Abstract
Classical approaches to routing problems often employ construction of trees and the use of heuristics to prevent combinatorial explosion. The algebraic approach presented herein, however, allows such explicit tree constructions to be avoided. Introduced here is the notion of generalized zeon algebras. Their inherent combinatorial properties make them useful for routing problems by implicitly pruning the underlying tree structures. Through the use of generalized idempotent algebras, max-min operators can be implemented for non-additive weights. Moreover, these algebras occur as subalgebras of Clifford algebras, lending them a natural connection to quantum probability and (by extension) to quantum computing.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellman R.: On a routing problem. Q. Appl. Math. 16, 87–90 (1958)
Ben Slimane, J., Song, Y-Q., Koubaa, A., Frikha, M.: Joint duty cycle scheduling, resource allocation and multi-constrained QoS routing algorithm. Lecture Notes in Computer Science, vol. 6811, Ad-hoc, Mobile, and Wireless Networks, pp. 29–43, 10th International Conference, ADHOC-NOW 2011, PADERBORN (Germany) (2011)
Ben Slimane, J., Schott, R., Song, Y-Q., Staples, G.S., Tsiontsiou, E.: Operator calculus algorithms for multi-constrained paths. Int. J. Math. Comput. Sci. 10, 69–104 (2015). http://ijmcs.future-in-tech.net/10.1/R-Jamila
Corley, H.W., Moon, I.D.: Shortest paths in networks with vector weights. J. Optim. Theory Appl. 46, 79–86 (1985). http://dx.doi.org/10.1007/BF00938761
Cruz-Sánchez, H., Staples, G.S., Schott, R., Song, Y-Q.: Operator calculus approach to minimal paths: precomputed routing in a store-and-forward satellite constellation. In: Proceedings of IEEE Globecom 2012, Anaheim, CA, USA, December 3–7, pp. 3455–3460. ISBN:978-1-4673-0919-6
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959). http://dx.doi.org/10.1007/BF01386390
Ford Jr, L.R., Fulkerson, D.R.: Flows in Networks, Princeton University Press, Princeton (1962)
Garroppo R.G., Giordano S., Tavanti L.: A survey on multi-constrained optimal path computation: exact and approximate algorithms. Comput. Netw. 54, 3081–3107 (2010)
Kuipers F.A.: Quality of Service Routing in the Internet: Theory, Complexity and Algorithms. Delft University Press, The Netherlands (2004)
Linwang Y., Zhaoyuan Y., Wen L.: Clifford algebra method for network expression, computation and algorithm construction. Math. Methods Appl. Sci. 37, 1428–1435 (2014)
Nefzi, B., Schott, R., Song, Y.-Q., Staples, G.S., Tsiontsiou, E.: An operator calculus approach for multi-constrained routing in wireless sensor networks. In: Proceedings of the 16th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 367–376. ACM, New York (2015). doi: 10.1145/2746285.2746301
Schott R., Staples G.S.: Operator Calculus on Graphs (Theory and Applications in Computer Science). Imperial College Press, London (2012)
Schott R., Staples G.S.: Nilpotent adjacency matrices and random graphs. Ars Comb. 98, 225–239 (2011)
Schott R., Staples G.S.: Nilpotent adjacency matrices, random graphs, and quantum random variables. J. Phys. A: Math. Theor. 41, 155205 (2008)
Sobrinho, J.L.: Algebra and algorithm for QoS path computation and hop-by-hop routing in the Internet. IEEE ACM Trans. Netw. 10, 541–550 (2002). http://dx.doi.org/10.1109/TNET.2002.801397
Staples G.S.: A new adjacency matrix for finite graphs. Adv. Appl. Clifford Algebras 18, 997–991 (2008)
Van Mieghem P., De Neve H., Kuipers F.A.: Hop-by-hop quality of service routing. Comput. Netw. 37, 407–423 (2001)
Van Mieghem P., Kuipers F.A.: Concepts of exact QoS routing Algorithms. IEEE/ACM Trans. Netw. 12, 851–864 (2004)
Yuan L., Yu Z., Luo W., Yi L., Lv G.: Geometric algebra for multidimension-unified geographical information system. Adv. Appl. Clifford Algebras 23, 497–518 (2013)
Zhaoyuan Y., Linwang Y., Wen L.: Clifford algebra and GIS spatial analysis algorithms—the case study of geographical network and voronoi analysis. Proc. GraVisMa 2, 155–158 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schott, R., Staples, G.S. Generalized Zeon Algebras: Theory and Application to Multi-Constrained Path Problems. Adv. Appl. Clifford Algebras 27, 45–57 (2017). https://doi.org/10.1007/s00006-015-0595-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-015-0595-0