1. Never-ending stories Kun-Mao Chao ( 趙坤茂 ) Dept. of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

2. Part I. Minimum spanning trees

3. 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|)

4. 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)

5. 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|))

6. 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)

7. Part II. Sequence Analysis (My story)

8. 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

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

10. 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

11. 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

12. 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)

13. 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

14. 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

15. 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

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

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

18. 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.

19. 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

20. Huang, X. and Chao, K. -M., 2003, “A Generalized Global Alignment Algorithm,” Bioinformatics, February issue. (Work done in May, 2002.) GAP3: Chaining local alignments

21. Part III.: Your stories (To be continued.)