Midterm 2 Review

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Midterm 2 Review

• Chris Coakley

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Topics

• Recurrence Relations
• Dynamic Programming
• NP-Complete

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Recurrence Relations

• Anything from lecture -- know it
• Master Method -- Still might apply
• T(n) T(n-2) O(nk)
• T(n) 2T(n/2 - 1) O(n)

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Dynamic Programming

• Matrix Multiply -- almost guaranteed
• Something analogous to problem 4 on HW 2
• Not likely to be like problem 7 on HW 2
• Rule of thumb An O(nk) time takes O(n(k-1))
  space when dynamic programming

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Dynamic Programming

• If T(n) O(nk), k gt 2
• Try sorting first
• Assume you need n(k-1) space
• Solution for 1st element
• Solution for 2 elements (all cases)
• ???
• Profit!

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Dynamic Programming

• Harder case Packed schedules or bins
• Go ahead and sort
• Start with size 1
• Find optimal solution for first element
• Add next element
• Increment size and repeat
• Not as likely to be on exam

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NP-Complete

• No reductions(!?)
• Should be restricted to lecture notes
• Cooks Theorem -- SAT is NP-Complete
• Examples
• 3SAT
• MAX independent set
• Subset Sum
• Partition

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NP Complete

• More Examples
• Vertex Cover
• Knapsack
• Bin Packing
• Hamiltonian Cycle
• TSP

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NP Complete

• Reduction
• Transform every instance of Known to Unknown in
  Polynomial Time
• Unknown is NP-Hard (is it NP-Complete?)

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Bad Reduction

• Unknown into Known
• Is 2SAT NP Complete?
• Reduce to 3SAT
• For each clause, add ( V x)
• At the end, add (!x V !x V !x)
• Call 3SAT( new_problem )
• 2SAT is NP-Hard?

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Good Reduction

• Unknown Bin packing (list, num, size)
• Known Partition (list)
• Reduction
• Add all items ? sum
• Bin_Packing(list, 2, sum/2)

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Knapsack DP

• Its NP-Complete
• Dynamic Programming Solution
• Case 2 -- packing a bin