Some Sequence of Wrapped Δ-Labellings for the Complete Bipartite Graph

ABSTRACT

The design of large disk array architectures leads to interesting combinatorial problems. Minimizing the number of disk operations when writing to consecutive disks leads to the concept of “cluttered orderings” which were introduced for the complete graph by Cohen*et al*. (2001). Mueller *et al*. (2005) adapted the concept of wrapped Δ-labellings to the complete bipartite case. In this paper, we give some sequence in order to generate wrapped Δ-labellings as cluttered orderings for the complete bipartite graph. New sequence we give is different from the sequences Mueller *et al*. gave, though the same graphs in which these sequences are labeled.

The design of large disk array architectures leads to interesting combinatorial problems. Minimizing the number of disk operations when writing to consecutive disks leads to the concept of “cluttered orderings” which were introduced for the complete graph by Cohen

Cite this paper

Adachi, T. and Kikuchi, D. (2015) Some Sequence of Wrapped Δ-Labellings for the Complete Bipartite Graph.*Applied Mathematics*, **6**, 195-205. doi: 10.4236/am.2015.61019.

Adachi, T. and Kikuchi, D. (2015) Some Sequence of Wrapped Δ-Labellings for the Complete Bipartite Graph.

References

[1] Hellerstein, L., Gibson, G., Karp, R., Katz, R. and Patterson, D. (1994) Coding Techniques for Handling Failures in Large Disk Arrays. Algorithmica, 12, 182-208.

http://dx.doi.org/10.1007/BF01185210

[2] Chen, P., Lee, E., Gibson, G., Katz, R. and Ptterson, D. (1994) RAID: High-Performance, Reliable Secondary Storage. ACM Computing Surveys, 26, 145-185.

http://dx.doi.org/10.1145/176979.176981

[3] Chee, Y., Colbourn, C. and Ling, A. (2000) Asymptotically Optimal Erasure-Resilient Codes for Large Disk Arrays. Discrete Applied Mathematics, 102, 3-36.

http://dx.doi.org/10.1016/S0166-218X(99)00228-0

[4] Cohen, M. and Colbourn, C. (2000) Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays. Lecture Notes in Computer Science, 1776, 95-104.

http://dx.doi.org/10.1007/10719839_10

[5] Cohen, M. and Colbourn, C. (2001) Ordering Disks for Double Erasure Codes. Proceedings of the Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures, 229-236.

[6] Cohen, M., Colbourn, C. and Froncek, D. (2001) Cluttered Orderings for the Complete Graph. Lecture Notes in Computer Science, 2108, 420-431.

http://dx.doi.org/10.1007/3-540-44679-6_48

[7] Cohen, M. and Colbourn, C. (2004) Ladder Orderings of Pairs and RAID Performance. Discrete Applied Mathematics, 138, 35-46.

http://dx.doi.org/10.1016/S0166-218X(03)00268-3

[8] Adachi, T. and Uehara, H. (2014) Construction of Wrapped ρ-Labellings for RAID. Journal of Mathematics and System Science, 4, 750-754.

[9] Mueller, M., Adachi, T. and Jimbo, M. (2005) Cluttered Orderings for the Complete Bipartite Graph. Discrete Applied Mathematics, 152, 213-228.

http://dx.doi.org/10.1016/j.dam.2005.06.005

[10] Adachi, T. (2007) Optimal Ordering of the Complete Tripartite Graph K9,9,9. Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis, 1-10.

[11] Bosák, J. (1990) Decompositions of Graphs. Kluwer Academic Publishers, Dordrecht.

[1] Hellerstein, L., Gibson, G., Karp, R., Katz, R. and Patterson, D. (1994) Coding Techniques for Handling Failures in Large Disk Arrays. Algorithmica, 12, 182-208.

http://dx.doi.org/10.1007/BF01185210

[2] Chen, P., Lee, E., Gibson, G., Katz, R. and Ptterson, D. (1994) RAID: High-Performance, Reliable Secondary Storage. ACM Computing Surveys, 26, 145-185.

http://dx.doi.org/10.1145/176979.176981

[3] Chee, Y., Colbourn, C. and Ling, A. (2000) Asymptotically Optimal Erasure-Resilient Codes for Large Disk Arrays. Discrete Applied Mathematics, 102, 3-36.

http://dx.doi.org/10.1016/S0166-218X(99)00228-0

[4] Cohen, M. and Colbourn, C. (2000) Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays. Lecture Notes in Computer Science, 1776, 95-104.

http://dx.doi.org/10.1007/10719839_10

[5] Cohen, M. and Colbourn, C. (2001) Ordering Disks for Double Erasure Codes. Proceedings of the Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures, 229-236.

[6] Cohen, M., Colbourn, C. and Froncek, D. (2001) Cluttered Orderings for the Complete Graph. Lecture Notes in Computer Science, 2108, 420-431.

http://dx.doi.org/10.1007/3-540-44679-6_48

[7] Cohen, M. and Colbourn, C. (2004) Ladder Orderings of Pairs and RAID Performance. Discrete Applied Mathematics, 138, 35-46.

http://dx.doi.org/10.1016/S0166-218X(03)00268-3

[8] Adachi, T. and Uehara, H. (2014) Construction of Wrapped ρ-Labellings for RAID. Journal of Mathematics and System Science, 4, 750-754.

[9] Mueller, M., Adachi, T. and Jimbo, M. (2005) Cluttered Orderings for the Complete Bipartite Graph. Discrete Applied Mathematics, 152, 213-228.

http://dx.doi.org/10.1016/j.dam.2005.06.005

[10] Adachi, T. (2007) Optimal Ordering of the Complete Tripartite Graph K9,9,9. Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis, 1-10.

[11] Bosák, J. (1990) Decompositions of Graphs. Kluwer Academic Publishers, Dordrecht.