Affiliation:
1. School of Intelligent Systems Engineering, Sun Yat-Sen University, Shen Zhen, China
2. Department of Computer Science and Engineering, HKUST, Hong Kong, China
3. Freie Universität Berlin, Berlin, Germany
Abstract
Ailon et al. [SICOMP’11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances
x
1
, ... ,
x
n
follow some unknown
product distribution
. That is,
x
i
is drawn independently from a fixed unknown distribution 𝒟
i
. After spending
O
(
n
1+ε
) time in a learning phase, the subsequent expected running time is
O
((
n
+
H
)/ε), where
H
∊ {
H
S
,
H
DT
}, and
H
S
and
H
DT
are the entropies of the distributions of the sorting and DT output, respectively. In this article, we allow dependence among the
x
i
’s under the
group product distribution
. There is a hidden partition of [1,
n
] into groups; the
x
i
’s in the
k
th group are fixed unknown functions of the same hidden variable
u
k
; and the
u
k
’s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map
u
k
to
x
i
’s are well-behaved. After an
O
(poly(
n
))-time training phase, we achieve
O
(
n
+
H
S
) and
O
(
n
α (
n
) +
H
DT
) expected running times for sorting and DT, respectively, where α (⋅) is the inverse Ackermann function.
Funder
National Natural Science Foundation of China
Research Grants Council, Hong Kong, China
ERC
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)