Computability

1
Computability

• History. More examples. NP Hard.
• Homework Presentation topics due after
  Thanksgiving.

2
Recommendation

• The Man Who Invented the Computer The Biography
  of John Atanasoff, Digital Pioneer by Jane
  Smiley
• Book is much better than titletouches on many
  people Turing, Flowers (engineer at Bletchley
  Park), Von Neumann, Mauchly, Eckert, Zuse.
  Includes Appendix on mathematics topics.

3
Very, very brief on History

• Atanasoff (Iowa State) failed to file patent on
  his ABC (Atanasoff Berry computer).
• Mauchly, Eckert (U of Penn) did.
• Von Neumann (Princeton, government) didn't
  believe in patents, ownership. Possibly/probably
  used Turing's idea for his architecture.
• Much later, Honeywell, et al, succeeded in
  overturning patent, largely based on Mauchly's
  failure to acknowledge his use of Atanasoff's
  work.

4
History, book, cont.

• Jane Smiley very good on suggesting connections,
  interdependencies, including the factor of World
  War II helped and hindered effort.
• Another mysterious suicide Clifford Berry, who
  worked with/for Atanasoff.
• Turing's ideas to use binary, do symbolic
  processing, important. May have inspired Von
  Neumann, others.
• Turing's own work to produce an actual computer
  failed.
• Presentation?

5
Your examples of NP Complete problems?

• ?

6
Coloring

• A coloring of a graph is an assignment of colors
  to nodes so that no two adjacent nodes (nodes
  connected by an edge) have the same color.
• Claim 3COLORGthe nodes of G can be colored by
  3 colors is NP-Complete.

7
Maps

• Consider a map of connected regions (countries)
  drawn in a plane. Claim 4 colors is enough to
  color map so no adjacent countries share the same
  color.
• Proved using a computer aided in proof by Appel
  and Haken (1977). Other, more formal proofs,
  followed.
• Problem is NP-complete.

8
Path finding

• Finding a collision free path of a robot through
  a crowded workspace
• Many versions
• 2-d, restrict to convex polygons as the 'robot'
  and the obstacles
• 2-d, allow more complex shapes as obstacles
• .
• 3-D, allow 6-degrees of freedom (angle) of robot

9
More on path finding

• AKA piano mover's, moving sofa, moving ladder,
  etc.
• Schwartz Sharir algorithm that solves a
  two-dimensional case of the following problem
  which arises in robotics Given a body B, and a
  region bounded by a collection of walls, either
  find a continuous motion connecting two given
  positions and orientations of B during which B
  avoids collision with the walls, or else
  establish that no such motion exists. The
  algorithm is polynomial in the number of walls
  (O(n5) if n is the number of walls), but for
  typical wall configurations can run more
  efficiently.
• Other approaches.
• Opportunity for presentation

10
NP-hard

• A problem X is NP-hard if all problems in NP are
  reducible to it.
• The definition doesn't require X to be in NP.
• X may be more difficult than any problems in NP.
• Recall if A is reducible to B, then A is no more
  difficult (time consuming) than B. B may be
  harder (more time consuming) than some solutions
  of A.

11
Examples

• Some variants of path finding are NP-hard.
• Determining if a polynomial in several variables
  has an integral root is not even decidable. It
  is NP-hard.
• Tetris
• No time pressure, given list of shapes, determine
  best sequence of moves to maximize score,
  minimize height
• http//arxiv.org/abs/cs.CC/0210020

12
NP hard

• problems are at least as hard as the hardest
  problems in NP

13
Homework

• NP-hard examples
• Next week watch video
• After holiday, make proposals for presentations.