Abstract
In this paper we consider how coined quantum walks can be applied to graph matching problems. The matching problem is abstracted using an auxiliary graph that connects pairs of vertices from the graphs to be matched by way of auxiliary vertices. A coined quantum walk is simulated on this auxiliary graph and the quantum interference on the auxiliary vertices indicates possible matches. When dealing with graphs for which there is no exact match, the interference amplitudes together with edge consistencies are used to define a consistency measure. We have tested the algorithm on graphs derived from the NCI molecule database and found it to significantly reduce the space of possible matchings thereby allowing the graphs to be matched directly. An analysis of the quantum walk in the presence of structural errors between graphs is used as the basis of the consistency measure. We test the performance of this measure on graphs derived from images in the COIL-100 database.
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References
US National Cancer Institute Database (2006), http://resresources.nci.nih.gov/database.cfm?id=1231
Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: A fast quantum mechanical algorithm for database search. In: Proc. 28th ACM Symp on Theory of Computation, pp. 50–59. ACM Press, New York (1996)
Barrow, H.G., Burstall, R.M.: Subgraph isomorphism, matching relational structures and maximal cliques. Information Processing Letters 4(4), 83–84 (1976)
Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30(1–7), 107–117 (1998)
Emms, D., Hancock, E., Severini, S., Wilson, R.C.: A matrix representation of graphs and its spectrum as a graph invariantn. Electronic Journal of Combinatorics 13(1), R34 (2006)
Emms, D., Severini, S., Wilson, R.C., Hancock, E.: Coined quantum walks lift the co-spectrality of graphs and trees. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 332–345. Springer, Heidelberg (2005)
Gori, M., Maggini, M., Sarti, L.: Graph matching using random walks. In: IEEE 17th ICPR, August 2004 (2004)
Grover, L.: A fast quantum mechanical algorithm for database search. In: STOC ’96: Proc. 28th ACM Theory of computing, pp. 212–219. ACM Press, New York (1996)
Harris, C., Stephens, M.: A combined corner and edge detector. In: Proc. of 4th Alvey Vision Conference, Manchester, vol. 15, pp. 147–151 (1988)
Kempe, J.: Quantum random walks – an introductory overview. Contemporary Physics 44(4), 307–327 (2003)
Meila, M., Shi, J.: A random walks view of spectral segmentation (2001)
Nadler, B., Lafon, S., Coifman, R., Kevrekidis, I.: Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators. In: Advances in Neural Information Processing Systems 18, MIT Press, Cambridge (2006)
Nene, S.A., Nayar, S.K., Murase, H.: Columbia object image library (coil-100) (1996)
Nielson, M., Chuang, I.: Quantum Computing and Quantum Information. Cambridge University Press, Cambridge (2000)
Robles-Kelly, A., Hancock, E.: Graph edit distance from spectral seriation. IEEE Transactions on Pattern Analysis and Machine Intelligence 27, 365–378 (2005)
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Emms, D., Hancock, E.R., Wilson, R.C. (2007). A Correspondence Measure for Graph Matching Using the Discrete Quantum Walk. In: Escolano, F., Vento, M. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2007. Lecture Notes in Computer Science, vol 4538. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72903-7_8
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DOI: https://doi.org/10.1007/978-3-540-72903-7_8
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