We included a parallel version of NQL's nilpotent quotient algorithm using the ParGap-package of GAP; see ParGap. In this chapter, we outline the basic usage of this parallel part of the NQL-package. For further details on the parallel GAP-sessions we refer to the ParGap-manual ParGap. We note that the ParGap-package has some bottlenecks in practice. Nevertheless the significant speed-up of our computations on a multiple-core system shows that this is a reasonable extension of the NQL-package.medskip
For using the parallel version of the nilpotent quotient algorithm,
you will need to install the ParGap-package as described in its
manual ParGap. When using Version 1.1.2 of the ParGap-package,
you will need to apply the following patch to pargap/lib/masslave.g
as otherwise the ParGap-session may crash. On a linux machine you can
simply use patch < ../../nql/gap/pargap/patch
from within the directory
pargap/lib/
.
--- masslave.g 2001-11-16 13:17:04.000000000 +0100 +++ masslave.g 2009-05-06 12:20:19.000000000 +0200 @@ -467,8 +467,9 @@ if Length(deltas)>1 then max2 := Maximum(max2, deltas[Length(deltas)-1]); fi; max1 := deltas[Length(deltas)]; pos1 := Position( List(slaveArray, x->realtime-x.time), max1 ); - if max1 > slaveTaskTimeFactor and max1 > 30 - and slaveTaskTime[pos2].total > 60 then + if max1 > slaveTaskTimeFactor and + max1 > 30 and pos2 <> fail and + slaveTaskTime[pos2].total > 60 then Print("SLAVE ",pos1," SEEMS DEAD!!\n"); fi; end);
Now, you are ready for creating a ParGap-session and you can load
the NQL-package from within ParGap using RequirePackage
as usual.
The same methods as described previously are available. The following
example shows the application of the NilpotentQuotient
-method to
the Grigorchuk group on a quad-core machine. Note that the significant
speed-up of the nilpotent quotient algorithm is especially noticeable
for large nilpotent quotients. This parallel version of NQL successfully
computes some nilpotent quotients which normally took more than a month
to complete.
GAP4, Version: 4.4.12 of 17-Dec-2008, i686-pc-linux-gnu-gcc GAP4, Version: 4.4.12 of 17-Dec-2008, i686-pc-linux-gnu-gcc GAP4, Version: 4.4.12 of 17-Dec-2008, i686-pc-linux-gnu-gcc GAP4, Version: 4.4.12 of 17-Dec-2008, i686-pc-linux-gnu-gcc GAP4, Version: 4.4.12 of 17-Dec-2008, i686-pc-linux-gnu-gcc gap> TOPCnumSlaves; 4 gap> RequirePackage("NQL"); true gap> G:=ExamplesOfLPresentations(1); <L-presented group on the generators [ a, b, c, d ]> gap> SetInfoLevel(InfoNQL,1); gap> NilpotentQuotient(G,2); #I Class 1: 3 generators with relative orders: [ 2, 2, 2 ] #I Computing a polycyclic presentation for the covering group... #I Checking the consistency relations... master -> 1: (AGGLOM_TASK): [ [ -3, 1 ], [ -3, 2 ], [ -2, 1 ], [ 2, -1 ], [ 3, -1 ] ] master -> 2: (AGGLOM_TASK): [ [ 3, -2 ], [ 1 ], [ 2 ], [ 3 ] ] 1 -> master: [ [ 0, 0, 0, 0, 0, -2, 0 ], [ 0, 0, 0, 0, 0, 0, -2 ], [ 0, 0, 0, 0, -2, 0, 0 ], [ 0, 0, 0, 0, -2, 0, 0 ], [ 0, 0, 0, 0, 0, -2, 0 ] ] 2 -> master: [ [ 0, 0, 0, 0, 0, 0, -2 ], [ 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0 ] ] #I Broadcasting the slaves... #I Inducing the endomorphisms... master -> 1: 1 master -> 2: 2 master -> 3: 3 master -> 4: 4 3 -> master: [ 2, 1 ] UPDATE: [ 3, [ 2, 1 ] ] 1 -> master: [ 2, 1, 8, 1 ] UPDATE: [ 1, [ 2, 1, 8, 1 ] ] 2 -> master: [ 2, 1, 3, 1, 4, 1 ] UPDATE: [ 2, [ 2, 1, 3, 1, 4, 1 ] ] master -> 1: 5 master -> 2: 6 master -> 3: 7 4 -> master: [ 4, -1, 6, -1, 10, -1 ] UPDATE: [ 4, [ 4, -1, 6, -1, 10, -1 ] ] 1 -> master: [ 6, 1, 8, 2 ] UPDATE: [ 5, [ 6, 1, 8, 2 ] ] 2 -> master: [ 4, 2, 6, 1, 7, 1, 10, 1 ] UPDATE: [ 6, [ 4, 2, 6, 1, 7, 1, 10, 1 ] ] 3 -> master: [ 6, 1 ] UPDATE: [ 7, [ 6, 1 ] ] master -> 1: 8 master -> 2: 9 master -> 3: 10 1 -> master: [ 10, 1 ] UPDATE: [ 8, [ 10, 1 ] ] 2 -> master: [ ] UPDATE: [ 9, [ ] ] 3 -> master: [ 10, -1 ] UPDATE: [ 10, [ 10, -1 ] ] #I Broadcasting the slaves... #I Mapping the relations... master -> 1: 1 master -> 2: 2 master -> 3: 3 master -> 4: 4 1 -> master: [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] 2 -> master: [ 0, 0, 0, 2, 0, 1, 1, 0, 0, 1 ] 3 -> master: [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ] 4 -> master: [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ] master -> 1: 5 master -> 2: 6 master -> 3: 7 1 -> master: [ 0, 0, 0, 1, 0, 1, 1, 0, 0, 1 ] 2 -> master: [ 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ] 3 -> master: [ 0, 0, 0, 0, 0, 0, 0, 4, 6, 4 ] #I Start spinning... #I Extend the quotient system... #I Class 2: 2 generators with relative orders: [ 2, 2 ] Pcp-group with orders [ 2, 2, 2, 2, 2 ]
Note that the only difference in the parallel version of the NQL-package
is a parallel version of the operation ExtendQuotientSystem
. This latter
operation covers the induction step of the nilpotent quotient algorithm.
NQL manual