Advances in Steiner Trees pp 39-62 | Cite as

# Computing Shortest Networks with Fixed Topologies

Chapter

## 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]S. Altschul and D. Lipman. Trees, stars, and multiple sequence alignment,
*SIAM Journal on Applied Math.*, 49 (1989), pp. 197–209.MathSciNetMATHCrossRefGoogle Scholar - [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.MATHGoogle Scholar - [3]T.H. Cormen, C.E. Leiserson and R. Rivest.
*Introduction to Algorithms*, The MIT Press, Cambridge, MA, 1990.MATHGoogle Scholar - [4]J.S. Farris. Methods for computing wagner trees,
*Systematic Zoology*19, 1970, pp. 83–92.CrossRefGoogle Scholar - [5]J. Felsenstein.
*PHYLIP version 3.5c (Phylogeny Inference Package)*, Department of Genetics, University of Washington, Seattle, WA, 1993.Google Scholar - [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]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]D. Gusfield.
*Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology*, Cambridge University Press, 1997.MATHCrossRefGoogle Scholar - [9]D. Gusfield. Efficient methods for multiple sequence alignment with guaranteed error bounds,
*Bulletin of Mathematical Biology*, 55 (1993), pp. 141–154.MATHGoogle Scholar - [10]M. Hanan. On Steiner problem with rectilinear distance,
*SIAM Journal on Applied Mathematics*14, 1966, pp. 255–265.MathSciNetMATHCrossRefGoogle Scholar - [11]J.A. Hartigan. Minimum mutation fits to a given tree,
*Biometrics*29, 1973, 53–65.CrossRefGoogle Scholar - [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]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]F.K. Hwang, On Steiner minimal trees with rectilinear distance,
*SIAM J. Appl. Math.*30, 1976, pp. 104–114.MathSciNetMATHCrossRefGoogle Scholar - [15]F.K. Hwang. A linear time algorithm for full Steiner trees,
*Oper. Res. Lett.*4, 1986, pp. 235 – 23 7.Google Scholar - [16]F.K. Hwang and J.F. Weng. The shortest network under a given topology,
*Journal of Algorithms*13, 1992, pp. 468–488.MathSciNetMATHCrossRefGoogle Scholar - [17]F.K. Hwang and D.S. Richards. Steiner tree problems,
*Networks*22, 1992, pp. 55–89.MathSciNetMATHCrossRefGoogle Scholar - [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]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]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]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]B. Ma, L. Wang and M. Li. Fixed topology alignment with recombination,
*Proc. 9th Annual Combinatorial Pattern Matching Conf.*, 1998.Google Scholar - [23]W. Miehle. Link length minimization in networks,
*Oper. Res.*6, 1958, pp. 232–243.MathSciNetCrossRefGoogle Scholar - [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]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]D. Sankoff. Minimal mutation trees of sequences,
*SIAM Journal of Applied Mathematics*, 28 (1975), pp. 35–42.MathSciNetMATHCrossRefGoogle Scholar - [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.MathSciNetMATHCrossRefGoogle Scholar - [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]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]W.D. Smith. How to find Steiner minimal trees in Euclidean d-space,
*Algorithmica*7, 1992, pp. 137–177.MathSciNetMATHCrossRefGoogle Scholar - [31]E. Sweedyk and T. Warnow, The tree alignment problem is NPcomplete,
*Manuscript*, 1994.Google Scholar - [32]L. Trevisan. When Hamming meets Euclid: the approximability of geometric TSP and MST,
*Proc. 29th ACM STOC*, 1997, pp. 21–29Google Scholar - [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]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]L. Wang and D. Gusfield. Improved approximation algorithms for tree alignment,
*Journal of Algorithms*25, 1997, pp. 255–173.MathSciNetMATHCrossRefGoogle Scholar - [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]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]L. Wang and X. Jia, Fixed topology Steiner trees and spanning forests,
*Theoretical Computer Science*, to appear.Google Scholar - [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]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]M.S. Waterman and M.D. Perlwitz. Line geometries for sequence comparisons,
*Bull. Math. Biol.*, 46 (1984), pp. 567–577.MathSciNetMATHGoogle Scholar - [42]M.S. Waterman.
*Introduction to Computational Biology: Maps, sequences, and genomes*, Chapman and Hall, 1995.MATHGoogle Scholar - [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.MathSciNetMATHCrossRefGoogle Scholar - [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]G. Xue, private communication, 1998.Google Scholar

## Copyright information

© Springer Science+Business Media Dordrecht 2000