1. Prof. Busch - LSU1 Time Complexity

2. Prof. Busch - LSU2 Consider a deterministic Turing Machine which decides a language

3. Prof. Busch - LSU3 For any string the computation of terminates in a finite amount of transitions Accept or Reject Initial state

4. Prof. Busch - LSU4 Accept or Reject Decision Time = #transitions Initial state

5. Prof. Busch - LSU5 Consider now all strings of length = maximum time required to decide any string of length

6. Prof. Busch - LSU6 Max time to accept a string of length STRING LENGTH TIME

7. Prof. Busch - LSU7 Time Complexity Class: All Languages decidable by a deterministic Turing Machine in time

8. Prof. Busch - LSU8 Example: This can be decided in time

9. Prof. Busch - LSU9 Other example problems in the same class

10. Prof. Busch - LSU10 Examples in class:

11. Prof. Busch - LSU11 Examples in class: CYK algorithm Matrix multiplication

12. Prof. Busch - LSU12 Polynomial time algorithms: Represents tractable algorithms: for small we can decide the result fast constant

13. Prof. Busch - LSU13 It can be shown:

14. Prof. Busch - LSU14 The Time Complexity Class “tractable” problems polynomial time algorithms Represents:

15. Prof. Busch - LSU15 CYK-algorithm Class Matrix multiplication

16. Prof. Busch - LSU16 Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve

17. Prof. Busch - LSU17 Example: the Hamiltonian Path Problem Question: is there a Hamiltonian path from s to t? s t

18. Prof. Busch - LSU18 s t YES!

19. Prof. Busch - LSU19 Exponential time Intractable problem A solution: search exhaustively all paths L = { : there is a Hamiltonian path in G from s to t}

20. Prof. Busch - LSU20 The clique problem Does there exist a clique of size 5?

21. Prof. Busch - LSU21 The clique problem Does there exist a clique of size 5?

22. Prof. Busch - LSU22 Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: Variables Question: is the expression satisfiable? clauses

23. Prof. Busch - LSU23 Satisfiable: Example:

24. Prof. Busch - LSU24 Not satisfiable Example:

25. Prof. Busch - LSU25 Algorithm: search exhaustively all the possible binary values of the variables exponential

26. Prof. Busch - LSU26 Non-Determinism Language class: A Non-Deterministic Turing Machine decides each string of length in time

27. Prof. Busch - LSU27 Non-Deterministic Polynomial time algorithms:

28. Prof. Busch - LSU28 The class Non-Deterministic Polynomial time

29. Prof. Busch - LSU29 Example: The satisfiability problem Non-Deterministic algorithm: Guess an assignment of the variables Check if this is a satisfying assignment

30. Prof. Busch - LSU30 Time for variables: Total time: Guess an assignment of the variables Check if this is a satisfying assignment

31. Prof. Busch - LSU31 The satisfiability problem is an - Problem

32. Prof. Busch - LSU32 Observation: Deterministic Polynomial Non-Deterministic Polynomial

33. Prof. Busch - LSU33 Open Problem: WE DO NOT KNOW THE ANSWER

34. Prof. Busch - LSU34 Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER Open Problem: