Fall 2004 COMP 335 1

Time Complexity

•We use a multitape Turing machine

•We count the number of steps until a string is accepted

•We use the O(k) notation

Fall 2004 COMP 335 2

Example: }0:{ nbaL nn

Algorithm to accept a string :w

•Use a two-tape Turing machine

•Copy the on the second tape

•Compare the and

a

a b

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|)(|wO•Copy the on the second tape

•Compare the and

a

a b

Time needed:

|)(|wO

Total time: |)(|wO

}0:{ nbaL nn

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For string of length

time needed for acceptance:

}0:{ nbaL nn

n

)(nO

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Language class: )(nDTIME

A Deterministic Turing Machine accepts each string of length in time

)(nDTIME

1L2L 3L

)(nOn

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)(nDTIME

}0:{ nba nn

}{ww

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In a similar way we define the class

))(( nTDTIME

)(nTfor any time function:

Examples: ),...(),( 32 nDTIMEnDTIME

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Example: The membership problem for context free languages

}grammar by generated is :{ GwwL

)( 3nDTIMEL (CYK - algorithm)

Polynomial time

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Theorem:

)( knDTIME

)( 1knDTIME

)()( 1 kk nDTIMEnDTIME

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Polynomial time algorithms: )( knDTIME

Represent tractable algorithms:

For small we can compute theresult fast

k

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)( knDTIMEP for all k

The class P

•All tractable problems

•Polynomial time

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P

CYK-algorithm}{ nnba

}{ww

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Exponential time algorithms: )2( nDTIME

Represent intractable algorithms:

Some problem instancesmay take centuries to solve

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Example: the Hamiltonian 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()!( nDTIMEnDTIMEL

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}

Fall 2004 COMP 335 17

Example: The Satisfiability Problem

Boolean expressions inConjunctive Normal Form:

ktttt 321

pi xxxxt 321

Variables

Question: is expression satisfiable?

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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( nDTIMELFor variables: n

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

exponential

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Non-Determinism

Language class: )(nNTIME

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

)(nNTIME

1L2L 3L

)(nOn

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Example: }{wwL

Non-Deterministic Algorithmto accept a string :ww

•Use a two-tape Turing machine

•Guess the middle of the string and copy on the second tape

•Compare the two tapes

w

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|)(|wO

Time needed:

|)(|wO

Total time: |)(|wO

}{wwL

•Use a two-tape Turing machine

•Guess the middle of the string and copy on the second tape

•Compare the two tapes

w

Fall 2004 COMP 335 24

)(nNTIME

}{wwL

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In a similar way we define the class

))(( nTNTIME

)(nTfor any time function:

Examples: ),...(),( 32 nNTIMEnNTIME

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Non-Deterministic Polynomial time algorithms:

)( knNTIMEL

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)( knNTIMENP for all k

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 an - 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