Omega notation expresses the lower bound and measures best case time complexity. Theta notation represents both the upper and lower bounds and is used for average case analysis. Examples are provided to demonstrate how functions are bounded using these notations.

Lec 5 asymptotic notations and recurrencesAnkita Karia

This document introduces asymptotic notation and methods for analyzing algorithm runtimes. It discusses Big-O, Big-Omega, and Big-Theta notations for describing upper bounds, lower bounds, and tight bounds of a function. It also covers solving recurrence relations using substitution, recursion trees, and the master method. The master method provides a way to solve recurrences of the form T(n) = aT(n/b) + f(n) based on comparing f(n) to nlogba.

Asymptotic notationSaranya Natarajan

This document introduces asymptotic notations that are used to describe the time complexity of algorithms. It defines big O, big Omega, and big Theta notations, which describe the limiting behavior of functions. Big O notation provides an asymptotic upper bound, big Omega provides a lower bound, and big Theta provides a tight bound. Examples are given of different asymptotic efficiency classes like constant, logarithmic, linear, quadratic, and exponential time. Properties of asymptotic notations like transitivity, reflexivity, symmetry, and transpose symmetry are also covered.

Asymptotic notationsEhtisham Ali

1. Asymptotic notation such as Big-O, Omega, and Theta are used to describe the running time of algorithms as the input size n approaches infinity, rather than giving the exact running time. 2. Big-O notation gives an upper bound and describes worst-case running time, Omega notation gives a lower bound and describes best-case running time, and Theta notation gives a tight bound where the worst and best cases are equal up to a constant. 3. Common examples of asymptotic running times include O(1) for constant time, O(log n) for logarithmic time, O(n) for linear time, and O(n^2) for quadratic time.

Asymptotic notationsNikhil Sharma

The document discusses asymptotic notations that are used to describe the time complexity of algorithms. It introduces big O notation, which describes asymptotic upper bounds, big Omega notation for lower bounds, and big Theta notation for tight bounds. Common time complexities are described such as O(1) for constant time, O(log N) for logarithmic time, and O(N^2) for quadratic time. The notations allow analyzing how efficiently algorithms use resources like time and space as the input size increases.

asymptotic analysis and insertion sort analysisAnindita Kundu

This document discusses asymptotic analysis of algorithms. It introduces key concepts like algorithms, data structures, best/average/worst case running times, and asymptotic notations like Big-O, Big-Omega, and Big-Theta. These notations are used to describe the long-term growth rates of functions and provide upper/lower/tight bounds on the running time of algorithms as the input size increases. Examples show how to analyze the asymptotic running time of algorithms like insertion sort, which is O(n^2) in the worst case but O(n) in the best case.

Asymptotic notationsMamta Pandey

The document discusses asymptotic notations used to estimate the growth rate of functions. It defines asymptotic notations like Big-O, Big-Omega, and Theta that provide upper, lower, and tight bounds. Examples show how to determine if a function f(n) is O(g(n)), Ω(g(n)), or Θ(g(n)) based on constants. Little-o and little-omega notations describe bounds that are not asymptotically tight. Asymptotic notations are useful for analyzing time complexities of algorithms.

Asymptotic NotationLovely Professional University

Whenever we want to perform analysis of an algorithm, we need to calculate the complexity of that algorithm. But when we calculate complexity of an algorithm it does not provide exact amount of resource required. So instead of taking exact amount of resource we represent that complexity in a general form (Notation) which produces the basic nature of that algorithm. We use that general form (Notation) for the analysis process.

Big oThanhvinh Vo

Big O notation describes how efficiently an algorithm or function grows as the input size increases. It focuses on the worst-case scenario and ignores constant factors. Common time complexities include O(1) for constant time, O(n) for linear time, and O(n^2) for quadratic time. To determine an algorithm's complexity, its operations are analyzed, such as the number of statements, loops, and function calls.

Asymptotic Notationsohelranasweet

This document discusses analyzing algorithms and asymptotic notation. It defines running time as the number of primitive operations before termination. Examples are provided to illustrate calculating running time functions and classifying them by order of growth such as constant, logarithmic, linear, quadratic, and exponential time. Asymptotic notation such as Big-O, Big-Omega, and Big-Theta are introduced to classify functions by their asymptotic growth rates. Examples are given to demonstrate determining tight asymptotic bounds between functions. Recurrences are defined as equations describing functions in terms of smaller inputs and base cases, which are useful for analyzing recurrent algorithms.

Asymptotic Notation and ComplexityRajandeep Gill

Asymptotic analysisSoujanya V

This document provides an introduction to asymptotic analysis of algorithms. It discusses analyzing algorithms based on how their running time increases with the size of the input problem. The key points are: - Algorithms are compared based on their asymptotic running time as the input size increases, which is more useful than actual running times on a specific computer. - The main types of analysis are worst-case, best-case, and average-case running times. - Asymptotic notations like Big-O, Omega, and Theta are used to classify algorithms based on their rate of growth as the input increases. - Common orders of growth include constant, logarithmic, linear, quadratic, and exponential time.

Asymptotic analysisNisha Soms

Asymptotics 140510003721-phpapp02mansab MIRZA

The presentation covered time and space complexity, average and worst case analysis, and asymptotic notations. It defined key concepts like time complexity measures the number of operations, space complexity measures memory usage, and worst case analysis provides an upper bound on running time. Common asymptotic notations like Big-O, Omega, and Theta were explained, and how they are used to compare how functions grow relative to each other as input size increases.

Algorithm big oAshim Lamichhane

Big O notation is used in Computer Science to describe the performance or complexity of an algorithm. Big O specifically describes the worst-case scenario, and can be used to describe the execution time required or the space used (e.g. in memory or on disk) by an algorithm. For further information https://github.com/ashim888/dataStructureAndAlgorithm References: https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/asymptotic-notation http://web.mit.edu/16.070/www/lecture/big_o.pdf https://rob-bell.net/2009/06/a-beginners-guide-to-big-o-notation/ https://justin.abrah.ms/computer-science/big-o-notation-explained.html

AsymptoticTuhin Utsab Paul

Complexity of AlgorithmMuhammad Muzammal

This document discusses the complexity of algorithms and the tradeoff between algorithm cost and time. It defines algorithm complexity as a function of input size that measures the time and space used by an algorithm. Different complexity classes are described such as polynomial, sub-linear, and exponential time. Examples are given to find the complexity of bubble sort and linear search algorithms. The concept of space-time tradeoffs is introduced, where using more space can reduce computation time. Genetic algorithms are proposed to efficiently solve large-scale construction time-cost tradeoff problems.

ReconstructionCOMSATS Abbottabad

This lab report describes reconstructing a sampled signal to its original continuous time signal. The students implemented the discrete time Fourier transform (DTFT) to reconstruct the signal using different reconstruction time periods. Decreasing the reconstruction frequency from the sampling frequency resulted in a reconstructed signal that differed from the original. The students analyzed the reconstruction process and effects of varying the reconstruction frequency relative to the sampling frequency.

Computational ComplexityKasun Ranga Wijeweera

This document discusses computational complexity and analyzing the running time of algorithms. It defines big-O notation, which is used to classify algorithms according to their worst-case performance as the problem size increases. Examples are provided of algorithms with running times that are O(1), O(log N), O(N), O(N log N), O(N^2), O(N^3), and O(2^N). The growth rates of these functions from slowest to fastest are also listed.

Algorithm.pptTareq Hasan

The document discusses divide and conquer algorithms and merge sort. It provides details on how merge sort works including: (1) Divide the input array into halves recursively until single element subarrays, (2) Sort the subarrays using merge sort recursively, (3) Merge the sorted subarrays back together. The overall running time of merge sort is analyzed to be θ(nlogn) as each level of recursion contributes θ(n) work and there are logn levels of recursion.

Lecture 4 asymptotic notationsjayavignesh86

This document discusses asymptotic notations and their use in analyzing the time complexity of algorithms. It introduces the Big-O, Big-Omega, and Big-Theta notations for describing the asymptotic upper bound, lower bound, and tight bound of an algorithm's running time. The document explains that asymptotic notations allow algorithms to be compared by ignoring lower-order terms and constants, and focusing on the highest-order term that dominates as the input size increases. Examples are provided to illustrate the different orders of growth and the notations used to describe them.

Asymptotic notations(Big O, Omega, Theta )swapnac12

The document discusses different asymptotic notations used to characterize the complexity of algorithms: Big-O(O) notation provides an upper bound, Big-Omega(Ω) provides a lower bound, and Big-Theta(Θ) indicates the same order of growth. It defines each notation, explaining that Big-O represents f(n) growing less than or equal to g(n), Big-Omega represents f(n) growing greater than or equal to g(n), and Big-Theta represents f(n) growing equal to g(n). The document then discusses basics of probability theory, defining a sample space as the set of all possible outcomes of an experiment, with events being subsets of the sample space.

Algorithmic NotationsMuhammad Muzammal

ComplexityA. S. M. Shafi

The document discusses complexity analysis of algorithms. It defines time complexity as the calculation of the total time required for an algorithm to execute, and space complexity as the calculation of memory space required. Time and space complexity can be analyzed using asymptotic analysis, which studies how performance changes with increasing input size. Asymptotic notations like Big-O, Omega, and Theta are used to analyze best case, worst case, and average case time complexity. Big-O notation represents upper time bound, Omega lower time bound, and Theta both upper and lower time bound. Examples are given of functions and their time complexities using these notations.

Lecture 5: Asymptotic analysis of algorithmsVivek Bhargav

The document discusses asymptotic analysis of algorithms to analyze efficiency in terms of time and space complexity. It explains that the number of basic operations like arithmetic, data movement, and control operations determines time complexity, while space complexity depends on the number of basic data types used. Different algorithms for counting numbers with factorial divisible by 5 are analyzed to find their time complexity. The time complexity of an algorithm can be expressed using Θ notation, which describes the dominant term as the input size increases.

Introducción al Análisis y diseño de algoritmosluzenith_g

The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms to determine their time and space complexity, and how this involves determining how the resources required grow with the size of the problem. It provides examples of analyzing simple algorithms and determining whether they have linear, quadratic, or other complexity.

AlgorithmSyam Kumar

This document provides an introduction to algorithms and their analysis. It discusses algorithm design techniques like insertion sort and asymptotic analysis using big O notation to describe an algorithm's running time. Insertion sort is analyzed as an example, with its best, average, and worst case times shown to be O(n), O(n^2), and O(n^2) respectively, ignoring constant factors. The key concepts of asymptotic notation like big O, Ω, and Θ are also introduced to formally analyze an algorithm's growth rate.

AlgorithmAnirban Sarkar

This document provides an introduction to algorithms and their analysis. It discusses algorithm design techniques like insertion sort and asymptotic analysis using big O notation to describe an algorithm's running time. Insertion sort is analyzed as an example, with its best, average, and worst case times shown to be O(n), O(n^2), and O(n^2) respectively, where n is the input size. Common functions and standard asymptotic notations are also introduced.

More Related Content

What's hot (20)

Asymptotic notationsMamta Pandey

Asymptotic NotationLovely Professional University

Big oThanhvinh Vo

Asymptotic Notationsohelranasweet

Asymptotic Notation and ComplexityRajandeep Gill

Asymptotic analysisSoujanya V

Asymptotic analysisNisha Soms

Asymptotics 140510003721-phpapp02mansab MIRZA

Algorithm big oAshim Lamichhane

AsymptoticTuhin Utsab Paul

Complexity of AlgorithmMuhammad Muzammal

ReconstructionCOMSATS Abbottabad

Computational ComplexityKasun Ranga Wijeweera

Algorithm.pptTareq Hasan

Lecture 4 asymptotic notationsjayavignesh86

Asymptotic notations(Big O, Omega, Theta )swapnac12

Algorithmic NotationsMuhammad Muzammal

ComplexityA. S. M. Shafi

Lecture 5: Asymptotic analysis of algorithmsVivek Bhargav

Introducción al Análisis y diseño de algoritmosluzenith_g

Asymptotic notationsMamta Pandey

Asymptotic NotationLovely Professional University

Big oThanhvinh Vo

Asymptotic Notationsohelranasweet

Asymptotic Notation and ComplexityRajandeep Gill

Asymptotic analysisSoujanya V

Asymptotic analysisNisha Soms

Asymptotics 140510003721-phpapp02mansab MIRZA

Algorithm big oAshim Lamichhane

AsymptoticTuhin Utsab Paul

Complexity of AlgorithmMuhammad Muzammal

ReconstructionCOMSATS Abbottabad

Computational ComplexityKasun Ranga Wijeweera

Algorithm.pptTareq Hasan

Lecture 4 asymptotic notationsjayavignesh86

Asymptotic notations(Big O, Omega, Theta )swapnac12

Algorithmic NotationsMuhammad Muzammal

ComplexityA. S. M. Shafi

Lecture 5: Asymptotic analysis of algorithmsVivek Bhargav

Introducción al Análisis y diseño de algoritmosluzenith_g

Similar to Introducction to Algorithm (20)

AlgorithmSyam Kumar

AlgorithmAnirban Sarkar

This document provides an introduction to algorithms and their analysis. It discusses algorithm design techniques like insertion sort and asymptotic analysis using big O notation to describe an algorithm's running time. Insertion sort is analyzed as an example, with its best, average, and worst case times shown to be O(n), O(n^2), and O(n^2) respectively, where n is the input size. Common functions and standard asymptotic notations are also introduced.

Algorithm in Computer, Sorting and NotationsAbid Kohistani

The document discusses algorithms and their analysis. It begins by introducing algorithms and their importance in computer science. The problem of sorting is used as an example to introduce algorithms. Insertion sort is presented as a basic sorting algorithm, with examples of how it works. The analysis of algorithms is then discussed, including analyzing insertion sort's worst case running time, which grows quadratically as Θ(n^2). Asymptotic notation such as O, Ω, Θ is also introduced to analyze long term algorithm running times independent of machine details.

Asymptotic Notations.pptxSunilWork1

This document provides an overview of time and space complexity analysis, average and worst case analysis, and asymptotic notations. It defines key concepts like time complexity, space complexity, average case vs worst case analysis, and asymptotic notations like Big-O, Omega, and Theta. It explains why worst case analysis is generally used and the importance of asymptotics for analyzing how efficiently problems can be solved as input sizes increase.

algorithm_lec_1eregdsgdfgdgdfgdfgdfg.pptpartho5958

Algorithm analysis and designMegha V

This document provides an introduction to algorithm analysis and design. It defines what an algorithm is and lists some key properties it should have like being unambiguous, having a finite number of steps, and terminating. It discusses different ways to specify algorithms using natural language, flowcharts, or pseudocode. It also covers analyzing algorithm efficiency in terms of time and space complexity and introduces common asymptotic notations used like Big O, Big Omega, and Big Theta notation.

Asymptotic notations adaShaishavShah8

This document discusses different asymptotic notations used to represent the complexity of algorithms: - O(g(n)) represents asymptotic upper bound - function g(n) grows at least as fast as f(n) for large n. - Ω(g(n)) represents asymptotic lower bound - function g(n) grows no faster than f(n) for large n. - θ(g(n)) represents asymptotic tight bound - functions g(n) and f(n) grow at the same rate for large n. It also explains the differences between O and o notations, and between Ω and ω notations - where the former represents loose asymptotic bounds and the latter

Asymptotic NotationsRishabh Soni

Anlysis and design of algorithms part 1Deepak John

Lec1Nikhil Chilwant

The document discusses data structures and algorithms. It defines key concepts like algorithms, programs, data structures, and asymptotic analysis. It explains how to analyze algorithms to determine their efficiency, including analyzing best, worst, and average cases. Common notations for describing asymptotic running time like Big-O, Big-Omega, and Big-Theta are introduced. The document provides examples of analyzing sorting algorithms like insertion sort and calculating running times. It also discusses techniques for proving an algorithm's correctness like assertions and loop invariants.

Asymptotic Analysis.pptabhishekchakraborty261420

This document discusses different types of asymptotic analyses used to analyze algorithms: worst-case, average-case, and best-case. It also introduces asymptotic notation such as big-O, Ω, Θ, o, and ω notation used to describe upper and lower time complexity bounds of algorithms. Examples are provided to illustrate how to determine if a function belongs to a specific asymptotic notation class. Common functions and standard notations used in asymptotic analysis like logarithms and factorials are also covered.

Lec 2 algorithms efficiency complexityAnaya Zafar

An algorithm is a set of steps to solve a problem. Algorithm efficiency describes how fast an algorithm solves problems of different sizes. Time complexity is the most important measure of efficiency, measuring the number of steps an algorithm takes based on the size of the input. Common techniques to analyze time complexity include calculating the number of operations in loops and recursions. Asymptotic notation like Big-O describes the long-term growth rate of an algorithm's running time as the input size increases.

Intro to super. advance algorithm..pptxManishBaranwal10

DSArrupa2

This document discusses data structures and algorithms. It provides course objectives which include imparting concepts of data structures and algorithms, introducing searching and sorting techniques, and developing applications using suitable data structures. Course outcomes include understanding algorithm performance analysis, concepts of data structures, linear data structures, and identifying efficient data structure implementations. The document also covers algorithm basics, specifications, expressions, analysis techniques and asymptotic notations for analyzing algorithms.

Data Structures- Part2 analysis toolsAbdullah Al-hazmy

This document discusses algorithm analysis tools. It explains that algorithm analysis is used to determine which of several algorithms to solve a problem is most efficient. Theoretical analysis counts primitive operations to approximate runtime as a function of input size. Common complexity classes like constant, linear, quadratic, and exponential time are defined based on how quickly runtime grows with size. Big-O notation represents the asymptotic upper bound of a function's growth rate to classify algorithms.

Unit ii algorithmTribhuvan University

This document provides an overview of algorithms including definitions, characteristics, design, and analysis. It defines an algorithm as a finite step-by-step procedure to solve a problem and discusses their key characteristics like input, definiteness, effectiveness, finiteness, and output. The document outlines the design of algorithms using pseudo-code and their analysis in terms of time and space complexity using asymptotic notations like Big O, Big Omega, and Big Theta. Examples are provided to illustrate linear search time complexity and the use of different notations to determine algorithm efficiency.

complexity big oh notation notation.pptxnarifmsit18seecs

Design and Analysis of Algorithms Lecture NotesSreedhar Chowdam

This document discusses algorithms and their analysis. It begins by defining an algorithm and its key characteristics like being finite, definite, and terminating after a finite number of steps. It then discusses designing algorithms to minimize cost and analyzing algorithms to predict their performance. Various algorithm design techniques are covered like divide and conquer, binary search, and its recursive implementation. Asymptotic notations like Big-O, Omega, and Theta are introduced to analyze time and space complexity. Specific algorithms like merge sort, quicksort, and their recursive implementations are explained in detail.

DS Unit-1.pptx very easy to understand..KarthikeyaLanka1

The document discusses algorithms, including their definition, properties, analysis of time and space complexity, and examples of recursion and iteration. It defines an algorithm as a finite set of instructions to accomplish a task. Properties include inputs, outputs, finiteness, definiteness, and effectiveness. Time complexity is analyzed using big-O notation, while space complexity considers static and variable parts. Recursion uses function calls to solve sub-problems, while iteration uses loops. Examples include factorial calculation, GCD, and Towers of Hanoi solved recursively.

Asymptotic_Notations_v2_1709293258280.pptxprasanna220904

AlgorithmSyam Kumar

AlgorithmAnirban Sarkar

Algorithm in Computer, Sorting and NotationsAbid Kohistani

Asymptotic Notations.pptxSunilWork1

algorithm_lec_1eregdsgdfgdgdfgdfgdfg.pptpartho5958

Algorithm analysis and designMegha V

Asymptotic notations adaShaishavShah8

Asymptotic NotationsRishabh Soni

Anlysis and design of algorithms part 1Deepak John

Lec1Nikhil Chilwant

Asymptotic Analysis.pptabhishekchakraborty261420

Lec 2 algorithms efficiency complexityAnaya Zafar

Intro to super. advance algorithm..pptxManishBaranwal10

DSArrupa2

Data Structures- Part2 analysis toolsAbdullah Al-hazmy

Unit ii algorithmTribhuvan University

complexity big oh notation notation.pptxnarifmsit18seecs

Design and Analysis of Algorithms Lecture NotesSreedhar Chowdam

DS Unit-1.pptx very easy to understand..KarthikeyaLanka1

Asymptotic_Notations_v2_1709293258280.pptxprasanna220904

More from Rajamanickam Gomathijayam (9)

Computer Application in BusinessRajamanickam Gomathijayam

This document defines computers and their components. It discusses that computers are machines that can perform logical and arithmetic operations automatically through programming. It describes the different types of computers based on size and power, including personal computers, workstations, minicomputers, mainframes, and supercomputers. The document also outlines the typical parts of a computer system, characteristics of computers, generations of computers, types of computer memory, and common input, output, and storage devices.

Sololearn certRajamanickam Gomathijayam

Uml pptRajamanickam Gomathijayam

This document provides an introduction to the Unified Modeling Language (UML). UML is a standard modeling language that includes various diagram types like use case diagrams, class diagrams, sequence diagrams, and state machine diagrams. Class diagrams specifically show classes, fields, methods, and relationships between classes like inheritance, association, aggregation, and composition. Examples of class diagrams are provided to demonstrate how classes, attributes, methods, and relationships are depicted. Tools for creating UML diagrams are also listed.

Coding for php with mysqlRajamanickam Gomathijayam

Php with mysql pptRajamanickam Gomathijayam

Corel Draw IntroductionRajamanickam Gomathijayam

This document provides an introduction to using CorelDraw, including how to open the program, an overview of the main window components, and descriptions of some key tools. It discusses how to open CorelDraw and access recent documents or templates. It describes the title bar, menu bar, toolbars, rulers, drawing page, and other elements of the CorelDraw window. Finally, it gives an overview of some common drawing and editing tools in the toolbox like the pick, shape, zoom, and text tools and provides examples of activities and questions for using these tools.

Introduction to computer graphicsRajamanickam Gomathijayam

This document provides an introduction to computer graphics. It discusses that computer graphics deals with creating images using hardware, software, and applications. It describes the basic graphics system including input devices, the output device, and the frame buffer. It then discusses the display processor and how it stores graphics in a display list. Finally, it outlines several applications of computer graphics including computer-aided design, presentation graphics, computer art, entertainment, education and training, visualization, image processing, and graphical user interfaces.

Artificial IntelligenceRajamanickam Gomathijayam

The document discusses various artificial intelligence problem-solving techniques including generate-and-test, hill climbing, best-first search, problem reduction, constraint satisfaction, and means-ends analysis. It provides examples and algorithms for hill climbing, best-first search, and constraint satisfaction problems. The key techniques of generating candidate solutions, evaluating solutions, and searching the problem space are described across different problem domains.

Network ConceptsRajamanickam Gomathijayam

Computer Application in BusinessRajamanickam Gomathijayam

Sololearn certRajamanickam Gomathijayam

Uml pptRajamanickam Gomathijayam

Coding for php with mysqlRajamanickam Gomathijayam

Php with mysql pptRajamanickam Gomathijayam

Corel Draw IntroductionRajamanickam Gomathijayam

Introduction to computer graphicsRajamanickam Gomathijayam

Artificial IntelligenceRajamanickam Gomathijayam

Network ConceptsRajamanickam Gomathijayam

Recently uploaded (20)

How to Create A Todo List In Todo of Odoo 18Celine George

How to Configure Public Holidays & Mandatory Days in Odoo 18Celine George

Link your Lead Opportunities into Spreadsheet using odoo CRMCeline George

Junction Field Effect Transistors (JFET)GS Virdi

In this concise presentation, Dr. G.S. Virdi (Former Chief Scientist, CSIR-CEERI, Pilani) introduces the Junction Field-Effect Transistor (JFET)—a cornerstone of modern analog electronics. You’ll discover: Why JFETs? Learn how their high input impedance and low noise solve the drawbacks of bipolar transistors. JFET vs. MOSFET: Understand the core differences between JFET and MOSFET devices. Internal Structure: See how source, drain, gate, and the depletion region form a controllable semiconductor channel. Real-World Applications: Explore where JFETs power amplifiers, sensors, and precision circuits. Perfect for electronics students, hobbyists, and practicing engineers looking for a clear, practical guide to JFET technology.

Myopathies (muscle disorders) for undergraduateMohamed Rizk Khodair

Biophysics Chapter 3 Methods of Studying Macromolecules.pdfPKLI-Institute of Nursing and Allied Health Sciences Lahore , Pakistan.

This chapter provides an in-depth overview of the viscosity of macromolecules, an essential concept in biophysics and medical sciences, especially in understanding fluid behavior like blood flow in the human body. Key concepts covered include: ✅ Definition and Types of Viscosity: Dynamic vs. Kinematic viscosity, cohesion, and adhesion. ⚙️ Methods of Measuring Viscosity: Rotary Viscometer Vibrational Viscometer Falling Object Method Capillary Viscometer 🌡️ Factors Affecting Viscosity: Temperature, composition, flow rate. 🩺 Clinical Relevance: Impact of blood viscosity in cardiovascular health. 🌊 Fluid Dynamics: Laminar vs. turbulent flow, Reynolds number. 🔬 Extension Techniques: Chromatography (adsorption, partition, TLC, etc.) Electrophoresis (protein/DNA separation) Sedimentation and Centrifugation methods.

All About the 990 Unlocking Its Mysteries and Its Power.pdfTechSoup

APGAR SCORE BY sweety Tamanna Mahapatra MSc PediatricSweetytamannaMohapat

Ancient Stone Sculptures of India: As a Source of Indian HistoryVirag Sontakke

This Presentation is prepared for Graduate Students. A presentation that provides basic information about the topic. Students should seek further information from the recommended books and articles. This presentation is only for students and purely for academic purposes. I took/copied the pictures/maps included in the presentation are from the internet. The presenter is thankful to them and herewith courtesy is given to all. This presentation is only for academic purposes.

dynastic art of the Pallava dynasty south IndiaPrachiSontakke5

03#UNTAGGED. Generosity in architecture.MCH

What makes space feel generous, and how architecture address this generosity in terms of atmosphere, metrics, and the implications of its scale? This edition of #Untagged explores these and other questions in its presentation of the 2024 edition of the Master in Collective Housing. The Master of Architecture in Collective Housing, MCH, is a postgraduate full-time international professional program of advanced architecture design in collective housing presented by Universidad Politécnica of Madrid (UPM) and Swiss Federal Institute of Technology (ETH). Yearbook MCH 2024. Master in Advanced Studies in Collective Housing UPM - ETH

Redesigning Education as a Cognitive Ecosystem: Practical Insights into Emerg...Leonel Morgado

Rock Art As a Source of Ancient Indian HistoryVirag Sontakke

Cultivation Practice of Onion in Nepal.pptxUmeshTimilsina1

Rococo versus Neoclassicism. The artistic styles of the 18th centuryGema

pulse ppt.pptx Types of pulse , characteristics of pulse , Alteration of pulsesushreesangita003

How to Manage Purchase Alternatives in Odoo 18Celine George

Lecture 1 Introduction history and institutes of entomology_1.pptxArshad Shaikh

LDMMIA Reiki Yoga S5 Daily Living WorkshopLDM Mia eStudios

Happy May and Happy Weekend, My Guest Students. Weekends seem more popular for Workshop Class Days lol. These Presentations are timeless. Tune in anytime, any weekend. <<I am Adult EDU Vocational, Ordained, Certified and Experienced. Course genres are personal development for holistic health, healing, and self care. I am also skilled in Health Sciences. However; I am not coaching at this time.>> A 5th FREE WORKSHOP/ Daily Living. Our Sponsor / Learning On Alison: Sponsor: Learning On Alison: — We believe that empowering yourself shouldn’t just be rewarding, but also really simple (and free). That’s why your journey from clicking on a course you want to take to completing it and getting a certificate takes only 6 steps. Hopefully Before Summer, We can add our courses to the teacher/creator section. It's all within project management and preps right now. So wish us luck. Check our Website for more info: https://ldmchapels.weebly.com Get started for Free. Currency is Euro. Courses can be free unlimited. Only pay for your diploma. See Website for xtra assistance. Make sure to convert your cash. Online Wallets do vary. I keep my transactions safe as possible. I do prefer PayPal Biz. (See Site for more info.) Understanding Vibrations If not experienced, it may seem weird understanding vibes? We start small and by accident. Usually, we learn about vibrations within social. Examples are: That bad vibe you felt. Also, that good feeling you had. These are common situations we often have naturally. We chit chat about it then let it go. However; those are called vibes using your instincts. Then, your senses are called your intuition. We all can develop the gift of intuition and using energy awareness. Energy Healing First, Energy healing is universal. This is also true for Reiki as an art and rehab resource. Within the Health Sciences, Rehab has changed dramatically. The term is now very flexible. Reiki alone, expanded tremendously during the past 3 years. Distant healing is almost more popular than one-on-one sessions? It’s not a replacement by all means. However, its now easier access online vs local sessions. This does break limit barriers providing instant comfort. Practice Poses You can stand within mountain pose Tadasana to get started. Also, you can start within a lotus Sitting Position to begin a session. There’s no wrong or right way. Maybe if you are rushing, that’s incorrect lol. The key is being comfortable, calm, at peace. This begins any session. Also using props like candles, incenses, even going outdoors for fresh air. (See Presentation for all sections, THX) Clearing Karma, Letting go. Now, that you understand more about energies, vibrations, the practice fusions, let’s go deeper. I wanted to make sure you all were comfortable. These sessions are for all levels from beginner to review. Again See the presentation slides, Thx.

YSPH VMOC Special Report - Measles Outbreak Southwest US 5-3-2025.pptxYale School of Public Health - The Virtual Medical Operations Center (VMOC)

A measles outbreak originating in West Texas has been linked to confirmed cases in New Mexico, with additional cases reported in Oklahoma and Kansas. The current case count is 817 from Texas, New Mexico, Oklahoma, and Kansas. 97 individuals have required hospitalization, and 3 deaths, 2 children in Texas and one adult in New Mexico. These fatalities mark the first measles-related deaths in the United States since 2015 and the first pediatric measles death since 2003. The YSPH Virtual Medical Operations Center Briefs (VMOC) were created as a service-learning project by faculty and graduate students at the Yale School of Public Health in response to the 2010 Haiti Earthquake. Each year, the VMOC Briefs are produced by students enrolled in Environmental Health Science Course 581 - Public Health Emergencies: Disaster Planning and Response. These briefs compile diverse information sources – including status reports, maps, news articles, and web content– into a single, easily digestible document that can be widely shared and used interactively. Key features of this report include: - Comprehensive Overview: Provides situation updates, maps, relevant news, and web resources. - Accessibility: Designed for easy reading, wide distribution, and interactive use. - Collaboration: The “unlocked" format enables other responders to share, copy, and adapt seamlessly. The students learn by doing, quickly discovering how and where to find critical information and presenting it in an easily understood manner. CURRENT CASE COUNT: 817 (As of 05/3/2025) • Texas: 688 (+20)(62% of these cases are in Gaines County). • New Mexico: 67 (+1 )(92.4% of the cases are from Eddy County) • Oklahoma: 16 (+1) • Kansas: 46 (32% of the cases are from Gray County) HOSPITALIZATIONS: 97 (+2) • Texas: 89 (+2) - This is 13.02% of all TX cases. • New Mexico: 7 - This is 10.6% of all NM cases. • Kansas: 1 - This is 2.7% of all KS cases. DEATHS: 3 • Texas: 2 – This is 0.31% of all cases • New Mexico: 1 – This is 1.54% of all cases US NATIONAL CASE COUNT: 967 (Confirmed and suspected): INTERNATIONAL SPREAD (As of 4/2/2025) • Mexico – 865 (+58) ‒Chihuahua, Mexico: 844 (+58) cases, 3 hospitalizations, 1 fatality • Canada: 1531 (+270) (This reflects Ontario's Outbreak, which began 11/24) ‒Ontario, Canada – 1243 (+223) cases, 84 hospitalizations. • Europe: 6,814

How to Create A Todo List In Todo of Odoo 18Celine George

How to Configure Public Holidays & Mandatory Days in Odoo 18Celine George

Link your Lead Opportunities into Spreadsheet using odoo CRMCeline George

Junction Field Effect Transistors (JFET)GS Virdi

Myopathies (muscle disorders) for undergraduateMohamed Rizk Khodair

Biophysics Chapter 3 Methods of Studying Macromolecules.pdfPKLI-Institute of Nursing and Allied Health Sciences Lahore , Pakistan.

All About the 990 Unlocking Its Mysteries and Its Power.pdfTechSoup

APGAR SCORE BY sweety Tamanna Mahapatra MSc PediatricSweetytamannaMohapat

Ancient Stone Sculptures of India: As a Source of Indian HistoryVirag Sontakke

dynastic art of the Pallava dynasty south IndiaPrachiSontakke5

03#UNTAGGED. Generosity in architecture.MCH

Redesigning Education as a Cognitive Ecosystem: Practical Insights into Emerg...Leonel Morgado

Rock Art As a Source of Ancient Indian HistoryVirag Sontakke

Cultivation Practice of Onion in Nepal.pptxUmeshTimilsina1

Rococo versus Neoclassicism. The artistic styles of the 18th centuryGema

pulse ppt.pptx Types of pulse , characteristics of pulse , Alteration of pulsesushreesangita003

How to Manage Purchase Alternatives in Odoo 18Celine George

Lecture 1 Introduction history and institutes of entomology_1.pptxArshad Shaikh

LDMMIA Reiki Yoga S5 Daily Living WorkshopLDM Mia eStudios

YSPH VMOC Special Report - Measles Outbreak Southwest US 5-3-2025.pptxYale School of Public Health - The Virtual Medical Operations Center (VMOC)

Introducction to Algorithm

1. INTRODUCTION TO ALGORITHM Prepared by R. Gomthijayam Asst. Prof. of CA

2. Algorithms • The word algorithm comes from the name of a Persian mathematician Abu Ja’far Mohammed ibn-i Musa al Khowarizmi. • In computer science, this word refers to a special method useable by a computer for solution of a problem. The statement of the problem specifies in general terms the desired input/output relationship. • For example, sorting a given sequence of numbers into nondecreasing order provides fertile ground for introducing many standard design techniques and analysis tools.

3. The problem of sorting

4. Asymptotic Notation The notation we use to describe the asymptotic running time of an algorithm are defined in terms of functions whose domains are the set of natural numbers  ...,2,1,0N

5. O-notation • For a given function , we denote by the set of functions • We use O-notation to give an asymptotic upper bound of a function, to within a constant factor. • means that there existes some constant c s.t. is always for large enough n. )(ng ))(( ngO         0 0 allfor)()(0 s.t.andconstantspositiveexistthere:)( ))(( nnncgnf ncnf ngO ))(()( ngOnf  )(ncg)(nf

6. Ω-Omega notation • For a given function , we denote by the set of functions • We use Ω-notation to give an asymptotic lower bound on a function, to within a constant factor. • means that there exists some constant c s.t. is always for large enough n. )(ng ))(( ng         0 0 allfor)()(0 s.t.andconstantspositiveexistthere:)( ))(( nnnfncg ncnf ng ))(()( ngnf  )(nf )(ncg

7. -Theta notation • For a given function , we denote by the set of functions • A function belongs to the set if there exist positive constants and such that it can be “sand- wiched” between and or sufficienly large n. • means that there exists some constant c1 and c2 s.t. for large enough n. )(ng ))(( ng         021 021 allfor)()()(c0 s.t.and,,constantspositiveexistthere:)( ))(( nnngcnfng nccnf ng )(nf ))(( ng 1c 2c )(1 ngc )(2 ngc Θ ))(()( ngnf  )()()( 21 ngcnfngc 

8. Standard notations and common functions • Logarithms: )lg(lglglg )(loglog logln loglg 2 nn nn nn nn kk e    

9. Standard notations and common functions • Factorials For the Stirling approximation:                    ne n nn n 1 12! 0n )lg()!lg( )2(! )(! nnn n non n n   

10. Performance Measurement Needs  programming language  working program  computer  compiler and options to use javac -o

Introducction to Algorithm

Recommended

More Related Content

What's hot (20)

Similar to Introducction to Algorithm (20)

More from Rajamanickam Gomathijayam (9)

Recently uploaded (20)

Introducction to Algorithm