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Never-ending stories. Kun-Mao Chao ( 趙坤茂 ) Dept. of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: [email protected] WWW: http://www.csie.ntu.edu.tw/~kmchao. Part I. Minimum spanning trees. Minimum spanning trees (MST).

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Never ending stories

Never-ending stories

Kun-Mao Chao (趙坤茂)

Dept. of Computer Science and Information Engineering

National Taiwan University, Taiwan

E-mail: [email protected]

WWW: http://www.csie.ntu.edu.tw/~kmchao



Minimum spanning trees mst
Minimum spanning trees (MST)

  • Input : weighted graph G=(V,E)

  • Output: A subset of E of minimum weight which forms a tree on V.

  • Two famous textbook algorithms:

    • Kruskal’s algorithm (1956) O (|E| log |E|)

    • Prim’s algorithm (1957) O(|E| log |V|)


The history of mst
The history of MST

  • Boruvka algorithm (1926) O(|E| log |V|)

  • Jarnik’s algorithm (1930) O(|E| log |V|),Rediscovered by

    • Prim (1957)

    • Dijkstra (1959)


Improvements
Improvements

  • Yao (1975) O(|E| log log |V|)

  • Cheriton and Tarjan (1976) O(|E| log log |V|)

  • ...

  • Karger, Klein and Tarjan (1995) Randomized O(|E|)

  • Chazelle (2000) O(|E|.α(|E|, |V|))

  • Pettie and Ramachandran (2002)An optimal MST algorithm Ω(|E|) ~ O(|E|.α(|E|, |V|))


Some variants of weighted spanning trees
Some Variants of weighted spanning trees

  • The Minimum Routing Cost Spanning Tree Problem (MRCT): to minimize the sum over all pairs of vertices of the cost of the path between the pair in the tree.

    • NP-hard (Johnson, Lenstra and Rinnooy Kan, 1978)

    • 2-approximation (Wong, 1980)

    • 1.5-approximation (Wu, Chao and Tang, 1997)

    • PTAS (Wu, Lancia, Bafna, Chao, Ravi and Tang, 1998)



Chao, K. -M., Pearson, W. R. and Miller, W. , 1992, Aligning Two Sequences within a Specified Diagonal Band, Computer Applications in the Biosciences (CABIOS, now Bioinformatics), 8: 481-487.

FASTA’s Last Stage


Chao, K. -M., Hardison, R. C. and Miller, W. , 1993, Constrained Sequence Alignment, Bulletin of Mathematical Biology, 55: 503-524.

Band Arbitrary boundary lines


Chao, K. -M., Hardison, R. C. and Miller, W. , 1993, Locating Well-Conserved Regions within a Pairwise Alignment, Computer Applications in the Biosciences (CABIOS, now Bioinformatics), 9: 387-396.

Robust Measures


Hardison, R. C., Chao, K. -M., Adamkiewicz, M., Price, D., Jackson, J., Zeigler, T., Stojanovic, N. and Miller, W. , 1993, Positive and Negative Regulatory Elements of the Rabbit Embryonic -Globin Gene Revealed by an Improved Multiple Alignment Program and Functional Analysis, DNA Sequence, 4: 163-176.

Hardison, R. C., Chao, K. -M., Schwartz, S., Stojanovic, N., Ganetsky, M. and Miller, W. , 1994, Globin Gene Server: A Prototype E-Mail Database Server Featuring Extensive Multiple Alignments and Data Compilation for Electronic Genetic Analysis, Genomics, 21: 344-353.

Multiple alignment applications


Chao, K. -M., Hardison R. C. and Miller, W. , 1994, Recent Developments in Linear-Space Alignment Methods: a Survey, Journal of Computational Biology, 1: 271-291.

YAMA (Yet Another Multiple Aligner)


Chao, K. -M. and Miller, W. , 1995, Linear-Space Algorithms that Build Local Alignments from Fragments, Algorithmica, 13: 106-134.

falign: Somewhere between FASTA and BLAST


Chao, K. -M., Zhang, J., Ostell, J. and Miller, W. , 1995, A Local Alignment Tool for Very Long DNA Sequences, Computer Applications in the Biosciences (CABIOS, now Bioinformatics), 11: 147-153.

falign + constrained sequence alignment


Chao, K. -M., Zhang, J., Ostell, J. and Miller, W. , 1997, A Tool for Aligning Very Similar DNA sequences, Computer Applications in the Biosciences (CABIOS, now Bioinformatics), 13: 75-80.

Fast algorithms for very similar sequences


Chao, K. -M., 1998, “On Computing all Suboptimal Alignments,” Information Sciences, 105: 189-207.

Suboptimal alignments


Chao, K. -M., 1999, “Calign: Aligning Sequences with Restricted Affine Gap Penalties,” Bioinformatics, 15: 298-304.

cDNA vs. Genomic sequences


Lin, Y. -L., Jiang, T. and Chao, K. -M., 2002, “Efficient Algorithms for Locating the Length-Constrained Heaviest Segments, with Applications to Biomolecular Sequence Analysis,” Journal of Computer and System Sciences (JCSS), Accepted. (Work done in October, 2001.)

Algorithms for locating a maximum-sum or maximum-average region with length constraints.


Lin, Y. -L., Huang, X., Jiang, T. and Chao, K. -M., 2003, “MAVG: Locating Non-Overlapping Maximum Average Segments in a Given Sequence,” Bioinformatics, January issue. (Work done in April, 2002.)

A tool for locating k-best average regions


Huang, X. and Chao, K. -M., 2003, “A Generalized Global Alignment Algorithm,” Bioinformatics, February issue. (Work done in May, 2002.)

GAP3: Chaining local alignments


Part III.: Your stories

(To be continued.)


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