Constant time approximation scheme for largest well predicted subset
The largest well predicted subset problem is formulated for comparison of two predicted 3D protein structures from the same sequence. A 3D protein structure is represented by an ordered point set A ={ a 1 ,…, a n }, where each a i is a point in 3D space. Given two ordered point sets A ={ a 1 ,…, a n...
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| Vydáno v: | Journal of combinatorial optimization Ročník 25; číslo 3; s. 352 - 367 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Boston
Springer US
01.04.2013
Springer Science + Business Media |
| Témata: | |
| ISSN: | 1382-6905, 1573-2886 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | The largest well predicted subset problem is formulated for comparison of two predicted 3D protein structures from the same sequence. A 3D protein structure is represented by an ordered point set
A
={
a
1
,…,
a
n
}, where each
a
i
is a point in 3D space. Given two ordered point sets
A
={
a
1
,…,
a
n
} and
B
={
b
1
,
b
2
,…
b
n
} containing
n
points, and a threshold
d
, the
largest well predicted subset
problem is to find the rigid body transformation
T
for a largest subset
B
opt
of
B
such that the distance between
a
i
and
T
(
b
i
) is at most
d
for every
b
i
in
B
opt
. A meaningful prediction requires that the size of
B
opt
is at least
αn
for some constant
α
(Li et al. in CPM 2008,
2008
). We use LWPS(
A
,
B
,
d
,
α
) to denote the largest well predicted subset problem with meaningful prediction. An (1+
δ
1
,1−
δ
2
)-approximation for LWPS(
A
,
B
,
d
,
α
) is to find a transformation
T
to bring a subset
B
′⊆
B
of size at least (1−
δ
2
)|
B
opt
| such that for each
b
i
∈
B
′, the Euclidean distance between the two points distance (
a
i
,
T
(
b
i
))≤(1+
δ
1
)
d
. We develop a constant time (1+
δ
1
,1−
δ
2
)-approximation algorithm for LWPS(
A
,
B
,
d
,
α
) for arbitrary positive constants
δ
1
and
δ
2
. To our knowledge, this is the first constant time algorithm in this area. Li et al. (CPM 2008,
2008
) showed an
time randomized (1+
δ
1
)-distance approximation algorithm for the largest well predicted subset problem under meaningful prediction. We also study a closely related problem, the
bottleneck distance
problem, where we are given two ordered point sets
A
={
a
1
,…,
a
n
} and
B
={
b
1
,
b
2
,…
b
n
} containing
n
points and the problem is to find the smallest
d
opt
such that there exists a rigid transformation
T
with
distance
(
a
i
,
T
(
b
i
))≤
d
opt
for every point
b
i
∈
B
. A (1+
δ
)-approximation for the bottleneck distance problem is to find a transformation
T
, such that for each
b
i
∈
B
, distance (
a
i
,
T
(
b
i
))≤(1+
δ
)
d
opt
, where
δ
is a constant. For an arbitrary constant
δ
, we obtain a linear
O
(
n
/
δ
6
) time (1+
δ
)-algorithm for the bottleneck distance problem. The best known algorithms for both problems require super-linear time (Li et al. in CPM 2008,
2008
). |
|---|---|
| ISSN: | 1382-6905 1573-2886 |
| DOI: | 10.1007/s10878-010-9371-1 |