Computing Shortest Networks with Fixed Topologies

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.

[1]  F. Hwang.  On Steiner Minimal Trees with Rectilinear Distance , 1976 .

[2]  Tao Jiang,et al.  Aligning sequences via an evolutionary tree: complexity and approximation , 1994, STOC '94.

[3]  Todd Wareham,et al.  A Simplified Proof of the NP- and MAX SNP-Hardness of Multiple Sequence Tree Alignment , 1995, J. Comput. Biol..

[4]  W. Miehle.  Link-Length Minimization in Networks , 1958 .

[5]  G. Moore,et al.  A method for constructing maximum parsimony ancestral amino acid sequences on a given network. , 1973, Journal of theoretical biology.

[6]  M. Waterman,et al.  Line geometries for sequence comparisons , 1984 .

[7]  Yinyu Ye,et al.  An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications , 1997, SIAM J. Optim..

[8]  Xiaohua Jia,et al.  A group multicast routing algorithm by using multiple minimum Steiner trees , 1997, Comput. Commun..

[9]  Maria Luisa Bonet,et al.  Better methods for solving parsimony and compatibility , 1998, RECOMB '98.

[10]  M. Hanan,et al.  On Steiner’s Problem with Rectilinear Distance , 1966 .

[11]  Sandeep K. Gupta,et al.  Making the Shortest-Paths Approach to Sum-of-Pairs Multiple Sequence Alignment More Space Efficient in Practice (Extended Abstract) , 1995, CPM.

[12]  F. Hwang,et al.  A linear time algorithm for full steiner trees , 1986 .

[13]  David Sankoff,et al.  Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison , 1983 .

[14]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[15]  J. Hein,et al.  A tree reconstruction method that is economical in the number of pairwise comparisons used. , 1989, Molecular biology and evolution.

[16]  Bin Ma,et al.  Fixed Topology Alignment with Recombination , 1998, CPM.

[17]  J Hein,et al.  A new method that simultaneously aligns and reconstructs ancestral sequences for any number of homologous sequences, when the phylogeny is given. , 1989, Molecular biology and evolution.

[18]  Dana S. Richards,et al.  Steiner tree problems , 1992, Networks.

[19]  D. Gusfield.  Efficient methods for multiple sequence alignment with guaranteed error bounds , 1993 .

[20]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[21]  R. Ravi,et al.  Approximation Algorithms for Multiple Sequence Alignment Under a Fixed Evolutionary Tree , 1995, CPM.

[22]  David Sankoff,et al.  Locating the vertices of a steiner tree in an arbitrary metric space , 1975, Math. Program..

[23]  J. Hartigan.  MINIMUM MUTATION FITS TO A GIVEN TREE , 1973 .

[24]  Michael S. Waterman,et al.  Introduction to Computational Biology: Maps, Sequences and Genomes , 1998 .

[25]  Tao Jiang,et al.  On the Complexity of Multiple Sequence Alignment , 1994, J. Comput. Biol..

[26]  D. Sankoff.  Minimal Mutation Trees of Sequences , 1975 .

[27]  Frank K. Hwang,et al.  The Shortest Network under a Given Topology , 1992, J. Algorithms.

[28]  J. Farris.  Methods for Computing Wagner Trees , 1970 .

[29]  Xiaohua Jia,et al.  Fixed Topology Steiner Trees and Spanning Forests with Applications in Network Communications , 1997, COCOON.

[30]  Tao Jiang,et al.  A more efficient approximation scheme for tree alignment , 1997, RECOMB '97.

[31]  Luca Trevisan,et al.  When Hamming meets Euclid: the approximability of geometric TSP and MST (extended abstract) , 1997, STOC '97.

[32]  A. K. Wong,et al.  A survey of multiple sequence comparison methods. , 1992, Bulletin of mathematical biology.

[33]  Michael S. Waterman,et al.  Introduction to computational biology , 1995 .

[34]  S. Altschul,et al.  A tool for multiple sequence alignment. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[35]  D. Lipman,et al.  Trees, stars, and multiple biological sequence alignment , 1989 .

[36]  W. Fitch.  Toward Defining the Course of Evolution: Minimum Change for a Specific Tree Topology , 1971 .

[37]  Lusheng Wang,et al.  Improved Approximation Algorithms for Tree Alignment , 1996, J. Algorithms.