Space-Saving Strategies for Computing Δ -points

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 E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

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. Method 1: O(MN) time; O(MN) space S - S + N M

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

5. Method 3: O(MN) time; O(M+N) space N S - S + M

6. Method 4: O(MN log M) time; O(M+N log M) space N S - S + M

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

8. 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

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

10. Method 8: O(1/εMN) time; O(1/εM1+ε+ N) spaceHere we use ε= 1/2 to illustrate the idea. M 2M 3M N M O(N) M Solve each M1/2M problem M1/2 S - S + M

11. Methods Method 1: O(MN) time; O(MN) space Method 2: O(M2N) time; O(M+N) space Method 3: O(MN) time; O(M+N) space Method 4: O(MN log M) time; O(M+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(M+1/ εMεN) space Method 8: O(1/εMN) time; O(1/εM1+ε+ N) space

12. 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/εM1+ε+N) space