Time complexity
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This document discusses time complexity classes for deterministic and non-deterministic Turing machines. It begins by defining the class DTIME(f(n)) as the problems decidable by a deterministic Turing machine in time f(n). Examples of problems in lower classes like DTIME(n) and DTIME(n^2) are given. Polynomial time algorithms are classified as tractable and fall within the class P. The document then discusses intractable problems that require exponential time, like the Hamiltonian path and clique problems. It introduces the SAT problem and shows it requires exponential time. Non-determinism is then covered, defining the class NTIME(f(n)) and showing the SAT problem can be solved in non
Time complexity
- 1. © 2020 Wael Badawy 1 Time Complexity 1 DISCLAIMER: This video is optimized for HD large display using patented and patent-pending “Nile Codec”, the first Egyptian Video Codec for more information, PLEASE check https://NileCodec.com Also available as a PodCast
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- 3. © 2020 Wael Badawy
- 4. © 2020 Wael Badawy 4 M L Consider a deterministic Turing Machine which decides a language
- 5. © 2020 Wael Badawy 5 For any string the computation of terminates in a finite amount of transitions w M ! Accept or Reject w Initial state
- 6. © 2020 Wael Badawy 6 ! Accept or Reject w Decision Time = #transitions Initial state
- 7. © 2020 Wael Badawy 7 Consider now all strings of length n )(nTM = maximum time required to decide any string of length n
- 8. © 2020 Wael Badawy 8 ! )(nTM Max time to decide string of length n 1 2 3 4 n ! STRING LENGTH TIME
- 9. © 2020 Wael Badawy 9 Time Complexity Class: ))(( nTTIME All Languages decidable by a deterministic Turing Machine in time 1L 2L 3L ))(( nTO
- 10. © 2020 Wael Badawy 10 Example: }0:{1 ³= nbaL n This can be decided in time)(nO )(nTIME }0:{1 ³= nbaL n
- 11. © 2020 Wael Badawy 11 )(nTIME }0:{1 ³= nbaL n }0,:{ ³knabaabn }evenis:{ nbn Other example problems in the same class }3:{ knbn =
- 12. © 2020 Wael Badawy 12 )( 2 nTIME }0:{ ³nba nn Examples in class: }},{:{ bawww R Î }},{:{ bawww Î
- 13. © 2020 Wael Badawy 13 )( 3 nTIME Examples in class: }grammarfree-context bygeneratedis:,{2 G wwGL = }and matrices:,,{ 321 3213 MMM nnMMML =´ ´= CYK algorithm Matrix multiplication
- 14. © 2020 Wael Badawy 14 Polynomial time algorithms: )( k nTIME Represents tractable algorithms: for small we can decide the result fast k constant 0>k
- 15. © 2020 Wael Badawy 15 )( k nTIME )( 1+k nTIME )()( 1+ Ì kk nTIMEnTIMEIt can be shown:
- 16. © 2020 Wael Badawy 16 !k k nTIMEP )(= The Time Complexity Class P •“tractable” problems •polynomial time algorithms Represents:
- 17. © 2020 Wael Badawy 17 P CYK-algorithm }{ nn ba }{ww Class Matrix multiplication }{ ban
- 18. © 2020 Wael Badawy 18 Exponential time algorithms: )2( k n TIME Represent intractable algorithms: Some problem instances may take centuries to solve
- 19. © 2020 Wael Badawy 19 Example: the Hamiltonian Path Problem Question: is there a Hamiltonian path from s to t? s t
- 20. © 2020 Wael Badawy 20 s t YES!
- 21. © 2020 Wael Badawy 21 )2()!( k n TIMEnTIMEL »Î Exponential time Intractable problem A solution: search exhaustively all paths
- 22. © 2020 Wael Badawy 22 The clique problem Does there exist a clique of size k?
- 23. © 2020 Wael Badawy 23 The clique problem Does there exist a clique of size k?
- 24. © 2020 Wael Badawy 24 Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: ktttt ÙÙÙÙ !321 pi xxxxt ÚÚÚÚ= !321 Variables Question: is the expression satisfiable? clauses
- 25. © 2020 Wael Badawy 25 )()( 3121 xxxx ÚÙÚ Satisfiable: 1,1,0 321 === xxx 1)()( 3121 =ÚÙÚ xxxx Example:
- 26. © 2020 Wael Badawy 26 2121 )( xxxx ÙÙÚ Not satisfiable Example:
- 27. © 2020 Wael Badawy 27 e}satisfiablisexpression:{ wwL = )2( k n TIMEL Î Algorithm: search exhaustively all possible binary values of the variables exponential
- 28. © 2020 Wael Badawy Non-Determinism 28 Language class: ))(( nTNTIME A Non-Deterministic Turing Machine decides each string of length in time 1L 2L 3L ))(( nTO n
- 29. 29 0q iq jq … … … … accept accept reject reject )(nT (deepest leaf) depth All computations of on string w nw =|| M
- 30. © 2020 Wael Badawy 30 Non-Deterministic Polynomial time algorithms: )( k nNTIMELÎ
- 31. © 2020 Wael Badawy 31 !k k nNTIMENP )(= The class NP Non-Deterministic Polynomial time
- 32. © 2020 Wael Badawy 32 Example: The satisfiability problem Non-Deterministic algorithm: •Guess an assignment of the variables e}satisfiablisexpression:{ wwL = •Check if this is a satisfying assignment
- 33. © 2020 Wael Badawy 33 Time for variables:n )(nO e}satisfiablisexpression:{ wwL = Total time: •Guess an assignment of the variables •Check if this is a satisfying assignment )(nO )(nO
- 34. © 2020 Wael Badawy 34 e}satisfiablisexpression:{ wwL = NPLÎ The satisfiability problem is a - ProblemNP
- 35. © 2020 Wael Badawy 35 Observation: NPP Í Deterministic Polynomial Non-Deterministic Polynomial
- 36. © 2020 Wael Badawy 36 Open Problem: ?NPP = WE DO NOT KNOW THE ANSWER
- 37. © 2020 Wael Badawy 37 Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER Open Problem: ?NPP =