# Using Optimal Golomb Rulers for Minimizing Collisions in Closed Hashing

• Lars Lundberg
• Kamilla Klonowska
• Göran Gustafsson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3321)

## Abstract

We give conditions for hash table probing which minimize the expected number of collisions. A probing algorithm is determined by a sequence of numbers denoting jumps for an item during multiple collisions. In linear probing, this sequence consists of only ones – for each collision we jump to the next location. To minimize the collisions, it turns out that one should use the Golomb ruler conditions: consecutive partial sums of the jump sequence should be distinct. The commonly used quadratic probing scheme fulfils the Golomb condition for some cases. We define a new probing scheme – Golomb probing – that fulfills the Golomb conditions for a much larger set of cases. Simulations show that Golomb probing is always better than quadratic and linear and in some cases the collisions can be reduced with 25% compared to quadratic and with more than 50% compared to linear.

## References

1. 1.
Atkinson, L.V., Cornah, A.J.: Full period quadratic hashing. International Journal of Computer Mathematics 4(2), 177–189 (1974)
2. 2.
Bloom, G.S., Golomb, S.W.: Applications of Numbered, Undirected Graphs. Proceedings of the IEEE 65(4), 562–571 (April 1977)
3. 3.
Czech, Z.J., Havas, G., Majewski, B.S.: Perfect hashing. Theoretical Computer Science, 1–143 (1997)Google Scholar
4. 4.
Dimitromanolakis, A.: Analysis of the Golomb Ruler and the Sidon Set Problems, and Determination of large, near-optimal Golomb Rulers. Dept. of Electronic and Computer Engineering Technical University of Crete (June 2002)Google Scholar
5. 5.
Gasch, S.: Dealing with Collisions, http://www.fearme.com/misc/alg/node30.html
6. 6.
Hayes, B.: Computing Science: Collective Wisdom. American Scientist 98(2), 118–122 (1998)Google Scholar
7. 7.
Klonowska, K., Lundberg, L., Lennerstad, H.: Using Golomb Rulers for Optimal Recovery Schemes in Fault Tolerant Distributed Computing. In: Proceedings of the 17th International Parallel and Distributed Processing Symposium (IPDPS 2003), Nice, France (April 2003)Google Scholar
8. 8.
Lundberg, L., Haggander, D., Klonowska, K., Svahnberg, C.: Recovery schemes for high availability and high performance distributed real-time computing. In: Proceedings of the 17th International Parallel and Distributed Processing Symposium (IPDPS 2003), Nice, France (April 2003)Google Scholar
9. 9.
Lundberg, L., Svahnberg, C.: Optimal Recovery Schemes for High-Availability Cluster and Distributed Computing. J. Parallel Distrib. Comput. 61(11), 1680–1691 (2001)
10. 10.
11. 11.

## Authors and Affiliations

• Lars Lundberg
• 1