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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. 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

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  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

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

  3. Time needed: • Copy the on the second tape • Compare the and Total time:

  4. For string of length time needed for acceptance:

  5. Language class: A Deterministic Turing Machine accepts each string of length in time

  6. In a similar way we define the class for any time function: Examples:

  7. Example: The membership problem for context free languages (CYK - algorithm) Polynomial time

  8. Theorem:

  9. Polynomial time algorithms: Represent tractable algorithms: For small we can compute the result fast

  10. The class for all • Polynomial time • All tractable problems

  11. CYK-algorithm

  12. Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve

  13. Example: the Hamiltonian Problem s t Question: is there a Hamiltonian path from s to t?

  14. s t YES!

  15. 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

  16. Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: Variables Question: is expression satisfiable?

  17. Example: Satisfiable:

  18. Example: Not satisfiable

  19. For variables: exponential Algorithm: search exhaustively all the possible binary values of the variables

  20. Non-Determinism Language class: A Non-Deterministic Turing Machine accepts each string of length in time

  21. 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

  22. 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:

  23. In a similar way we define the class for any time function: Examples:

  24. Non-Deterministic Polynomial time algorithms:

  25. The class for all Non-Deterministic Polynomial time

  26. The satisfiability problem Example: Non-Deterministic algorithm: • Guess an assignment of the variables • Check if this is a satisfying assignment

  27. Time for variables: • Guess an assignment of the variables • Check if this is a satisfying assignment Total time:

  28. The satisfiability problem is an - Problem

  29. Observation: Deterministic Polynomial Non-Deterministic Polynomial

  30. Open Problem: WE DO NOT KNOW THE ANSWER

  31. Open Problem: Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER

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