Internet EngineeringInternet Engineering

Czesław SmutnickiCzesław Smutnicki

Discrete Mathematics Discrete Mathematics – – Computational ComplexityComputational Complexity

CONTENTS

• Asymptotic notation• Decision/optimization problems• Calculation models• Turing machines• Problem, instances, data coding• Complexity classes• Polynomial-time algorithms• Theory of NP-completness• Approximate methods• Quality measures of approximation• Analysis of quality measures• Calculation cost• Competitive analysis (on-line algorithms)• Inapproximality theory

ASYMPTOTIC NOTATION – symbol O(n)

))(()( ngOnf

00 )()(0,,0 nnngcnfNnc

)(253 22 nOnn

)(8 23 nOnn

)(3 2nOn

Definition

Examples

ASYMPTOTIC NOTATION – symbol (n)

))(()( ngnf

00 )()(0,,0 nnnfngcNnc

)(253 22 nnn

)(8 23 nnn

)(3 2nn

Definition

Examples

ASYMPTOTIC NOTATION – symbol (n)

))(()( ngnf

021021 )()()(0,,0, nnngcnfngcNncc

)(253 22 nnn )(8 23 nnn )(12 2nn

Definition

Examples

ASYMPTOTIC NOTATION - symbol o(n)

0)(

)(lim ng

nfn

))(()( ngonf

Definition

Examples

)(3 2non )(253 22 nonn

ASYMPTOTIC NOTATION - symbol (n)

))(()( ngnf

)(

)(lim

ng

nfn

Definition

Examples

)(8 23 nnn )(253 22 nnn

DECISION/OPTIMIZATION PROBLEMS

• decision problem: answer yes-no2-partition problem: given numbers . Does a set

exist such that

• optimization problem: find min or max of the goal function valueknapsack problem: given numbers , and . Find the set such that ,

• any optimization problem can be transformed into decision problemknapsack problem: given numbers , , and . Does a set exist such that ,

nnnaaa ,...,, 21

},...,2,1{ nNI INi iIi i aa \

12 n naaa ,...,, 21 nccc ,...,, 21b

},...,2,1{ nNI NIIi ic max baIi i

22 n naaa ,...,, 21 nccc ,...,, 21

b y},...,2,1{ nNI ycIi i baIi i

CALCULATION MODELS

• Simple machine

• Finite-state machine

• Automata: Mealy

Moore

• Deterministic/non-deterministic finite automata

OIffOI oo :),,,(

S

oi

i o

OSIfSSIfffSOI osos :,:),,,,,(

OSIfSSIfsffSOI osoos :,:),,,,,,(

OSfSSIfsffSOI osoos :,:),,,,,,(

OSIfSIfsffSOI oS

soos :,2:),,,,,,(

DETERMINISTIC TURING MACHINE

SssSAfASAfSSAfsfffSA nymosomos ,},1,0,1{:,:,:),,,,,,(

s

0 1 2 3 4-1-2 …

NON-DETERMINISTIC TURING MACHINE

s

0 1 2 3 4-1-2 …

SssSAfASAfSAfsfffSA nymoS

somos ,},1,0,1{:,:,2:),,,,,,(

CODING

• Instance I/ Problem P• Decimal coding of I• Binary coding of I• Unary coding of I• Data string x(I)• Size N(I) of the instance I• Coding of numbers and structural elements

32logloglogloglog)( 110 ncaybnIN ni ii

32lglglglglg)( 12 ncaybnIN ni ii

32)( 11 ncaybnIN ni ii

COMPUTATIONAL COMPLEXITY FUNCTION

} whereI, instance thesolve tonecessary

machine computing of steps elementary ofnumber theis:max{)(

N(I)n

ttnfA

DEPENDS ON:• Coding rule• Model of calculations (DTM)

FUNDAMENTAL COMPLEXITY CLASSES

Polynomial time algorithm O(p(n)), p – polynomial, solvable by DTM, P class

Exponential time algorithm

NP class, solvable in O(p(n)) on NDTM = solvable in O(2p(n)) on DTM

10 60

n 10-5 s 6·10-5 s

n3 10-3 s 2·10-1 s

n5 10-1 s 13 m

2n 10-3 s 3366 y

NP COMPLETE PROBLEMS

21: PPf

yes)( yes 222 IfPI

)))((( 2INpO

POLYNOMIAL TIME TRANSFORMATION

PROBLEM P1 IS NP-COMPLETE IF P1 BELONGS TO NP CLASS AND FOR ANY P2 FROM NP CLASS, P2 IS POLYNOMIALLY TRANSFORMABLE TO P1

12 PP

PROBLEM IS PSEUDO-POLYNOMIAL (NPI CLASS) IF ITS COMPUTATIONAL COMPLEXITY FUNCTION IS A POLYNOMIAL OF N(I) AND MAX(I)

COMPLEXITY CLASSES

NP CLASS

P CLASS

NPI CLASS NP COMPLETE CLASS

STRONGLY NP COMPLETE CLASS

Thank you for your attention

DISCRETE MATHEMATICSCzesław Smutnicki