Affiliation:
1. University of North Carolina Greensboro
Abstract
Let
f
: {0, 1}
n
→ {0, 1}. Let
μ
be a product probability measure on {0, 1}
n
. For
ϵ
≥ 0, we define
Dϵ
(
f
), the
ϵ-approximate
decision tree complexity of
f
, to be the minimum depth of a decision tree
T
with
μ
(
T
(
x
) ≠
f
(
x
)) ≤
ϵ
. For
j
= 0 or 1 and for
δ
≥ 0, we define
Cj,δ
(
f
), the
δ-approximate
j
-certificate complexity of
f
, to be the minimum certificate complexity of a set
A
⊆
Ω
with
μ
(
AΔf
−1
(
j
)) ≤
ϵ
. Note that if
μ
(
x
) > 0 for all
x
then
D
0
(
f
) =
D
(
f
) and
Cj
,0
(
f
) =
Cj
(
f
) are the ordinary decision tree and
j
-certificate complexities of
f
, respectively. We extend the well-known result,
D
(
f
) ≤
C
1
(
f
)
C
0
(
f
) [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all
ϵ
> 0 there exists a
δ
> 0 and a constant
K
=
K
(
ϵ
,
δ
) > 0 such that for all
n
,
μ
,
f
,
Dϵ
(
f
) ≤
K C
1,
δ
(
f
)
C
0,
δ
(
f
). We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science