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Computing Shortest Networks with Fixed Topologies

  • Tao Jiang
  • Lusheng Wang
Part of the Combinatorial Optimization book series (COOP, volume 6)

Abstract

We discuss the problem of computing a shortest network interconnecting a set of points under a fixed tree topology, and survey the recent algorithmic and complexity results in the literature covering a wide range of metric spaces, including Euclidean, rectilinear, space of sequences with Hamming and edit distances, communication networks, etc. It is demonstrated that the problem is polynomial time solvable for some spaces and NP-hard for the others. When the problem is NPhard, we attempt to give approximation algorithms with guaranteed relative errors.

Keywords

Internal Node Steiner Tree Regular Point Edit Distance Steiner Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. Altschul and D. Lipman. Trees, stars, and multiple sequence alignment, SIAM Journal on Applied Math., 49 (1989), pp. 197–209.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    S. C. Chan, A. K. C. Wong and D. K. T. Chiu. A survey of multiple sequence comparison methods, Bulletin of Mathematical Biology, 54, 4 (1992), pp. 563–598.zbMATHGoogle Scholar
  3. [3]
    T.H. Cormen, C.E. Leiserson and R. Rivest. Introduction to Algorithms, The MIT Press, Cambridge, MA, 1990.zbMATHGoogle Scholar
  4. [4]
    J.S. Farris. Methods for computing wagner trees, Systematic Zoology 19, 1970, pp. 83–92.CrossRefGoogle Scholar
  5. [5]
    J. Felsenstein. PHYLIP version 3.5c (Phylogeny Inference Package), Department of Genetics, University of Washington, Seattle, WA, 1993.Google Scholar
  6. [6]
    W.M. Fitch. Towards defining the course of evolution: minimum change for a specific tree topology, Systematic Zoology 20, 1971, pp. 406–416.CrossRefGoogle Scholar
  7. [7]
    S. Gupta, J. Kececioglu, and A. Schaffer. Making the shortest-paths approach to sum-of-pairs multiple sequence alignment more space efficient in practice, Proceedings of the 6ith Symposium on Combinatorial Pattern Matching, Springer LNCS 937, 1995, pp. 128–143.Google Scholar
  8. [8]
    D. Gusfield. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology, Cambridge University Press, 1997.zbMATHCrossRefGoogle Scholar
  9. [9]
    D. Gusfield. Efficient methods for multiple sequence alignment with guaranteed error bounds, Bulletin of Mathematical Biology, 55 (1993), pp. 141–154.zbMATHGoogle Scholar
  10. [10]
    M. Hanan. On Steiner problem with rectilinear distance, SIAM Journal on Applied Mathematics 14, 1966, pp. 255–265.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    J.A. Hartigan. Minimum mutation fits to a given tree, Biometrics 29, 1973, 53–65.CrossRefGoogle Scholar
  12. [12]
    J. Hein. A tree reconstruction method that is economical in the number of pairwise comparisons used, Mol. Biol. Evol., 6, 6 (1989), pp. 669–684.Google Scholar
  13. [13]
    J. Hein. A new method that simultaneously aligns and reconstructs ancestral sequences for any number of homologous sequences, when the phylogeny is given, Mol. Biol. Evol., 6 (1989), pp. 649–668.Google Scholar
  14. [14]
    F.K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math. 30, 1976, pp. 104–114.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    F.K. Hwang. A linear time algorithm for full Steiner trees, Oper. Res. Lett. 4, 1986, pp. 235 – 23 7.Google Scholar
  16. [16]
    F.K. Hwang and J.F. Weng. The shortest network under a given topology, Journal of Algorithms 13, 1992, pp. 468–488.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    F.K. Hwang and D.S. Richards. Steiner tree problems, Networks 22, 1992, pp. 55–89.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    X. Jia and L. Wang, Group multicast routing using multiple minimum Steiner trees, Journal of Computer Communications, pp. 750–758, 1997.Google Scholar
  19. [19]
    T. Jiang, E. L. Lawler and L. Wang, Aligning sequences via an evolutionary tree: complexity and approximation, Proc. 26th ACM Symp. on Theory of Computing, pp. 760–769, 1994.Google Scholar
  20. [20]
    J. Lipman, S.F. Altschul, and J.D. Kececioglu. A tool for multiple sequence alignment, Proc. Nat. Acid Sci. U.S.A., 86, pp.4412–4415, 1989.CrossRefGoogle Scholar
  21. [21]
    F. Liu and T. Jiang. Tree Alignment and Reconstruction (TAAR) V1.0, Department of Computer Science, McMaster University, Hamilton, Ontario, Canada, 1998. The software is available via WWW at http://www.dcss.mcmaster.ca/~fliu/taar download.htmlGoogle Scholar
  22. [22]
    B. Ma, L. Wang and M. Li. Fixed topology alignment with recombination, Proc. 9th Annual Combinatorial Pattern Matching Conf., 1998.Google Scholar
  23. [23]
    W. Miehle. Link length minimization in networks, Oper. Res. 6, 1958, pp. 232–243.MathSciNetCrossRefGoogle Scholar
  24. [24]
    G.W. Moore, J. Barnabas and M. Goodman. A method for constructing maximum parsimony ancestral amino acid sequences on a given network, Journal of Theoretical Biology 38, 1973, pp. 459–485.CrossRefGoogle Scholar
  25. [25]
    R. Ravi and J. Kececioglu. Approximation algorithms for multiple sequence alignment under a fixed evolutionary tree, Proc. 5th Annual Symposium on Combinatorial Pattern Matching, 1995, pp. 330–339.Google Scholar
  26. [26]
    D. Sankoff. Minimal mutation trees of sequences, SIAM Journal of Applied Mathematics, 28 (1975), pp. 35–42.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    D. Sankoff and P. Rousseau. Locating the vertices of a Steiner tree in an arbitrary metric space, Mathematical Programming 9, 1975, pp. 240–246.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    D. Sankoff, R. J. Cedergren and G. Lapalme. Frequency of insertiondeletion, transversion, and transition in the evolution of 5S ribosomal RNA, J. Mol. Evol. 7 (1976), pp. 133–149.CrossRefGoogle Scholar
  29. [29]
    D. Sankoff and R. Cedergren. Simultaneous comparisons of three or more sequences related by a tree, in D. Sankoff and J. Kruskal, editors, Time warps, string edits, and macromolecules: the theory and practice of sequence comparison, pp. 253–264, Addison Wesley, 1983.Google Scholar
  30. [30]
    W.D. Smith. How to find Steiner minimal trees in Euclidean d-space, Algorithmica 7, 1992, pp. 137–177.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    E. Sweedyk and T. Warnow, The tree alignment problem is NPcomplete, Manuscript, 1994.Google Scholar
  32. [32]
    L. Trevisan. When Hamming meets Euclid: the approximability of geometric TSP and MST, Proc. 29th ACM STOC, 1997, pp. 21–29Google Scholar
  33. [33]
    L. Wang and T. Jiang. On the complexity of multiple sequence alignment, Journal of Computational Biology 1, 1994, pp. 337–348.CrossRefGoogle Scholar
  34. [34]
    L. Wang, T. Jiang and E.L. Lawler. Approximation algorithms for tree alignment with a given phylogeny, Algorithmica 16, 1996, pp. 302–315.MathSciNetCrossRefGoogle Scholar
  35. [35]
    L. Wang and D. Gusfield. Improved approximation algorithms for tree alignment, Journal of Algorithms 25, 1997, pp. 255–173.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    L. Wang, T. Jiang, and Dan Gusfield. A more efficient approximation scheme for tree alignment, Proc. 1 st Annual International Conference on Computational Molecular Biology, 1997, pp. 310–319.Google Scholar
  37. [37]
    L. Wang and X. Jia, Fixed topology Steiner trees and spanning forests with application in network communication, Proc. 3rd Annual Computing and Combinatorics Conf., 1997, pp. 373–382.Google Scholar
  38. [38]
    L. Wang and X. Jia, Fixed topology Steiner trees and spanning forests, Theoretical Computer Science, to appear.Google Scholar
  39. [39]
    H. T. Wareham, A simplified proof of the NP-hardness and MAX SNPhardness of multiple sequence tree alignment, Journal of Computational Biology 2, pp. 509–514, 1995.CrossRefGoogle Scholar
  40. [40]
    M. Bonet, M. Steel, T. Warnow, and S. Yooseph. Better methods for solving parsimony and compatibility, Proc. 2nd Annual International Conference on Computational Molecular Biology, 1998, pp. 40–49.Google Scholar
  41. [41]
    M.S. Waterman and M.D. Perlwitz. Line geometries for sequence comparisons, Bull. Math. Biol., 46 (1984), pp. 567–577.MathSciNetzbMATHGoogle Scholar
  42. [42]
    M.S. Waterman. Introduction to Computational Biology: Maps, sequences, and genomes, Chapman and Hall, 1995.zbMATHGoogle Scholar
  43. [43]
    G. Xue and Y. Ye. An efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM J. Optim. 7, 1997, pp. 1017–1036.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    G. Xue and D.Z. Du. An O(n log n) average time algorithm for cornputing the shortest network under a given topology, to appear in Algorithmica, 1997.Google Scholar
  45. [45]
    G. Xue, private communication, 1998.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Tao Jiang
    • 1
  • Lusheng Wang
    • 2
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.Department of Computer ScienceCity University of Hong KongKowloonHong Kong

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