1. Space-Saving Strategies for Computing Δ-points
Kun-Mao Chao (趙坤茂)
Department of Computer Science and Information Engineering
National Taiwan University, Taiwan
WWW:

2. Δ-points S-(i, j): the best score of a path from (0, 0) to (i, j).
S+(i, j): the best score of a path from (i, j) to (M, N).
Δ-points: S-(i, j) + S+(i, j) >= Δ
S -
S +

3. The leftmost/rightmost Δ-paths
For simple scoring schemes, finding the leftmost Δ-path and the rightmost Δ-path is easy.
For affine gap penalties, it is more complicated.

4. Two alignments may not intersect!

5. Method 1: O(MN) time; O(MN) space

6. Method 2: O(M2N) time; O(N) space
Each row takes O(MN) time. In total, O(M) x O(MN) = O(M2N)
S + M

7. Method 3: O(MN) time; O(N) space

8. Method 4: O(MN log M) time; O(N log M) space

9. Method 4: O(MN log M) time; O(N log M) space (cont’d)…
O(log M) layers M
O(N)
O(N)
O(N)
O(N)
O(N)

10. The computation of S-(i, j) and S+(i, j) inside a block

11. Method 5: O(MN log min {M, N}) time; O(M+N) space

12. Method 6: O(MN log log min {M, N}) time; O(M+N) space
Real Size
1/25
1/23 N
1/210
1/25
1/22 M
1/29
1/219

13. Method 7: O(1/ε MN) time; O(1/ε MεN) space Here we use ε= 1/2 to illustrate the idea.
Solve each M1/2N problem
M1/2
S -
S + M

14. Method 8: O(1/εMN) time; O(1/ε M1+ε+ N) space Here we use ε= 1/2 to illustrate the idea.
O(N) M
Solve each M1/2M problem
M1/2
S -
S + M

15. Methods Method 1: O(MN) time; O(MN) space
Method 2: O(M2N) time; O(M) space
Method 3: O(MN) time; O(M) space
Method 4: O(MN log M) time; O(N log M) space
Method 5: O(MN log min {M, N}) time; O(M+N) space
Method 6: O(MN log log min {M, N}) time; O(M+N) space
Method 7: O(1/εMN) time; O(1/ ε MεN) space
Method 8: O(1/εMN) time; O(1/ε M1+ε+ N) space

16. Bonus points O(MN) time; O(M+N) space
o(MN log log min {M, N}) time; O(M+N) space
O(1/εMN) time; o(1/ε M1+ε+N) space