Least-Squares Fitting of Two 3-D Point Sets
IEEE Transactions on Pattern Analysis and Machine Intelligence
Least-Squares Estimation of Transformation Parameters Between Two Point Patterns
IEEE Transactions on Pattern Analysis and Machine Intelligence
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
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The Local/Global Alignment (Zemla, 2003), or LGA, is a popular method for the comparison of protein structures. One of the two components of LGA requires us to compute the longest common contiguous segments between two protein structures. That is, given two structures A=(a1, …, an) and B=(b1, …, bn) where ak, bk∈ℝ3, we are to find, among all the segments f=(ai,…,aj) and g=(bi,…,bj) that fulfill a certain criterion regarding their similarity, those of the maximum length. We consider the following criteria: (1) the root mean square deviation (RMSD) between f and g is to be within a given t∈ℝ; (2) f and g can be superposed such that for each k, i≤k≤j, ||ak−bk||≤t for a given t∈ℝ. We give an algorithm of $O(n\log n+n\mbox{\it \textbf{l}})$ time complexity when the first requirement applies, where $\mbox{\it \textbf{l}}$ is the maximum length of the segments fulfilling the criterion. We show an FPTAS which, for any ε∈ℝ, finds a segment of length at least l, but of RMSD up to (1+ε)t, in O(nlogn+n/ε) time. We propose an FPTAS which for any given ε∈ℝ, finds all the segments f and g of the maximum length which can be superposed such that for each k, i≤k≤j, ||ak−bk||≤(1+ε) t, thus fulfilling the second requirement approximately. The algorithm has a time complexity of O(nlog2n/ε5) when consecutive points in A are separated by the same distance (which is the case with protein structures).