GAP (lpres) - Chapter 6: On a parallel nilpotent quotient algorithm

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6 On a parallel nilpotent quotient algorithm
6.1 Usage

### 6 On a parallel nilpotent quotient algorithm

We included a parallel version of lpres's nilpotent quotient algorithm using the ParGAP-package of GAP; see [Coo04]. In this chapter, we outline the basic usage of this parallel part of the lpres-package. For further details on the parallel GAP-sessions we refer to the ParGAP-manual [Coo04]. 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 lpres-package.

#### 6.1 Usage

For using the parallel version of the nilpotent quotient algorithm, you will need to install the ParGAP-package as described in its manual [Coo04]. 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 lpres-package from within ParGAP using `LoadPackage' 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 lpres 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> LoadPackage("LPRES");
    true
    gap> G:=ExamplesOfLPresentations(1);
    <L-presented group on the generators [ a, b, c, d ]>
    gap> SetInfoLevel(InfoLPRES,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 lpres-package is a parallel version of the operation `ExtendQuotientSystem'. This latter operation covers the induction step of the nilpotent quotient algorithm.

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