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## PowerPoint Slideshow about ' Time Complexity' - hamish-medina

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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

COMP 335

Algorithm to accept a string :

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

COMP 335

Example: The membership problem

for context free languages

(CYK - algorithm)

Polynomial time

COMP 335

COMP 335

Represent tractable algorithms:

For small we can compute the

result fast

COMP 335

COMP 335

Represent intractable algorithms:

Some problem instances

may take centuries to solve

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

exponential

Algorithm:

search exhaustively all the possible

binary values of the variables

COMP 335

Non-Determinism

Language class:

A Non-Deterministic Turing Machine

accepts each string of length

in time

COMP 335

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

- 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

Example:

Non-Deterministic algorithm:

- Guess an assignment of the variables

- Check if this is a satisfying assignment

COMP 335

- Guess an assignment of the variables

- Check if this is a satisfying assignment

Total time:

COMP 335

Example: Does the Satisfiability problem

have a polynomial time

deterministic algorithm?

WE DO NOT KNOW THE ANSWER

COMP 335

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