On the General Chain Pair Simplification Problem

Authors Chenglin Fan, Omrit Filtser, Matthew J. Katz, Binhai Zhu

Thumbnail PDF


  • Filesize: 0.58 MB
  • 14 pages

Document Identifiers

Author Details

Chenglin Fan
Omrit Filtser
Matthew J. Katz
Binhai Zhu

Cite AsGet BibTex

Chenglin Fan, Omrit Filtser, Matthew J. Katz, and Binhai Zhu. On the General Chain Pair Simplification Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr'echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.
  • chain simplification
  • discrete Frechet distance
  • dynamic programming
  • geometric arrangements
  • protein structural resemblance


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Pankaj K. Agarwal, Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. Computing the discrete Fréchet distance in subquadratic time. SIAM J. Comput., 43(2):429-449, 2014. Google Scholar
  2. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Internat. J. Comput. Geometry Appl., 5:75-91, 1995. Google Scholar
  3. Sergey Bereg, Minghui Jiang, Wencheng Wang, Boting Yang, and Binhai Zhu. Simplifying 3D polygonal chains under the discrete Fréchet distance. In Proc. 8th Latin American Theoretical Informatics Sympos., LATIN'08, pages 630-641, 2008. Google Scholar
  4. Karl Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th IEEE Annual Sympos. on Foundations of Computer Science, FOCS'14, pages 661-670, 2014. Google Scholar
  5. Karl Bringmann and Wolfgang Mulzer. Approximability of the discrete Fréchet distance. JoCG, 7(2):46-76, 2016. Google Scholar
  6. Kevin Buchin, Maike Buchin, Wouter Meulemans, and Wolfgang Mulzer. Four soviets walk the dog - with an application to Alt’s conjecture. In Proc. 25th Annual ACM-SIAM Sympos. on Discrete Algorithms, SODA'14, pages 1399-1413, 2014. Google Scholar
  7. Anne Driemel and Sariel Har-Peled. Jaywalking your dog: Computing the Fréchet distance with shortcuts. SIAM J. Comput., 42(5):1830-1866, 2013. Google Scholar
  8. Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Information Systems Dept., Technical University of Vienna, 1994. Google Scholar
  9. Chenglin Fan, Omrit Filtser, Matthew J. Katz, Tim Wylie, and Binhai Zhu. On the chain pair simplification problem. In Algorithms and Data Structures - 14th Internat. Symp., WADS 2015, pages 351-362, 2015. Google Scholar
  10. Michael Godau. A natural metric for curves - computing the distance for polygonal chains and approximation algorithms. In STACS 91, 8th Annual Sympos. on Theoretical Aspects of Computer Science, pages 127-136, 1991. Google Scholar
  11. Minghui Jiang, Ying Xu, and Binhai Zhu. Protein structure-structure alignment with discrete Fréchet distance. J. Bioinformatics and Computational Biology, 6(1):51-64, 2008. Google Scholar
  12. Tim Wylie, Jun Luo, and Binhai Zhu. A practical solution for aligning and simplifying pairs of protein backbones under the discrete Fréchet distance. In Proc. Internat. Conf. Computational Science and Its Applications, ICCSA'11, Part III, pages 74-83, 2011. Google Scholar