Space-Saving Strategies for Computing Δ-points

Kun-Mao Chao (趙坤茂 )Department of Computer Science an

d Information EngineeringNational Taiwan University, Taiwan

E-mail: kmchao@csie.ntu.edu.tw

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

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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 +

M

N

4

Method 2: O(M2N) time; O(M+N) space

S -

S +

Each row takes O(MN) time. In total, O(M) x O(MN) = O(M2N)

M

N

5

Method 3: O(MN) time; O(M+N) space

S -

S +

M

N

6

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

S -

S +

M

N

7

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

M

N

8

Method 6: O(MN log log min {M, N}) time; O(M+N) space

M

N

1/22

1/23

1/25

1/29

1/25

1/210

1/219

Real Size

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Method 7: O(1/ε MN) time; O(M+ 1/ε MεN) space

Here we use ε= 1/2 to illustrate the idea.

S -

S +

M

N

M1/2 Solve each M1/2N problem

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Method 8: O(1/εMN) time; O(1/ε M1+ε+ N) space

Here we use ε= 1/2 to illustrate the idea.

M

NM 2M 3M

S -

S +

M

M

M1/2 Solve each M1/2M problem

O(N)

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Methods

Method 1: O(MN) time; O(MN) spaceMethod 2: O(M2N) time; O(M+N) spaceMethod 3: O(MN) time; O(M+N) spaceMethod 4: O(MN log M) time; O(M+N log M) spaceMethod 5: O(MN log min {M, N}) time; O(M+N) sp

aceMethod 6: O(MN log log min {M, N}) time; O(M+

N) spaceMethod 7: O(1/εMN) time; O(M+1/ ε MεN) spaceMethod 8: O(1/εMN) time; O(1/ε M1+ε+ N) space

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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