1. Fall 2004COMP 3351 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. Fall 2004COMP 3352 Example: Algorithm to accept a string : Use a two-tape Turing machine Copy the on the second tape Compare the and

3. Fall 2004COMP 3353 Copy the on the second tape Compare the and Time needed: Total time:

4. Fall 2004COMP 3354 For string of length time needed for acceptance:

5. Fall 2004COMP 3355 Language class: A Deterministic Turing Machine accepts each string of length in time

6. Fall 2004COMP 3356

7. Fall 2004COMP 3357 In a similar way we define the class for any time function: Examples:

8. Fall 2004COMP 3358 Example: The membership problem for context free languages (CYK - algorithm) Polynomial time

9. Fall 2004COMP 3359 Theorem:

10. Fall 2004COMP 33510 Polynomial time algorithms: Represent tractable algorithms: For small we can compute the result fast

11. Fall 2004COMP 33511 for all The class All tractable problems Polynomial time

12. Fall 2004COMP 33512 CYK-algorithm

13. Fall 2004COMP 33513 Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve

14. Fall 2004COMP 33514 Example: the Hamiltonian Problem Question: is there a Hamiltonian path from s to t? s t

15. Fall 2004COMP 33515 s t YES!

16. Fall 2004COMP 33516 Exponential time Intractable problem A solution: search exhaustively all paths L = { : there is a Hamiltonian path in G from s to t}

17. Fall 2004COMP 33517 Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: Variables Question: is expression satisfiable?

18. Fall 2004COMP 33518 Satisfiable: Example:

19. Fall 2004COMP 33519 Not satisfiable Example:

20. Fall 2004COMP 33520 For variables: Algorithm: search exhaustively all the possible binary values of the variables exponential

21. Fall 2004COMP 33521 Non-Determinism Language class: A Non-Deterministic Turing Machine accepts each string of length in time

22. Fall 2004COMP 33522 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

23. Fall 2004COMP 33523 Time needed: Total time: Use a two-tape Turing machine Guess the middle of the string and copy on the second tape Compare the two tapes

24. Fall 2004COMP 33524

25. Fall 2004COMP 33525 In a similar way we define the class for any time function: Examples:

26. Fall 2004COMP 33526 Non-Deterministic Polynomial time algorithms:

27. Fall 2004COMP 33527 for all The class Non-Deterministic Polynomial time

28. Fall 2004COMP 33528 Example: The satisfiability problem Non-Deterministic algorithm: Guess an assignment of the variables Check if this is a satisfying assignment

29. Fall 2004COMP 33529 Time for variables: Total time: Guess an assignment of the variables Check if this is a satisfying assignment

30. Fall 2004COMP 33530 The satisfiability problem is an - Problem

31. Fall 2004COMP 33531 Observation: Deterministic Polynomial Non-Deterministic Polynomial

32. Fall 2004COMP 33532 Open Problem: WE DO NOT KNOW THE ANSWER

33. Fall 2004COMP 33533 Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER Open Problem: