Abstract
We consider how continuous-time quantum walks can be used for graph matching. We focus in detail on both exact and inexact graph matching, and consider in depth the problem of measuring graph similarity. We commence by constructing an auxiliary graph, in which the two graph to be matched are co-joined by a layer of indicator nodes (one for each potential correspondence between a pair of nodes). We simulate a continuous time quantum walk in parallel on the two graphs. The layer of connecting indicator nodes in the auxiliary graph allow quantum interference to take place between the two walks. The interference amplitudes on the indicator nodes are determined by differences in the two walks. We show how these interference amplitudes can be used to compute graph edit distances without explicitly determining node correspondences.
Chapter PDF
Similar content being viewed by others
References
Babai, L., Grigoryev, D.Y., Mount, D.M.: Isomorphism of graphs with bounded eigenvalue multiplicity. In: STOC 1982: Proceedings of the fourteenth annual ACM symposium on Theory of computing, pp. 310–324. ACM Press, New York (1982)
Barrow, H.G., Burstall, R.M.: Subgraph isomorphism, matching relational structures and maximal cliques. Information Processing Letters 4(4), 83–84 (1976)
Beals, R.: Quantum computation of Fourier transforms over symmetric groups. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computation (STOC), Texas, pp. 48–53. ACM Press, New York (1997)
Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: STOC 2003: Proc. 35th ACM symposium on TOC, pp. 59–68. ACM Press, New York (2003)
McKay, B.D.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)
Emms, D., Hancock, E.R., Wilson, R.C.: Graph similarity using interfering quantum walks. In: Kropatsch, W.G., Kampel, M., Hanbury, A. (eds.) CAIP 2007. LNCS, vol. 4673, pp. 823–831. Springer, Heidelberg (2007)
Emms, D., Wilson, R.C., Hancock, E.R.: Graph Embedding Using Quantum Commute Times. In: Escolano, F., Vento, M. (eds.) GbRPR. LNCS, vol. 4538, pp. 371–382. Springer, Heidelberg (2007)
Ettinger, M., Høyer, P.: A quantum observable for the graph isomorphism problem (1999)
Ettinger, M., Høyer, P.: The hidden subgroup problem and permutation group theory (2004)
Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. 28th Annual ACM Symposium on the Theory of Computation, pp. 212–219. ACM Press, New York (1996)
Hopcroft, J.E., Wong, J.K.: A linear time algorithm for isomorphism of planar graphs. In: Proceedings of the 6th Annual ACM Symposium on the Theory of Computing (STOC 1974), pp. 172–184 (1974)
Jozsa, R.: Quantum factoring, discrete logarithms, and the hidden subgroup problem. Computing in Science and Engineering 03(2), 34–43 (2001)
Kempe, J.: Quantum random walks—an introductory overview. Contemporary Physics 44(4), 307–327 (2003)
Köbler, J.: On graph isomorphism for restricted graph classes. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 241–256. Springer, Heidelberg (2006)
Miyazaki, T.: The complexity of Mckay’s canonical labeling algorithm (1996)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)
Ullmann, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emms, D., Wilson, R.C., Hancock, E.R. (2008). Graph Edit Distance without Correspondence from Continuous-Time Quantum Walks. In: da Vitoria Lobo, N., et al. Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2008. Lecture Notes in Computer Science, vol 5342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89689-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-89689-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89688-3
Online ISBN: 978-3-540-89689-0
eBook Packages: Computer ScienceComputer Science (R0)