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Suppose to the contrary that v1 , . . , vn is a cycle in the underlying graph. Since a cannot be on this cycle and by the above observation, each of the cycle’s vertices is reachable from a. , let v1 be the farthest from a in Gi among all cycle vertices. Then one can obtain a smaller solution by removing one of the edges (v1 , v2 ), (vn , v1 ) (depending on their orientations), a contradiction. As noted earlier, when k = 1 the MKL problem is equivalent to the minimum directed Steiner tree problem, which is known to be NP-complete [5].

In case of two fragments from each protein structure, we call them remote fragment pair. Here, we require the pair of remote fragments to be of the same size and to be at least three residues away from each other to avoid modeling information redundant to the local fragments. 1. Since the initial set of ROTRANs may contain a great deal of noise, we try to filter out most of the noise with a star-like k-median clustering algorithm in the second step. Assuming that we know the maximum distance between the Towards Reliable Automatic Protein Structure Alignment 21 median of a cluster and any member of the same cluster, an approximate clustering is applied using a neighbor graph: each vertex represents a rotation matrix, and two vertices are connected if and only if the distance between them is at most .

Pp. 25–40 (1996) 3. : SARFing the PDB. Protein Eng. 9(9), 727–732 (1996) 4. : Structural alignment of large-size proteins via lagrangian relaxation. In: RECOMB 2002: Proceedings of the Sixth Annual International Conference on Computational Biology, pp. 100–108. ACM, New York (2002) 5. : Proust: a comparison method of threedimensional structure of proteins using indexing techniques. Journal of Computational Biology 11, 1061–1072 (2004) Towards Reliable Automatic Protein Structure Alignment 31 6.

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Algorithms for Comm. Systs and Their Applns by N. Benvenuto, et al.,

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