Online Dynamic Programming Speedups

• Published in: Theory of computing systems Vol. 45; no. 3; pp. 429 - 445
• Main Authors: Bar-Noy, Amotz, Golin, Mordecai J., Zhang, Yan
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01-10-2009; Springer Nature B.V
• Subjects: Algorithms; Computer Science; Dynamic programming; Internet; Studies; Theory of Computation; Monge property; Dynamic programming
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-009-9166-x

Consider the dynamic program
h
(
n
)=min 
1≤
j
≤
n
a
(
n
,
j
), where
a
(
n
,
j
) is some formula that may (online) or may not (offline) depend on the previously computed
h
(
i
), for
i
<
n
. The goal is to compute all
h
(
n
), for 1≤
n
≤
N
. It is well known that, if
a
(
n
,
j
) satisfy the
Monge
property, then the SMAWK algorithm (Aggarwal et al., Algorithmica 2(1):195–208,
1987
) can solve the offline problem in
O
(
N
) time; a 
Θ
(
N
) speedup over the naive algorithm.
In this paper we extend this speedup to the online case, that is, to compute
h
(
n
) in the order
n
=1,2,…,
N
when (i) we do not know the values of
a
(
n
′,
j
) for
n
′>
n
before
h
(
n
) has been computed and (ii) do not know the problem size
N
in advance. We show that if
a
(
n
,
j
) satisfy a stronger, but sometimes still natural, property than the Monge one, then each
h
(
n
) can be computed in online fashion in
O
(1) amortized time. This maintains the speedup online, in the sense that the total time to compute all
h
(
n
) is
O
(
N
). We also show how to compute each
h
(
n
) in the worst case
O
(log 
N
) time, while maintaining the amortized time bound.
For
a
(
n
,
j
) satisfying our stronger property, our algorithm is also simpler than the standard SMAWK algorithm for solving the offline case. We illustrate our technique on two examples from the literature; the first is the
D
-median problem on a line, and the second comes from mobile wireless paging.