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

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. C G G A T C A T -3 -6 -9 -12 -15 -18 -21 -24 8 5 2 -1 -4 -7 -10 -13 3
Match: 8
Mismatch: -5
Gap symbol: -3
C G G A T C A T -3 -6 -9
-12
-15
-18
-21
-24 8 5 2 -1 -4 -7
-10
-13 3 7 4 1 -2 -5 9 6 10 -8
-11
-14 14
C T T A A C T
optimal score

4. C T T A A C – T C G G A T C A T 8 – 5 –5 +8 -5 +8 -3 +8 = 14
8 – 5 – = 14
C G G A T C A T -3 -6 -9
-12
-15
-18
-21
-24 8 5 2 -1 -4 -7
-10
-13 3 7 4 1 -2 -5 9 6 10 -8
-11
-14 14
C T T A A C T

5. C G G A T C A T -3 -6 -9 -12 -15 -18 -21 -24 8 5 2 -1 -4 -7 -10 -13 3
Match: 8
Mismatch: -5
Gap symbol: -3
S- Matrix
C G G A T C A T -3 -6 -9
-12
-15
-18
-21
-24 8 5 2 -1 -4 -7
-10
-13 3 7 4 1 -2 -5 9 6 10 -8
-11
-14 14
C T T A A C T

6. -21 C G G A T C A T -18 -15 C T T A A C T -12 -9 -6 -3 -24 Match: 8
Mismatch: -5
Gap symbol: -3
S+ Matrix
C G G A T C A T
-21
-18
-15
-12 -9 -6 -3
-24
C T T A A C T

7. Match: 8
Mismatch: -5
Gap symbol: -3
S+ Matrix
C G G A T C A T 14 3 6 8 10 12 1
-10
-21 11 13 2 4 -7
-18 5 16 7 -4
-15 -1
-12 9 15 18 -9 -2 -6
-13 -3
-24
C T T A A C T

8. C G G A T C A T C T T A A C T Match: 8 Mismatch: -5 Gap symbol: -3
S- and S+ Matrix
C G G A T C A T 14 -3 3 -6 6 -9 8
-12 10
-15 12
-18 1
-21
-10
-24 5 2 11 -1 13 -4 -7 4
-13 16 7 -2 -5 9 15 18 -8
-11
-14
C T T A A C T

9. C G G A T C A T C T T A A C T Match: 8 Mismatch: -5 S- and S+ Matrix
Gap symbol: -3
S- and S+ Matrix
C G G A T C A T 14 -3 3 -6 6 -9 8
-12 10
-15 12
-18 1
-21
-10
-24 5 2 11 -1 13 -4 -7 4
-13 16 7 -2 -5 9 15 18 -8
-11
-14
C T T A A C T

10. Match: 8
Mismatch: -5
Gap symbol: -3
S- + S+ Matrix
C G G A T C A T 14 -1 -2 -3
-17
-31
-45 13 12 11 1
-16
-15
-30
-29
C T T A A C T

11. Match: 8
Mismatch: -5
Gap symbol: -3
S- + S+ Matrix
Δ = 14
C G G A T C A T 14 -1 -2 -3
-17
-31
-45 13 12 11 1
-16
-15
-30
-29
C T T A A C T

12. Match: 8
Mismatch: -5
Gap symbol: -3
S- + S+ Matrix
Δ = 13
C G G A T C A T 14 -1 -2 -3
-17
-31
-45 13 12 11 1
-16
-15
-30
-29
C T T A A C T

13. 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.

14. Two alignments may not intersect!

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

16. 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

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

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

19. 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)

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

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

22. C G G A T C A T -3 -6 -9 -12 -15 -18 -21 -24 8 5 2 -1 -4 -7 -10 -13 3
Match: 8
Mismatch: -5
Gap symbol: -3
S- Matrix
C G G A T C A T -3 -6 -9
-12
-15
-18
-21
-24 8 5 2 -1 -4 -7
-10
-13 3 7 4 1 -2 -5 9 6 10 -8
-11
-14 14
C T T A A C T

23. Method 5: O(MN log min {M, N}) time; O(M+N) space (cont’d)…
O(log min {M, N}) layers M
4(M+N)
2(M+N)
M+N
1/2(M+N)
1/4(M+N)

24. 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

25. Method 6 (cont’d) The width at layer i is M/22i+i-1
Partition Lines
Number of Cuts M 4N 22 1
M/22 2N
2/1/22 = 23 2
M/22/23 = M/25 N
1/1/25 = 25 3
M/25/25 = M/210
N/2
1/2/1/210 = 29 4
M/210/29 = M/219
N/22
1/22/1/219 = 217 5
M/219/217 = M/236
N/23
1/23/1/236 = 233 6
M/236/233 = M/269
N/24
1/24/1/269 = 265

26. 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

27. 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

28. 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

29. 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