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Computability

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

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Computability

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  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 title—touches 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: 3COLOR={G|the 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.

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