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(n 5 ) 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.