1. Golomb Rulers Richard Stroop Virginia Tech Department of Computer Engineering 4/29/11

2. History  The term "Golomb Ruler" is derived from work by Professor Solomon W. Golomb of the University of Southern California.  He is famous for inventing polyominoes, which inspired the game Tetris.

3. What it is?  A Golomb ruler is a ruler with marks at integer positions such that the distance between any two marks is not the same as any other two marks.  Expressed as the points where the marks are made, starting at zero and ending at the length of the ruler.  Example: 0, 6, 15, 43

4. Is it Really?  Difference Triangles  Are used to show clearly if a ruler is a Golomb ruler or not.  Example: 0 6 15 43 ----------------- 6 9 28 15 37 43 All of the values in the bottom triangle are different so this is a Golomb ruler

5. Difference Triangles  This ruler does not work because the distance between 8- 0 and 16-8 are both 8: 0 6 8 16 ---------------- 6 2 8 8 10 16

6. Differences Triangles  Is this a Golomb ruler? 0 1 4 6 11 -------------------- 1 3 2 5 4 5 7 6 10 11

7. Perfection…  A Perfect Golomb ruler contains all of the distances from 0 to the length of the ruler, still without repeating any distances between any pair of marks.  Perfect rulers:  These are the only perfect Golomb rulers! OrderLengthMarks 100 210, 1 330, 1, 3 460, 1, 4, 6

8. Optimize!  Instead of looking for perfect Golomb rulers, Optimal Golomb rulers are sought out.  They can be optimal in terms of maximum order for a given length:  Length 5: 0, 5 -> 0, 1, 5 = 0, 2, 5 = 0, 3, 5 = 0, 4, 5  Length 6: 0, 1, 6 -> 0, 1, 4, 6 = 0, 2, 5, 6  Or minimally short for a given order:  Order 4: 0, 3, 8, 14 -> 0, 1, 4, 6  Order 6: 0, 1, 4, 10, 12, 25 -> 0, 1, 4, 10, 12, 17

9. Challenge, Accepted  Most of the efforts today are being done to optimize the for the smallest length for a given order  SOLVE:  Find the optimal ruler for order 5 (without using the internet)  First one done gets a prize!  I will have my Altera board search for order 12  And my python script solve for order 10  Anyone that beats my script gets a prize too!  Download: code.google.com/p/ccm-golomb-rulers

10. Algorithms  Scientific American  First proposed algorithm  Token Passing  Faster  Shift  Fastest on hardware  Tree – Uses way too much memory  Mine

11. Token Passing

12. Shift  Currently the best!  The algorithm that is running on my FPGA is based on this.  Developed by D. McCracken at Duke University.  You have to know the upper bound or it will search forever.

13. Shift Flowchart:

14. My Code – code.google.com/p/ccm-golomb-rulers  Python  Short and Simple  Uses Push and Pop to emulate GE1  10 marks with 55 as the upper bound takes 46 seconds  11 marks with 72 as the upper bound takes 26 minutes  FPGA  Used memory so I didn’t have to create a push and pop controller  12 marks with 85 as the upper bound takes 86 seconds  13 marks with 106 as the upper bound takes 35 minutes

15. The Best Code – GE3  Technical University of Crete  Optimizes out sections that really don’t need to be checked (past halfway)  Easily parallelized  Multiple Shifts at once!  Can run over a server on multiple FPGAs, giving it the same power as distributed.net  Speed ups of 500 on Virtex5 compared to a Core 2 Duo processor  Projected time to complete N=27 is 7 years, if we had 1000 FPGAs instead 1000 computers working on this problem every day, it would only take 5 days!

16. Applications  Linear Arrays  Maximum accuracy is gained when no two antennas are the same distance apart.  Radio Communications  3rd and 5th Order Interference is canceled.  X-Ray Crystallography  Removes ambiguities.  Self-Orthogonal Codes  Uses the difference triangle as parity encoding vectors.

17. Bibliography  W. T. Rankin. “Optimal Golomb Rulers: An Exhaustive Parallel Search Implementation”. MS Thesis, Department of Electrical Engineering, Duke University, 1993.  A. Dollas, W. Rankin, and D. McCracken. “New Algorithms for Golomb Ruler Evaluation and the Proof of the 19-Mark Ruler”. IEEE Transactions on Information Theory, January 1998.  A. Dollas, E. Sotiriades, and A. Emmanouelides. “Architecture and design of Ge1, A FCCM for golomb ruler derivation”. IEEE Symposium on Field-Programmable Custom Computing Machines, pp. 48–56. April 1998.  E. Sotiriades, A. Dollas, P. Athanas. “Hardware - Software Codesign and Parallel Implementation of a Golomb Ruler Derivation Engine”. Proceedings, International Symposium on Field Programmable Custom Computing Machines,FCCM 2000, pp. 227-235. IEEE Computer Society, 2000.  P. Malakonakis, E. Sotiriades, A Dollas. “GE3: a single FPGA client-server architecture for Golomb ruler derivation”. 2010 International Conference of Field-Programmable Technology, pp.470-473. 2010.  Distributed.net, Massive Internet search for Optimal Golomb Rulers, www.distributed.net.www.distributed.net  Wikipedia articles on Golomb Rulers and related content.  WolframAlpha search on Golomb Rulers.