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: |
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Springer US
01.04.2013
Springer Science + Business Media |
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| ISSN: | 1382-6905, 1573-2886 |
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| Abstract | 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
). |
|---|---|
| AbstractList | 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
). |
| Author | Fu, Bin Wang, Lusheng |
| Author_xml | – sequence: 1 givenname: Bin surname: Fu fullname: Fu, Bin email: binfu@cs.panam.edu organization: Department of Computer Science, University of Texas–Pan American – sequence: 2 givenname: Lusheng surname: Wang fullname: Wang, Lusheng organization: Department of Computer Science, City University of Hong Kong |
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| Cites_doi | 10.1093/bioinformatics/16.9.776 10.1002/prot.20264 10.1007/3-540-45253-2_6 10.1145/506147.506150 10.1093/nar/gkg571 10.1109/TPAMI.1987.4767965 10.1007/978-3-540-44827-3_1 |
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| Keywords | Approximation scheme Constant time Randomization |
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| References | Lancia, Istrail (CR6) 2003 Goodrich, Mitchell, Orletsky (CR4) 1994 Ambühl, Chakraborty, Gärtner (CR1) 2000 Indyk, Motwani (CR5) 1999 Li, Bu, Xu, Li (CR8) 2008 Arun, Huang, Blostein (CR2) 1987; 9 Siew, Elofsson, Rychlewski, Fischer (CR10) 2000; 16 Zemla (CR11) 2003; 31 Zhang, Skolnick (CR12) 2004; 57 Choi, Goyal (CR3) 2004 Motwani, Raghavan (CR9) 2000 Li, Ma, Wang (CR7) 2002; 49 R Motwani (9371_CR9) 2000 Y Zhang (9371_CR12) 2004; 57 KS Arun (9371_CR2) 1987; 9 C Ambühl (9371_CR1) 2000 SC Li (9371_CR8) 2008 P Indyk (9371_CR5) 1999 V Choi (9371_CR3) 2004 A Zemla (9371_CR11) 2003; 31 M Goodrich (9371_CR4) 1994 M Li (9371_CR7) 2002; 49 G Lancia (9371_CR6) 2003 N Siew (9371_CR10) 2000; 16 |
| References_xml | – volume: 16 start-page: 776 issue: 9 year: 2000 end-page: 785 ident: CR10 article-title: Maxsub: an automated measure for the assessment of protein structure prediction quality publication-title: Bioinformatics doi: 10.1093/bioinformatics/16.9.776 – start-page: 103 year: 1994 end-page: 112 ident: CR4 article-title: Practical methods for approximate geometric pattern matching under rigid motions publication-title: SOCG 1994 – volume: 57 start-page: 702 year: 2004 end-page: 710 ident: CR12 article-title: Scoring function for automated assessment of protein structure template quality publication-title: Proteins doi: 10.1002/prot.20264 – start-page: 52 year: 2000 end-page: 63 ident: CR1 article-title: Computing largest common point sets under approximate congruence publication-title: ESA 2000 doi: 10.1007/3-540-45253-2_6 – start-page: 457 year: 1999 end-page: 465 ident: CR5 article-title: Geometric matching under noise: combinatorial bounds and algorithms publication-title: SODA 1999 – year: 2000 ident: CR9 publication-title: Randomized algorithms – volume: 49 start-page: 157 issue: 2 year: 2002 end-page: 171 ident: CR7 article-title: On the closest string and substring problems publication-title: J ACM doi: 10.1145/506147.506150 – volume: 31 start-page: 3370 issue: 13 year: 2003 end-page: 3374 ident: CR11 article-title: LGA: a method for folding 3d similarities in protein structures publication-title: Nucleic Acids Res doi: 10.1093/nar/gkg571 – start-page: 44 year: 2008 end-page: 55 ident: CR8 article-title: Finding largest well-predicted subset of protein structure models publication-title: CPM 2008 – volume: 9 start-page: 698 issue: 5 year: 1987 end-page: 700 ident: CR2 article-title: Least-squares fitting of two 3-d point sets publication-title: IEEE Trans Pattern Anal Mach Intell doi: 10.1109/TPAMI.1987.4767965 – start-page: 1 year: 2003 end-page: 33 ident: CR6 article-title: Protein structure comparison: algorithms and applications publication-title: Mathemat methods for protein struct analysis and design doi: 10.1007/978-3-540-44827-3_1 – start-page: 285 year: 2004 end-page: 296 ident: CR3 article-title: A combinatorial shape matching algorithm for rigid protein docking publication-title: CPM 2004 – start-page: 44 volume-title: CPM 2008 year: 2008 ident: 9371_CR8 – start-page: 103 volume-title: SOCG 1994 year: 1994 ident: 9371_CR4 – volume: 9 start-page: 698 issue: 5 year: 1987 ident: 9371_CR2 publication-title: IEEE Trans Pattern Anal Mach Intell doi: 10.1109/TPAMI.1987.4767965 – volume: 31 start-page: 3370 issue: 13 year: 2003 ident: 9371_CR11 publication-title: Nucleic Acids Res doi: 10.1093/nar/gkg571 – volume: 16 start-page: 776 issue: 9 year: 2000 ident: 9371_CR10 publication-title: Bioinformatics doi: 10.1093/bioinformatics/16.9.776 – volume: 49 start-page: 157 issue: 2 year: 2002 ident: 9371_CR7 publication-title: J ACM doi: 10.1145/506147.506150 – volume-title: Randomized algorithms year: 2000 ident: 9371_CR9 – volume: 57 start-page: 702 year: 2004 ident: 9371_CR12 publication-title: Proteins doi: 10.1002/prot.20264 – start-page: 285 volume-title: CPM 2004 year: 2004 ident: 9371_CR3 – start-page: 52 volume-title: ESA 2000 year: 2000 ident: 9371_CR1 doi: 10.1007/3-540-45253-2_6 – start-page: 457 volume-title: SODA 1999 year: 1999 ident: 9371_CR5 – start-page: 1 volume-title: Mathemat methods for protein struct analysis and design year: 2003 ident: 9371_CR6 doi: 10.1007/978-3-540-44827-3_1 |
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| Snippet | 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... |
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| SubjectTerms | Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation |
| Title | Constant time approximation scheme for largest well predicted subset |
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