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: http://www.csie.ntu.edu.tw/~kmchao

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 +

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

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

Method 3: O(MN) time; O(N) 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 Real Size 1/25 1/23 N 1/210 1/25 1/22 M 1/29 1/219

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

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

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

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