Definitions Problem Reduction NP Completeness Proofs Readings and References

Definitions problem reduction np completeness proofs

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Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References Turing Machine (2/3) Turing sought the most primitive model of a computing device Device should have some basic capabilities as human computer! Tape Stores input, output, and intermediate results 1 arbitrarily long strip, divided into cells Finite alphabet of symbols ... A # C & Y B A # ... Tape head Points to one cell of tape Reads a symbol from active cell Overwrites a symbol to active cell Moves left or right one cell at a time COMP 6651 / Fall 2013 , Dr. B. Jaumard 14
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. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References Turing Machine (3/3) In a deterministic Turing machine , the set of rules prescribes at most one action to be performed for any given situation. A non-deterministic Turing machine (NTM) , by contrast, may have a set of rules that prescribes more than one action for a given situation. Example : a non-deterministic Turing machine may have both If you are in state 2 and you see an A , change it to a B and move left If you are in state 2 and you see an A , change it to a C and move right. in its rule set. At each step in the computation performed by an NTM, instead of having a single transition rule, there are multiple rules that can be invoked. To determine if the NTM accepts or rejects, you look at all possible branches of the computation. So if there are, say, exactly 2 transitions to choose from at each step, and each computation branch has a total of N steps, then there will be 2N total branches to consider. COMP 6651 / Fall 2013 , Dr. B. Jaumard 15
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. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References Problem Reduction Suppose you have a problem P2 which you know how to solve, e.g., by using algorithm A2. Suppose you are given another problem P1 that seems similar to P2. How might you solve P1? You could try to solve P1 from scratch. You could try to borrow elements of A2. You could try to find a reduction from P1 to P2. A reduction of P1 to P2: Transforms inputs to P1 into inputs to P2; Runs A2 (which solves P2) as a black-box ; and Interprets the outputs from A2 as answers to P1. COMP 6651 / Fall 2013 , Dr. B. Jaumard 16
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. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References Reduction (2/3) More formally, A problem P1 is reducible to a problem P2 if there is a function f that takes any input x to P1 and transforms it to an input f ( x ) of P2, such that the solution to P2 on f ( x ) is the solution to P1 on x . COMP 6651 / Fall 2013 , Dr. B. Jaumard 17
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. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References Reduction: An example (3/3) Matrix Multiplication Parameters: Two matrices, M1 and M2 Returns: The result of multiplying M1 and M2 together. Squaring a Matrix Parameters: A matrix, M. Returns: The result of squaring M.
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  • Fall '09
  • Computational complexity theory, NP-complete problems, NP-complete, Professor B. Jaumard

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