1. 1 Sorting by Transpositions Based on the First Increasing Substring Concept Advisor: Professor R.C.T. Lee Speaker: Ming-Chiang Chen

2. 2 The Genome Rearrangement Problem

3. 3 The genome rearrangement problem is defined as follows: Given two sequences representing two species, find the smallest number of a specific operation needed to transform a sequence to the other sequence. The Genome Rearrangement Problem

4. 4 Without losing generality, the genome rearrangement can be modeled by a combinatorial problem of sorting. Genome X: 3 1 2 4  ………  Genome Y: 4 1 3 2 Y: 4 1 3 2  Y’: 1 2 3 4 X: 3 1 2 4  X’: 3 2 4 1 Genome X’: 3 2 4 1  ………  Genome Y’: 1 2 3 4 Re-index

5. 5 The Genome Rearrangement Problem There are different problems defined by different operations such as sorting by reversals, sorting by transpositions, etc. Note that the problem is not an ordinary sorting problem, but an optimization problem, because the purpose is to find the minimum number of operations needed.

6. 6 Sorting by Transpositions Transposition: Swap two adjacent substrings of any length without changing the order in the permutation. Examples: 3 1 5 2 4  3 2 4 1 5 5 3 2 6 4 7 1  4 7 5 3 2 6 1

7. 7 Sorting by Transpositions Examples: 1 4 2 6 3 5  1 2 3 4 5 6 1 4 2 6 3 5 1 3 5 4 2 6 1 3 4 2 5 6 1 3 2 4 5 6 1 2 3 4 5 6 The number of transpositions performed is 4. 1 4 2 6 3 5 1 2 6 3 4 5 1 2 3 4 5 6 The number of transpositions performed is 2.

8. 8 Sorting by Transpositions Sorting by Transpositions: The goal is to determine the minimum number of transpositions needed to transform a sequence into another one.

9. 9 Previous Work [BP98] Sorting by Transpositions, Bafna, V. and Pevzner, P. A., SIAM Journal on Discrete Mathematics, Vol. 11, No. 2, 1998, pp. 224-240. The Method is designed based on the cycle graph. Example:

10. 10 Our Approach The increasing substring: for a sequence , a substring is an increasing substring of maximal length satisfying. Example: A permutation: 2 4 1 6 3 5 Three increasing substrings: 2 4, 1 6, and 3 5.

11. 11 Our Approach Note that we always ignore the sorted parts in the leftmost and rightmost sides of the permutation when finding the increasing substring. Example: A permutation: 1 2 4 5 3 6 Two increasing substrings: 4 5, and 3. 1 2, and 6 are not considered since they are sorted in both sides of the permutation.

12. 12 Our Approach A sorted permutation contains only one increasing substring. Example:1 2 3 4 5 6 7 The goal is to sort the input permutation into a permutation with only one increasing substring. Sorting by transpositions corresponds to enlarging the increasing substring, or decreasing the number of increasing substrings to one.

13. 13 Our Approach For the first increasing substring, we perform a transposition on the first increasing substring and its adjacent substring which ends with. The idea is to move the substring containing to the left of the first increasing substring, and thus the first increasing substring is augmented.

14. 14 Our Approach For the first increasing substring, the element must exist to the right of this substring. Otherwise, the first increasing substring must include, or is not the first increasing substring. Examples: 3 4 1 5 2 2 3 4 1 5

15. 15 Conclusions The proposed concept about increasing substring is very simple. The method based on the concept works for sorting by transpositions and is very easy to understand and implement.

16. 16 ~ Thank you ~