Time%20Complexity

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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

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Example
Algorithm to accept a string

• Use a two-tape Turing machine
• Copy the on the second tape
• Compare the and

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Time needed

• Copy the on the second tape
• Compare the and

Total time
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For string of length time needed for acceptance
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Language class
A Deterministic Turing Machine accepts each
string of length in time
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In a similar way we define the class
for any time function
Examples
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Example The membership problem
for context free languages
(CYK - algorithm)
Polynomial time
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Theorem
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Polynomial time algorithms
Represent tractable algorithms
For small we can compute the result fast
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The class
for all

• Polynomial time

• All tractable problems

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CYK-algorithm
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Exponential time algorithms
Represent intractable algorithms
Some problem instances may take centuries to solve
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Example the Hamiltonian Problem
s
t
Question is there a Hamiltonian path
from s to t?
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s
t
YES!
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A solution search exhaustively all paths
L ltG,s,tgt there is a Hamiltonian path
in G from s to t
Exponential time
Intractable problem
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Example The Satisfiability Problem
Boolean expressions in Conjunctive Normal Form
Variables
Question is expression satisfiable?
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Example
Satisfiable
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Example
Not satisfiable
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For variables
exponential
Algorithm search exhaustively all the
possible binary values of the variables
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Non-Determinism
Language class
A Non-Deterministic Turing Machine accepts each
string of length in time
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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

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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
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(No Transcript)
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In a similar way we define the class
for any time function
Examples
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Non-Deterministic Polynomial time algorithms
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The class
for all
Non-Deterministic Polynomial time
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The satisfiability problem
Example
Non-Deterministic algorithm

• Guess an assignment of the variables

• Check if this is a satisfying assignment

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Time for variables

• Guess an assignment of the variables

• Check if this is a satisfying assignment

Total time
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The satisfiability problem is an - Problem
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Observation
Deterministic Polynomial
Non-Deterministic Polynomial
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Open Problem
WE DO NOT KNOW THE ANSWER
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Open Problem
Example Does the Satisfiability problem
have a polynomial time
deterministic algorithm?
WE DO NOT KNOW THE ANSWER