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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 COMP 335

  2. Example: Algorithm to accept a string : • Use a two-tape Turing machine • Copy the on the second tape • Compare the and COMP 335

  3. Time needed: • Copy the on the second tape • Compare the and Total time: COMP 335

  4. For string of length time needed for acceptance: COMP 335

  5. Language class: A Deterministic Turing Machine accepts each string of length in time COMP 335

  6. COMP 335

  7. In a similar way we define the class for any time function: Examples: COMP 335

  8. Example: The membership problem for context free languages (CYK - algorithm) Polynomial time COMP 335

  9. Theorem: COMP 335

  10. Polynomial time algorithms: Represent tractable algorithms: For small we can compute the result fast COMP 335

  11. The class for all • Polynomial time • All tractable problems COMP 335

  12. CYK-algorithm COMP 335

  13. Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve COMP 335

  14. Example: the Hamiltonian Problem s t Question: is there a Hamiltonian path from s to t? COMP 335

  15. s t YES! COMP 335

  16. 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

  17. Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: Variables Question: is expression satisfiable? COMP 335

  18. Example: Satisfiable: COMP 335

  19. Example: Not satisfiable COMP 335

  20. For variables: exponential Algorithm: search exhaustively all the possible binary values of the variables COMP 335

  21. Non-Determinism Language class: A Non-Deterministic Turing Machine accepts each string of length in time COMP 335

  22. 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

  23. 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

  24. COMP 335

  25. In a similar way we define the class for any time function: Examples: COMP 335

  26. Non-Deterministic Polynomial time algorithms: COMP 335

  27. The class for all Non-Deterministic Polynomial time COMP 335

  28. The satisfiability problem Example: Non-Deterministic algorithm: • Guess an assignment of the variables • Check if this is a satisfying assignment COMP 335

  29. Time for variables: • Guess an assignment of the variables • Check if this is a satisfying assignment Total time: COMP 335

  30. The satisfiability problem is an - Problem COMP 335

  31. Observation: Deterministic Polynomial Non-Deterministic Polynomial COMP 335

  32. Open Problem: WE DO NOT KNOW THE ANSWER COMP 335

  33. 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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