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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. Example:. Algorithm to accept a string :. Use a two-tape Turing machine Copy the on the second tape Compare the and. Time needed:.

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Time complexity
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

COMP 335


Example:

Algorithm to accept a string :

  • Use a two-tape Turing machine

  • Copy the on the second tape

  • Compare the and

COMP 335


Time needed:

  • Copy the on the second tape

  • Compare the and

Total time:

COMP 335


For string of length

time needed for acceptance:

COMP 335


Language class:

A Deterministic Turing Machine

accepts each string of length

in time

COMP 335



In a similar way we define the class

for any time function:

Examples:

COMP 335


Example: The membership problem

for context free languages

(CYK - algorithm)

Polynomial time

COMP 335


Theorem:

COMP 335


Polynomial time algorithms:

Represent tractable algorithms:

For small we can compute the

result fast

COMP 335


The class

for all

  • Polynomial time

  • All tractable problems

COMP 335


CYK-algorithm

COMP 335


Exponential time algorithms:

Represent intractable algorithms:

Some problem instances

may take centuries to solve

COMP 335


Example: the Hamiltonian Problem

s

t

Question: is there a Hamiltonian path

from s to t?

COMP 335


s

t

YES!

COMP 335


A solution: search exhaustively all paths

L = {<G,s,t>: there is a Hamiltonian path

in G from s to t}

Exponential time

Intractable problem

COMP 335


Example: The Satisfiability Problem

Boolean expressions in

Conjunctive Normal Form:

Variables

Question: is expression satisfiable?

COMP 335


Example:

Satisfiable:

COMP 335


Example:

Not satisfiable

COMP 335


For variables:

exponential

Algorithm:

search exhaustively all the possible

binary values of the variables

COMP 335


Non determinism
Non-Determinism

Language class:

A Non-Deterministic Turing Machine

accepts each string of length

in time

COMP 335


Example:

Non-Deterministic Algorithm

to accept a string :

  • Use a two-tape Turing machine

  • Guess the middle of the string

  • and copy on the second tape

  • Compare the two tapes

COMP 335


Time needed:

  • Use a two-tape Turing machine

  • Guess the middle of the string

  • and copy on the second tape

  • Compare the two tapes

Total time:

COMP 335



In a similar way we define the class

for any time function:

Examples:

COMP 335



The class

for all

Non-Deterministic Polynomial time

COMP 335


The satisfiability problem

Example:

Non-Deterministic algorithm:

  • Guess an assignment of the variables

  • Check if this is a satisfying assignment

COMP 335


Time for variables:

  • Guess an assignment of the variables

  • Check if this is a satisfying assignment

Total time:

COMP 335



Observation:

Deterministic

Polynomial

Non-Deterministic

Polynomial

COMP 335


Open Problem:

WE DO NOT KNOW THE ANSWER

COMP 335


Open Problem:

Example: Does the Satisfiability problem

have a polynomial time

deterministic algorithm?

WE DO NOT KNOW THE ANSWER

COMP 335


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