end if end for - -10 2012 XVIELAVIO ,ValedosVinhedos,RS,BrazilFebruary5 CarlileLavor ,AntonioMu

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

interval Branch & Prune

0: IBP(i, n, d , nbranches) if (x

is a duplicated atom) then

assign to x

the same coordinates of its previous copy;

IBP(i + 1,n,d ,nbranches);

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Instance with 3 amino acids

We firstly consider this simple molecule.

Bond lengths and bond angles are constant, torsion angles can

vary into a predefined interval.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Preliminary experiments

layer atom duplicated? branches w/out pruning branches with pruning atom

1 N 1 no 1 1 N 1

2 H 1 no 1 1 H 1

3 H 1 no 1 1 H 1

4 CA 1 no 2 2 CA 1

5 N 1 yes 2 2 N 1

6 HA 1 no 24 18 HA 1

7 CA 1 yes 24 18 CA 1

8 C 1 no 48 36 C 1

9 N 2 no 576 360 N 2

10 CA 2 no 1152 720 CA 2

11 H 2 no 2304 10 H 2

12 N 2 yes 2304 10 N 2

13 CA 2 yes 2304 10 CA 2

14 HA 2 no 27648 70 HA 2

15 C 2 no 55296 140 C 2

16 CA 2 yes 55296 140 CA 2

17 N 3 no 663552 1400 N 3

18 C 2 yes 663552 1400 C 2

19 CA 3 no 1327104 2800 CA 3

20 H 3 no 2654208 4 H 3

21 N 3 yes 2654208 4 N 3

22 CA 3 yes 2654208 4 CA 3

23 HA 3 no 31850496 9 HA 3

24 C 3 no 63700992 18 C 3

25 CA 3 yes 63700992 18 CA 3

26 O 3 no 764411904 52 O 3

27 C 3 yes 764411904 52 C 3

28 O 3 no 9172942848 10 O 3

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The first symmetry

It is possible to prove that:

If the DMDGP problem has a solution, then it has at least another symmetric solution.

Pairs of solutions of the DMDGP are symmetric.

Theorem

Let x : G → R ³ be a solution for the DMDGP, defined by the

torsion angles ω _1,4 , . . . , ω _n ₋ _3,n . If we invert the sign of sin ω _i ₋ _3,i ,

for i = 4, ..., n, then we obtain a new solution x ⁰ : G → R ³ for the

DMDGP.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The other symmetries

We recently proved that, for each vertex i belonging to this set B = {v ∈ V :6 ∃(u, w) s.t. u + 3 < v ≤ w}

there is a symmetry on the binary tree.

In other words, for each i ∈ B, there are two symmetric branches on the tree having as root a common position for x i − 1 .

Theorem

With probability 1, the number of solutions of the DMDGP is a

power of 2.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

An instance with 2 symmetries (and 4 solutions)

The set B related to this small instance is {4, 6}. Feasible branches are marked in light yellow.

If one solution is already available, BP can exploit B in order to

guide the search towards the feasible branches of the tree. This is

the idea behind symBP.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| #Sol #DDF time #Sol #DDF time

lavor_10 10 29 4 44 0.00 4 2 0.00

lavor_20 20 77 64 86 0.00 64 7 0.00

lavor_30 30 161 64 592 0.01 64 18 0.00

lavor_40 40 194 2048 30720 0.05 2048 19 0.01 lavor_50 50 203 1024 46728 0.07 1024 49 0.01

lavor_60 60 357 256 71352 0.12 256 89 0.01

lavor_70 70 591 16 232 0.00 16 67 0.00

lavor_100 100 605 2 4460924 7.55 2 815010 1.37

lavor_200 200 1844 32 35298 0.09 32 394 0.01

lavor_300 300 2505 4 38364 0.07 4 9378 0.03

Experiments with Lavor instances.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| |B| #Sol time #Sol time

lavor_400 400 2600 35 10

⁵

8 10

⁵

2 lavor_500 500 4577 44 10

⁵

310 10

⁵

4 lavor_600 600 5933 25 10

⁵

30 10

⁵

7 lavor_700 700 5679 71 10

⁵

7050 10

⁵

134 All these experiments are stopped when 100,000 solutions are found. The

cardinality of B is so large that neither BP nor symBP are able to enumerate all

solutions.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| #Sol #DDF time #Sol #DDF time

1m40 1224 20382 2 2488 0.03 2 115 0.02

1bpm 1443 14292 2 2890 0.04 2 519 0.02

1mqq 2032 19564 2 4066 0.08 2 737 0.04

3b34 2790 29188 2 5664 0.15 2 1798 0.10

1epw 3861 35028 2 11974 0.45 2 3269 0.21

Experiments with PDB instances.

No documento -10 2012 XVIELAVIO ,ValedosVinhedos,RS,BrazilFebruary5 CarlileLavor ,AntonioMucherino,LeoLiberti,NelsonMaculan TheDiscretizableMolecularDistanceGeometryProblem (páginas 43-55)