GVSU CIS 263

Week 10 / Day 1

NP

• NP means “Non-deterministic polynomial time”.
  • Algorithm will run in polynomial time, if the algorithm can make an optimal guess at each stage.
  • Easier to think of it as an algorithm whose answer can be verified in polynomial time.
• Of course, all algorithms that run in polynomial time (i.e, “P”) are in NP.
• List some problems not known to be in P:
  • Travelling salesman problem
  • Satisfiability
  • Colorability. (Not the four-color map problem!)
  • Max clique
  • Bin packing
• Similar problems can go from very easy to very hard
  • Schedule jobs to minimize waiting time is in P.
  • Minimize final completion time on a multi-processor is in NP.
• Cook-Levin theorem

Decideability

• The Halting Problem
  • boolean will_halt(string code, string input)
  • What does will_halt(troublemaker, troublemaker) return?

boolean troublemaker(string code, string input) {
   if (will_halt(code, input)) {
      while (true);
   } else {
     return false;
   }
}

• The https://en.wikipedia.org/wiki/Post_correspondence_problem(Post correspondence problem)
  • Consider an alphabet of symbols (e.g., “a”, and “b”)
  • Define two sets of n words
    • A: a ab bba
    • B: baa aa bb
  • Can you find a sequence of indexes i1, i2, i3, … such that such that Ai1, Ai2, Ai3, … = Bi1, Bi2, Bi3, …
  • In this case (3, 2, 3, 1) is a solution.
  • In general, no algorithm is guaranteed to find a solution.
• Determining whether a player has a winning strategy in Magic the Gathering is undecidable.

Week 10 / Day 2

Greedy algorithms

• What is a “greedy” algorithm. Give examples
  • Giving change
  • Job scheduling to minimize average wait time
    • Just schedule shortest job first.
    • Extends to multi-processor system. Given P processors, just sort by time then schedule in batches of P.
    • Note: Minimizing total completion time, although it sounds similar is much more difficult.
  • Big picture:
    • Small changes to problem definition can make big changes in running time.
    • Knowing what problems are NP-complete, and how to do reductions can help you avoid spending a lot of time trying to find an ideal solution when none exists (in practice).
  • Huffman Codes
• Huffman Codes
  • Using 8 bits for each ASCII character is not optimal because some letters are used much more often than others.
  • Let more frequently used letters have shorter codes.
  • Combine trees with lowest total frequency.
  • This algorithm provides optimal tree
  • This is a greedy algorithm because we can make an optimal step by looking at individual nodes (rather than having to consider the complete global picture).
  • Contrast with Max Clique, where there is no “greedy” choice that works (e.g., looking at biggest vertexes first doesn’t help.)
• Where greedy algorithms don’t work: Bin Packing
  • Pack N items of given size into as few containers as possible.
  • (Normalize containers to size 1. Package sizes must then be < 1.)
  • Two versions:
    • Online: Each item must be assigned a container as it arrives.
    • Offline: Get to see the entire set of items before making a decision.
• Is there an online bin packing algorithm that will give an optimal solution (i.e., a solution as good as the optimal off-line solution?)
  • No. Consider M items of 1/2 - ε followed by M items of 1/2 + ε. An optimal on-line algorithm must put each of the first M items in a separate bin, so each of the next M can be paired accordingly. However, a different workload with only the small items must be packed two to a container. The online algorithm has no way of knowing which workload it has currently.
  • Any online algorithm can be forced to use at least 4/3 of the optimal number of bins. Why
    • Have students try to come up with worst-case
    • Think in terms of the “adversary”. Given any online bin packing algorithm, the adversary can construct an input to force an output 4/3 of of optimal or worse.
    • Suppose the workload starts out with 100 “small” boxes (size 1/2 - ε) What must algorithm do so adversary can’t just top and call “foul?”
      • If algorithm pairs the small boxes up, the adversary can switch to large boxes, which will eventually become sub-optimal. (For example, if you stack for small boxes in two piles, adversary will then send 4 large boxes, giving you 6 instead of optimal 4. FAIL.)
      • If algorithm doesn’t pair small boxes, adversary just declars end of input when current state is sub-optimal.
• Next fit: It either fits in previous box or starts a new box: O(n)
  • What is running time for N items? O(n).
  • How bad can this get? 2x. Look at pairs of filled bins. Sum of each pair must be at least 1, otherwise they would have been combined.
  • Is the 2x factor “tight” (i.e., is there a sequence of package sizes that results in a 2opt number of bins?)
    • Have students try to come up with worst-case
    • Alternate boxes of size 1/2 and 2/N. Optimally uses N/4 + 1 boxes. But algorithm will use N/2 boxes.
• First fit: Place item into first box into which it fits.
  • What is running time for N items? Naively O(n^2). Can be done in O(n log n)
  • What is worst case?
    • Have students try to come up with worst-case
    • about 1.7opt.
    • Consider 6M of 1/7 + ε followed by 6M of 1/3 + ε followed by 6M of 1/2 + ε
      • Will use 10M bins (M for the 1/7, 3M for the 1/3 and 6M for the 1/2)
      • Optimal is 6M.
  • Average case for random inputs is only 1.02. Pretty good!.
• Best fit: Place in tightest space.
  • Sounds good; but, has similar worst cases.