Computability
History. More examples. NP Hard.
Homework: Presentation topics
due after Thanksgiving.
Recommendation
• The Man Who Invented the Computer:
The Biography of John Atanasoff,
Digital Pioneer by Jane Smiley
• Book is much better than title—touches on
many people: Turing, Flowers (engineer at
Bletchley Park), Von Neumann, Mauchly,
Eckert, Zuse. Includes Appendix on
mathematics topics.
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.
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?
Your examples of NP Complete
problems?
• ?
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: 3COLOR={G|the nodes of G can
be colored by 3 colors} is NP-Complete.
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.
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
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
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.
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
NP hard
… problems are at least as hard as the
hardest problems in NP
Homework
• NP-hard examples
• Next week: watch video
• After holiday, make proposals for
presentations.