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Time Complexity

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ML

Consider a deterministic Turing Machinewhich decides a language

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For any string the computation ofterminates in a finite amount of transitions

w M

Acceptor Reject w

Initialstate

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Acceptor Reject w

Decision Time = #transitions

Initialstate

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Consider now all strings of lengthn

)(nTM = maximum time required to decide any string of length n

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)(nTM

Max time to accept a string of length n

1 2 3 4 n STRING LENGTH

TIM

E

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Time Complexity Class: ))(( nTTIME

All Languages decidable by a deterministic Turing Machinein time

1L 2L

3L

))(( nTO

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Example: }0:{1 nbaL n

This can be decided in time)(nO

)(nTIME

}0:{1 nbaL n

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)(nTIME

}0:{1 nbaL n

}0,:{ knabaabn

}even is :{ nbn

Other example problems in the same class

}3:{ knbn

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)( 2nTIME

}0:{ nba nn

Examples in class:

}},{:{ bawwwR

}},{:{ bawww

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)( 3nTIMEExamples in class:

} grammar free-context by generated is :,{2

GwwGL

} and matrices :,,{

321

3213

MMMnnMMML

CYK algorithm

Matrix multiplication

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Polynomial time algorithms: )( knTIME

Represents tractable algorithms:for small we can decidethe result fast

k

constant 0k

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)( knTIME

)( 1knTIME

)()( 1 kk nTIMEnTIMEIt can be shown:

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k

knTIMEP )(

The Time Complexity Class P

•“tractable” problems

•polynomial time algorithmsRepresents:

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P

CYK-algorithm

}{ nnba }{ww

Class

Matrix multiplication

}{ ban

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Exponential time algorithms: )2( knTIME

Represent intractable algorithms:Some problem instancesmay take centuries to solve

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Example: the Hamiltonian Path Problem

Question: is there a Hamiltonian path from s to t?

s t

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s t

YES!

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)2()!( knTIMEnTIMEL

Exponential time

Intractable problem

A solution: search exhaustively all paths

L = {<G,s,t>: there is a Hamiltonian path in G from s to t}

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The clique problem

Does there exist a clique of size 5?

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The clique problem

Does there exist a clique of size 5?

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Example: The Satisfiability Problem

Boolean expressions inConjunctive Normal Form:

ktttt 321

pi xxxxt 321Variables

Question: is the expression satisfiable?

clauses

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)()( 3121 xxxx

Satisfiable: 1,1,0 321 xxx

1)()( 3121 xxxx

Example:

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2121 )( xxxx

Not satisfiable

Example:

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e}satisfiabl is expression :{ wwL

)2( knTIMEL

Algorithm: search exhaustively all the possible binary values of the variables

exponential

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Non-DeterminismLanguage class: ))(( nTNTIME

A Non-Deterministic Turing Machine decides each string of length in time

1L

2L3L

))(( nTOn

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0q

iq jq

… … …

… accept

accept

reject

reject

)(nT

(deepest leaf)

depth

All computations of on stringw

nw ||

M

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Non-Deterministic Polynomial time algorithms:

)( knNTIMEL

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k

knNTIMENP )(

The class NP

Non-Deterministic Polynomial time

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Example: The satisfiability problem

Non-Deterministic algorithm:•Guess an assignment of the variables

e}satisfiabl is expression :{ wwL

•Check if this is a satisfying assignment

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Time for variables: n

)(nO

e}satisfiabl is expression :{ wwL

Total time:

•Guess an assignment of the variables

•Check if this is a satisfying assignment )(nO

)(nO

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e}satisfiabl is expression :{ wwL

NPL

The satisfiability problem is a - ProblemNP

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Observation:

NPP

DeterministicPolynomial

Non-DeterministicPolynomial

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Open Problem: ?NPP

WE DO NOT KNOW THE ANSWER

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Example: Does the Satisfiability problem have a polynomial time deterministic algorithm?

WE DO NOT KNOW THE ANSWER

Open Problem: ?NPP