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CS20a: NP problems

CS20a: NP problems. The SAT problem Graph theory. Time-bounded TMs. All tapes are 2-way infinite M is DTIME(T(n)) if M is deterministic For any input of length n, M takes at most T(n) steps M is NTIME(T(n)) Nondeterministic case. NP problems.

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CS20a: NP problems

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  1. CS20a: NP problems • The SAT problem • Graph theory Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  2. Time-bounded TMs • All tapes are 2-way infinite • M is DTIME(T(n)) if • M is deterministic • For any input of length n, M takes at most T(n) steps • M is NTIME(T(n)) • Nondeterministic case Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  3. NP problems • A problem is in NP if it can be decided yes/no by a nondeterministic TM in O(nc) steps • n is the length of the input • c is a constant • How long does it take to decide a problem in NP? • It takes O(nc) time to explore each possible nondeterministic choice • There are an exponential number of choices • So it takes O(2n) time worst case • How long does it take to verify an NP execution? • There are O(nc) steps of size O(nc) • So it takes O(n2c) time Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  4. CNF satisfiability Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  5. Satisfiability algorithm • CNF should make the decision problem easier, since there is only conjunction, disjunction, negation • An algorithm in NP • Guess a truth assignment • Verify the assignment • An algorithm in P • Must be deterministic (no “guessing”) • Just figure out if the formula is true Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  6. Building an algorithm for 2SAT Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  7. 2SAT graph Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  8. Graph theory • A graph is a set of points (vertices) that are interconnected by a set of lines (edges) Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  9. Some common graphs Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  10. Formal definition of graphs Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  11. Graph definitions Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  12. A simple theorem Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  13. Path definitions Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  14. Path example Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  15. Connected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  16. Graph components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  17. Trees • A tree is a connected graph with no cycles • A forest is a set of trees Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  18. Directed graphs • A directed graph assigns a direction to the edges Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  19. Paths, cycles Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  20. Connected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  21. Graph algorithms: Depth-First-Search (DFS) • DFS does the following: • Assigns a unique number to each vertex • Forms a spanning tree (called the DFS spanning tree) • Basic algorithm • We need a stack of edges (initially empty) • A counter c (initially 1) Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  22. DFS algorithm Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  23. DFS: Initial search Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  24. DFS: Backtracking Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  25. DFS: Finalize Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  26. Biconnected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  27. Biconnected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  28. Transitive case (part 1) Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  29. Transitive case (part 2) Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  30. Articulation points Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  31. Articulation points Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  32. Using DFS to get biconnected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  33. DFS backedges Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  34. DFS biconnected components Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  35. Solving 2SAT • We can define DFS for directed graphs • Follow edges only in the forward direction • Use DFS to determine strongly-connected components • Strongly-connected component • For each u, v, there is a path from u to v, and from v to u Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  36. 2SAT graph Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

  37. 2SAT solution • The formula is satisfiable iff there is a strongly-connected component that contains A and not A for some letter A • If it does, then A implies not A and not A implies A (contradiction) • If not, we have to define a satisfying assignment (next time) Computation, Computers, and Programs Recursive functions http://www.cs.caltech.edu/courses/cs20/a/ March 10, 2014

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