Limits to Computation

1
Limits to Computation

• How do you analyze a new algorithm?
• Put it in the form of existing algorithms that
  you know the analysis.
• For example, given 2 arrays, pair the smallest in
  each array, the next smallest, etc.
• Can be done by sorting each array and pairing
  first, second, etc.
• No worse than O(n log n).

2
Algorithm Reduction

• But is is possible to do pairings faster than O(n
  log n)?
• Look at using pairings for sorting (without using
  sorting)
• The second item of the paired results from
  pairing will tell us the position of the first
  item when it is sorted.
• If we had a pairing algorithm that was faster
  than n log n, then we could take an arbitrary
  array, use an array of the integers 1,.., n
• So if there is a pairing algorithm of O(n) we can
  do sorting in O(n). There is a proof that this
  cannot be done. So pairing must take at least
  O(n log n).

3
Hard Problems

• A hard problem is a problem whose best know
  algorithm to solve it takes exponential time.
• A Hard Algorithm is a problem that runs in
  exponential time.
• It takes ?(cn) for some cgt1, where n is the
  problem size.

4
NP Problems

• Assume you have a computer that finds solutions
  by guessing the answer and checks to see if
  that answer is correct.
• Could be a super parallel computer that can test
  all possible solutions simultaneously.
• If the answer cannot be checked in polynomial
  time (such as the towers of Hanoi), it is not an
  NP problem.

5
NP - Completeness

• The algorithm using guessing is a
  nondeterministic algorithm.
• Hence NP Nondeterministic polynomial.
• Some NP problems do not have known efficient
  deterministic solutions.
• Every problem in this set reduces to every other
  problem in this set.
• If can solve one in polynomial time, can solve
  them all in polynomial time.

6
NP- Complete Problems

• Traveling Salesman
• K-Clique given an arbitrary undirected graph,
  is there a complete subgraph of at least k
  vertices.

7
Why know about NP-complete

• If the best algorithm you can find for your
  problem is exponential then you may try to find a
  reduction into a problem that is NP-complete.
• If you can, then you have shown that whereas your
  solution has not been shown to be the best, it is
  very difficult (and maybe impossible) to find a
  polynomial algorithm.

8
Heuristic

• What to do when you must find a solution to an
  NP-complete problem?
• Solve small problems
• Solve special cases
• Find an approximate solution one that is good,
  but not necessarily the best. (I.e., find a short
  path for the traveling salesman, not necessarily
  the shortest).
• Go the to next closest city.

9
Impossible Problems

• The halting problem given an arbitrary program
  P and the data D, determine (in finite time) if
  the program P will halt on data D.
• Can prove that such a program cannot be written

10
Halting Problem Proof

• Assume you have a halting program
• bool halt(P,D)
• Write a new program
• bool selfhalt(P), calls halt(P, P)
• This program will tell if the program P will halt
  when the input is the program P
• P is restricted to programs that take programs as
  data

11
Final Step

• Now write a function
• bool contrary(P)if selfhalt(P) while(TRUE)
• What happens when we do
• contrary(contrary)?
• Either contrary
• 1. Halts, but then it does not halt
• 2. Does not halt, but then it halts.
• Contradiction so cannot write halt.