Slowing down sorting networks to obtain faster sorting algorithms
Journal of the ACM (JACM)
Approximate matching of polygonal shapes (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Measuring the resemblance of polygonal curves
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Matching Polygonal Curves with Respect to the Fréchet Distance
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Fréchet distance for curves, revisited
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Simplifying 3D polygonal chains under the discrete Fréchet distance
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
ICCSA'11 Proceedings of the 2011 international conference on Computational science and its applications - Volume Part III
The approximability of the exemplar breakpoint distance problem
AAIM'06 Proceedings of the Second international conference on Algorithmic Aspects in Information and Management
A polynomial time solution for protein chain pair simplification under the discrete fréchet distance
ISBRA'12 Proceedings of the 8th international conference on Bioinformatics Research and Applications
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For protein structure alignment and comparison, a lot of work has been done using RMSD as the distance measure, which has drawbacks under certain circumstances. Thus, the discrete Fréchet distance was recently applied to the problem of protein (backbone) structure alignment and comparison with promising results. For this problem, visualization is also important because protein chain backbones can have as many as 500-600 $(\alpha)$-carbon atoms, which constitute the vertices in the comparison. Even with an excellent alignment, the similarity of two polygonal chains can be difficult to visualize unless the chains are nearly identical. Thus, the chain pair simplification problem (CPS-3F) was proposed in 2008 to simultaneously simplify both chains with respect to each other under the discrete Fréchet distance. The complexity of CPS-3F is unknown, so heuristic methods have been developed. Here, we define a variation of CPS-3F, called the constrained CPS-3F problem ($({\rm CPS\hbox{-}3F}^+)$), and prove that it is polynomially solvable by presenting a dynamic programming solution, which we then prove is a factor-2 approximation for CPS-3F. We then compare the $({\rm CPS\hbox{-}3F}^+)$ solutions with previous empirical results, and further demonstrate some of the benefits of the simplified comparisons. Chain pair simplification based on the Hausdorff distance (CPS-2H) is known to be NP-complete, and here we prove that the constrained version ($(\rm CPS\hbox{-}2H^+)$) is also NP-complete. Finally, we discuss future work and implications along with a software library implementation, named the Fréchet-based Protein Alignment & Comparison Toolkit (FPACT).