Jo Ellis-Monaghan St. Michaels College Colchester, VT 05439 e-mail: jellis-monaghan@smcvt website: academics.smcvt/je

1. 4 3 3 2 1 4 1 2 4 1 3 2 2 Crazy Cubes • Jo Ellis-Monaghan • St. Michaels College • Colchester, VT 05439 • e-mail: jellis-monaghan@smcvt.edu • website: http://academics.smcvt.edu/jellis-monaghan andOther Applications of Graphs and Networks

2. A multiple edge A loop A A A B B B C C C D D D Graphs and Networks A Graph or Network is a set of vertices (dots) with edges (lines) connecting them. Two vertices are adjacent if there is a line between them. The vertices A and B above are adjacent because the edge AB is between them. An edge is incident to each of the vertices which are its end points. The degree of a vertex is the number of edges sticking out from it.

3. The Kevin Bacon Gameor6 Degrees of separation http://www.spub.ksu.edu/issues/v100/FA/n069/fea-making-bacon-fuqua.html Total number of linkable actors: 631275Weighted total of linkable actors: 1860181Average Bacon number: 2.947 Kevin Bacon is not even among the top 1000 most connected actors in Hollywood (1222th). Average Connery Number: 2.706 Data from The Oracle of Bacon at UVA

4. The small world phenomenon • Stanley Milgram sent a series of traceable letters from people in the Midwest to one of two destinations in Boston. The letters could be sent only to someone whom the current holder knew by first name. Milgram kept track of the letters and found a median chain length of about six, thus supporting the notion of "six degrees of separation." http://mathforum.org/mam/04/poster.html

5. Social Networks • Stock Ownership (2001 NY Stock Exchange) • Children’s Social Network • Social Network of Sexual Contacts http://mathforum.org/mam/04/poster.html

6. Infrastructure and Robustness Scale Free Number of vertices Vertex degree JetBlue Distributed Number of vertices Vertex degree MapQuest

7. Start with any matching Switch matching to nonmatching and vice versa A maximal matching! Maximal Matchings in Bipartite Graphs A Bipartite Graph Start at an unmatched vertex on the left End at an unmatched vertex on the right Find an alternating path

8. A E B D C A E B D C Conflict Scheduling Draw edges between classes with conflicting times Color so that adjacent vertices have different colors. Minimum number of colors = minimum required classrooms.

9. Conflict Scheduling • Register Allocation • Assign variables to hardware registers during program execution. Variables conflict with each other if one is used both before and after the other within a short period of time (for instance, within a subroutine). Minimize the use of non-register memory. • Vertices: the different variables • Edges: between variables which conflict with each other • Colors: assignment of registers • Need at least as many registers as the minimum number of colors required! • Frequency Assignment • Assign frequencies to mobile radios and other users of the electromagnetic spectrum. Two customers that are sufficiently close must be assigned different frequencies, while those that are distant can share frequencies. Minimize the number of frequencies. • Vertices: users of mobile radios • Edges: between users whose frequencies might interfere • Colors: assignments of different frequencies • Need at least as many frequencies as the minimum number of colors required!

10. Rectilinear pattern recognitionjoint work with J. Cohn (IBM), R. Snapp and D. Nardi (UVM) • IBM’s objective is to check a chip’s design and find all occurrences of a simple pattern to: • Find possible error spots • Check for already patented segments • Locate particular devices for updating The Haystack The Needle…

11. Pre-Processing BEGIN /* GULP2A CALLED ON THU FEB 21 15:08:23 2002 */ EQUIV 1 1000 MICRON +X,+Y MSGPER -1000000 -1000000 1000000 1000000 0 0 HEADER GYMGL1 'OUTPUT 2002/02/21/14/47/12/cohn' LEVEL PC LEVEL RX CNAME ULTCB8AD CELL ULTCB8AD PRIME PGON N RX 1467923 780300 1468180 780300 1468180 780600 + 1469020 780600 1469020 780300 1469181 780300 1469181 + 781710 1469020 781710 1469020 781400 1468180 781400 + 1468180 781710 1467923 781710 PGON N PC 1468500 782100 1468300 782100 1468300 781700 + 1468260 781700 1468260 780300 1468500 780300 1468500 + 780500 1468380 780500 1468380 781500 1468500 781500 RECT N PC 1467800 780345 1503 298 ENDMSG (Raw data format) Two different layers/rectangles are combined into one layer that contains three shapes; one rectangle (purple) and two polygons (red and blue) Algorithm is cutting edge, and not currently used for this application in industry.

12. Linear time subgraph search for target Both target pattern and entire chip are encoded like this, with the vertices also holding geometric information about the shape they represent. Then we do a depth-first search for the target subgraph. The addition information in the vertices reduces the search to linear time, while the entire chip encoding is theoretically N2 in the number of faces, but practically NlogN.

13. Netlist Layout How do we convert this… … into this?

14. Gate Pin Wire Netlist • A set S of vertices ( the pins) hundreds of thousands. • A partition P1 of the pins (the gates) 2 to 1000 pins per gate, average of about 3.5. • A partition P2 of the pins (the wires) again 2 to 1000 pins per wire, average of about 3.5. • A maximum permitted delay between pairs of pins. Example

15. The Wires

16. The Wiring Space Placement layer-gates/pins go here Vias (vertical connectors) Horizontal wiring layer Up to 12 or so layers Vertical wiring layer

17. The general idea • Place the pins so that pins are in their gates on the placement layer with non-overlapping gates. • Place the wires in the wiring space so that the delay constrains on pairs of pins are met, where delay is proportional to minimum distance within the wiring, and via delay is negligible

18. B D G A F C E H Lots of Problems…. • Identify Congestion • Identify dense substructures from the netlist • Develop a congestion ‘metric’ Congested area Congested area What would be good What often happens

19. Automate Wiring Small Configurations • Some are easy to place and route • Simple left to right logic • No / few loops (circuits) • Uniform, low fan-out • Statistical models work • Some are very difficult • E.g. ‘Crossbar Switches’ • Many loops (circuits) • Non-uniform fan-out • Statistical models don’t work

20. Biomolecular constructions • Nano-Origami: Scientists At Scripps Research Create Single, Clonable Strand Of DNA That Folds Into An Octahedron • A group of scientists at The Scripps Research Institute has designed, constructed, and imaged a single strand of DNA that spontaneously folds into a highly rigid, nanoscale octahedron that is several million times smaller than the length of a standard ruler and about the size of several other common biological structures, such as a small virus or a cellular ribosome. http://www.sciencedaily.com/releases/2004/02/040212082529.htm

21. Assuring cohesion A problem from biomolecular computing—physically constructing graphs by ‘zipping together’ single strands of DNA (not allowed) N. Jonoska, N. Saito, ’02

22. AGGCTC AGGCT GGCTC CTACT TCTAC CTCTA TTCTA DNA sequencing It is very hard in general to “read off’ the sequence of a long strand of DNA. Instead, researchers probe for “snippets” of a fixed length, and read those. The problem then becomes reconstructing the original long strand of DNA from the set of snippets.

23. Enumerating the reconstructions • This leads to a directed graph with the same number of in-arrows as out arrows at each vertex. • The number of reconstructions is then equal to the number of paths through the graph that traverse all the edges in the direction of their arrows.

24. R G W W R B R G R B R W 4 3 Conquering the crazy cubes • The cubes from my puzzle are represented below. B R W R G B G W G G W B 2 1

25. Build the Model • We will model each cube with a multigraph. • The vertices will correspond to the four colors and we connect the corresponding vertices u and v if there is a pair of opposite faces colored u and v. 2 1 3 4

26. 4 3 3 2 1 4 1 2 4 1 3 2 2 Build the Model • Now construct a single multigraph with 4 vertices and the 12 edges, labeling each edge by the cube associated with it. R W 2 1 B G 4 3

27. 4 3 3 2 1 4 1 2 4 1 3 2 2 Characterize a Solution • Suppose the puzzle has a solution. How would it be represented on the final multigraph? One subgraph will represent the front and back of the tower and a second subgraph will represent the sides of the tower. Using an edge in a subgraph corresponds to a positioning of the cube (either front/back or sides).

28. Characterize Front/Back and Left/Right subgraphs What are the restrictions on the subgraphs and how do they relate to the solution? 1. Uses all four vertices. (all four colors) 2. Must contain four edges, one from each cube. (orient each cube) 3. No edge can be used more than once. (can’t use same orientation twice) • Each vertex must be of degree 2. (use that color twice, • one front & one back, or one left & one right)

29. 4 3 3 2 1 4 1 2 4 1 3 2 2 Two Subgraphs satisfying the conditions • Here are two such subgraphs: 3 2 4 2 1 4 3 1 Sides Front/Back Now, stack the cubes using these faces as the front/back and sides. Since each edge represents an orientation, label the edges to determine the orientation.

30. 4 3 3 2 1 4 1 2 4 1 3 2 2 Build the solution stack • Now, stack the cubes using these faces as the front/back and sides. b b r 3 2 f f r l l 4 1 2 4 b r b 3 r f l 1 f l Sides Front/Back ? Is there another solution? ? Is it possible to find a set of subgraphs that use the loops ?

31. “crossed cycle” “loop and triangle” “square cycle” “four loops” “two loops and a C2” “two C2’s” All subgraphs with 4 edges and 4 vertices of degree 2 • Only six types of subgraphs meet the solution requirements:

32. Crossed Cycles Loop and a Triangle “Same”

33. Three loops “Same” Arrangements Two double loops

34. So, look for these types of subgraphs “crossed cycle” “loop and triangle” “square cycle” “four loops” “two loops and a C2” “two C2’s” Crazy Cubes/Instant Insanity slides modified from: Sarah Graham--http://academics.smcvt.edu/jellis-monaghan/combo2/special_topics_presentations.htm Judy Lalani-- http://math.hope.edu/lalani/ScienceDay.ppt