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Understanding the Space and Time Hierarchy Theorems in Computational Complexity

This lecture delves into the concepts of space and time hierarchy theorems, exploring space-constructible functions and their implications. It explains how a function is fully space-constructible if a deterministic Turing machine can compute its space requirements for large inputs. The lecture also examines the differences between classes such as DSPACE and DTIME, highlighting critical functions and their relationships. Additionally, it outlines decision problems in the complexity class P and discusses notable problems in EXP, including the Traveling Salesman Problem.

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Understanding the Space and Time Hierarchy Theorems in Computational Complexity

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  1. Lecture 4 Hierarchy Theorem

  2. Space Hierarchy Theorem

  3. Space-constructible function • s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with |x|=n, SpaceM(x) = s(n).

  4. Space Hierarchy If • s2(n) is a fully space-constructible function, • s1(n)/s2(n) → 0 as n → infinity, • s1(n) > log n, then DSPACE(s2(n)) DSPACE(s1(n)) ≠ Φ

  5. Input tape (read only) Storage tapes Output tape (possibly, write only)

  6. Input tape (read only) Storage tapes Output tape (possibly, write only)

  7. Claim Proof

  8. Claim Proof

  9. Time Hierarchy

  10. Time-constructible function • t(n) is fully time-constructible if there exists a DTM such that for sufficiently large n and any input x with |x|=n, TimeM(x) = t(n).

  11. Time Hierarchy If • t1(n) > n+1, • t2(n) is fully time-constructible, • t1(n) log t1(n) /t2(n) → 0 as n → infinity, then DTIME(t2(n)) DTIME(t1(n)) ≠ Φ

  12. Model Independent Classes c • P = U c>0 DTIME(n ) • EXP = U c > 0 DTIME(2 ) • EXPOLY = U c > 0 DTIME(2 ) • PSPACE = U c > 0 DSPACE(n ) cn c n c

  13. P c EXP

  14. EXP ≠ PSAPACE

  15. PSPACE≠EXP

  16. PSPACE c EXPOLY

  17. Problems in P • Are they in P? Sorting minimum spanning tree shortest path maximum flow

  18. Problems in P • None of following is in P? Sorting minimum spanning tree shortest path maximum flow • They are all polynomial-time computable functions

  19. P contains only languages or decision problems • A decision problem is a problem who has only two answers, YES and NO. • A decision problem can be described by a language consisting of all inputs at which YES answer would be obtained.

  20. Every optimization problem has a decision version Minimum spanning tree Decision version of minimum spanning tree

  21. For optimization problem with integer value, the decision version is equivalent to it.

  22. Problem in EXP • Traveling Salesman Problem • Minimum Vertex Cover • Hamiltonian Cycle • Satisfiability • Partition

  23. Edmonds Conjecture in 1965 • Traveling Salesman Problem cannot be solved in polynomial time.

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