Analysis of Algorithms

2-1Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd
ed., Ch. 2
COMSATS INSTITUTE OFCOMSATS INSTITUTE OF
INFORMATION & TECHNOLOGY,INFORMATION & TECHNOLOGY,
VEHARIVEHARI
Subject Code : CSC201
Subject Teacher : Sir Muhammad Afzal
Name Of Subject : DESIGN & ANALYSIS OF ALGORITHMS
Name of Unit : ANALYSIS OF ALGORITHMS
Topic : Introduction, Analysis, Efficeincy of algorithms
Name of Students : Mubariz Hamza Aslam (FA15-BCS-152)
Muzamil Hayat Khan (FA15-BCS-080)

ed., Ch. 2
CONTENTSCONTENTS
IINTRODUCTION TO ALGORITHMSNTRODUCTION TO ALGORITHMS
ANALYSIS OF ALGORITHMANALYSIS OF ALGORITHM
EFFIECIENCY OF ALGORITHMEFFIECIENCY OF ALGORITHM
BEST CASEBEST CASE
WORST CASEWORST CASE
AVERAGE CASEAVERAGE CASE
EXAMPLESEXAMPLES
Sub: DESIGN & ANALYSIS OF ALGORITHM
Topic: Introduction,Analysis,Effieciency Of Algorithm

ed., Ch. 2
INTRODUCTION OFINTRODUCTION OF
ALGORITHMSALGORITHMS
• An Algorithm is a Step by Step solution of aAn Algorithm is a Step by Step solution of a
specific mathematical or computer relatedspecific mathematical or computer related
problemproblem
• There are different types of AlgorithmsThere are different types of Algorithms
• GreedyGreedy
• DynamicDynamic
• Divide & Conquer ,etcDivide & Conquer ,etc
• They can be expressed in different ways.They can be expressed in different ways.
• Pseudo codePseudo code
• By using simple English statements(Steps)By using simple English statements(Steps)

ed., Ch. 2
ANALYSIS OF ALGORITHMSANALYSIS OF ALGORITHMS
• Issues:Issues:
• correctnesscorrectness
• time efficiencytime efficiency
• space efficiencyspace efficiency
• optimalityoptimality
• Approaches:Approaches:
• theoretical analysistheoretical analysis
• empirical analysisempirical analysis

THEORETICAL ANALYSIS OFTHEORETICAL ANALYSIS OF
TIME EFFICIENCYTIME EFFICIENCY
Time efficiency is analyzed by determining the
number of repetitions of the basic operation as
a function of input size
• Basic operation: the operation that
contributes most towards the running time of
the algorithm
T(n) ≈ copC(n)running time
execution time
for basic operation
Number of times
basic operation is
executed
input size

INPUT SIZE AND BASIC OPERATIONINPUT SIZE AND BASIC OPERATION
EXAMPLESEXAMPLES
ProblemProblem Input size measureInput size measure Basic operationBasic operation
Searching for keySearching for key
in a list ofin a list of nn itemsitems
Number of list’sNumber of list’s
items, i.e.items, i.e. nn
Key comparisonKey comparison
Multiplication ofMultiplication of
two matricestwo matrices
Matrix dimensions orMatrix dimensions or
total number oftotal number of
elementselements
Multiplication ofMultiplication of
two numberstwo numbers
Checking primalityChecking primality
of a given integerof a given integer
nn
n’n’size = number ofsize = number of
digits (in binarydigits (in binary
representation)representation)
DivisionDivision
Typical graphTypical graph
problemproblem
#vertices and/or#vertices and/or
edgesedges
Visiting a vertexVisiting a vertex
or traversing anor traversing an
edgeedge

EMPIRICAL ANALYSIS OF TIMEEMPIRICAL ANALYSIS OF TIME
EFFICIENCYEFFICIENCY
• Select a specific (typical) sample of inputsSelect a specific (typical) sample of inputs
• Use physical unit of time (e.g., milliseconds)Use physical unit of time (e.g., milliseconds)
oror
Count actual number of basic operation’s executionsCount actual number of basic operation’s executions
• Analyze the empirical dataAnalyze the empirical data

BEST-CASE, AVERAGE-CASE,BEST-CASE, AVERAGE-CASE,
WORST-CASEWORST-CASE
For some algorithms efficiency depends on form ofFor some algorithms efficiency depends on form of
input:input:
• Worst case: CWorst case: Cworstworst((nn) – maximum over inputs of size) – maximum over inputs of size nn
• Best case: CBest case: Cbestbest((nn) – minimum over inputs of size) – minimum over inputs of size nn
• Average case: CAverage case: Cavgavg((nn) – “average” over inputs of size) – “average” over inputs of size nn
• Number of times the basic operation will be executed on typical inputNumber of times the basic operation will be executed on typical input
• NOT the average of worst and best caseNOT the average of worst and best case
• Expected number of basic operations considered as a random variable underExpected number of basic operations considered as a random variable under
some assumption about the probability distribution of all possible inputssome assumption about the probability distribution of all possible inputs

EXAMPLE: SEQUENTIAL SEARCHEXAMPLE: SEQUENTIAL SEARCH
• Worst caseWorst case
• Best caseBest case
• Average caseAverage caseSub: DESIGN & ANALYSIS OF ALGORITHM

TYPES OF FORMULAS FOR BASICTYPES OF FORMULAS FOR BASIC
OPERATION’S COUNTOPERATION’S COUNT
• Exact formulaExact formula
e.g., C(e.g., C(nn) =) = nn((nn-1)/2-1)/2
• Formula indicating order of growth with specificFormula indicating order of growth with specific
multiplicative constantmultiplicative constant
e.g., C(e.g., C(nn)) ≈≈ 0.50.5 nn22
• Formula indicating order of growth with unknownFormula indicating order of growth with unknown
multiplicative constantmultiplicative constant
e.g., C(e.g., C(nn)) ≈≈ cncn22

ORDER OF GROWTHORDER OF GROWTH
• Most important: Order of growth within a constantMost important: Order of growth within a constant
multiple asmultiple as nn ∞→∞→
• Example:Example:
• How much faster will algorithm run on computer that is twice as fast?How much faster will algorithm run on computer that is twice as fast?
• How much longer does it take to solve problem of double input size?How much longer does it take to solve problem of double input size?

ASYMPTOTIC ORDER OFASYMPTOTIC ORDER OF
GROWTHGROWTH
A way of comparing functions that ignores constant factors andA way of comparing functions that ignores constant factors and
small input sizessmall input sizes
• O(O(gg((nn)): class of functions)): class of functions ff((nn) that grow) that grow no fasterno faster thanthan gg((nn))
• ΘΘ((gg((nn)): class of functions)): class of functions ff((nn) that grow) that grow at same rateat same rate asas gg((nn))
• ΩΩ((gg((nn)): class of functions)): class of functions ff((nn) that grow) that grow at least as fastat least as fast asas gg((nn))

BIG-OHBIG-OH

BIG-OMEGABIG-OMEGA

BIG-THETABIG-THETA

ORDERS OF GROWTH OF SOMEORDERS OF GROWTH OF SOME
IMPORTANT FUNCTIONSIMPORTANT FUNCTIONS
• All logarithmic functions logAll logarithmic functions logaa nn belong to the same classbelong to the same class
ΘΘ(log(log nn) no matter what the logarithm’s base) no matter what the logarithm’s base aa > 1 is> 1 is
• All polynomials of the same degreeAll polynomials of the same degree kk belong to the samebelong to the same
class:class: aakknnkk
++ aakk-1-1nnkk-1-1
+ … ++ … + aa00 ∈∈ ΘΘ((nnkk
))
• Exponential functionsExponential functions aann
have different orders of growthhave different orders of growth
for differentfor different aa’s’s
• orderorder loglog n <n < orderorder nnαα
((αα>0) < order>0) < order aann
< order< order nn! < order! < order nnnn

BASIC ASYMPTOTIC EFFICIENCYBASIC ASYMPTOTIC EFFICIENCY
CLASSESCLASSES
11 constantconstant
loglog nn logarithmiclogarithmic
nn linearlinear
nn loglog nn n-n-loglog-n-n
nn22
quadraticquadratic
nn33
cubiccubic
22nn
exponentialexponential
nn!! factorialfactorial

TIME EFFICIENCY OF NON-RECURSIVETIME EFFICIENCY OF NON-RECURSIVE
ALGORITHMSALGORITHMS
General Plan for AnalysisGeneral Plan for Analysis
• Decide on parameterDecide on parameter nn indicatingindicating input sizeinput size
• Identify algorithm’sIdentify algorithm’s basic operationbasic operation
• DetermineDetermine worstworst,, averageaverage, and, and bestbest cases for input of sizecases for input of size nn
• Set up a sum for the number of times the basic operation isSet up a sum for the number of times the basic operation is
executedexecuted
• Simplify the sum using standard formulas and rulesSimplify the sum using standard formulas and rules

EXAMPLE 1: MAXIMUM ELEMENTEXAMPLE 1: MAXIMUM ELEMENT

ed., Ch. 2
REFERENCESREFERENCES
Books:-
1. Introduction to design & analysis of
algorithm
By Anany levitin, Pearson Publication
2. Design & Analysis of Algorithm
By A.A.Puntambekar , Technical Publication
3. Introduction to Algorithms
By Thomas Cormen , MIT Press
Web Resources:-
1.Google
2.Wikipedia

Analysis of Algorithms

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