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15 Parallel computation
15.1 An embarassingly parallel computation
15.2 A non-embarassingly parallel computation
15.3 Parallel persistent homology

### 15 Parallel computation

#### 15.1 An embarassingly parallel computation

The following example creates fifteen child processes and uses them simultaneously to compute the second integral homology of each of the 2328 groups of order 128. The final command shows that

H_2(G, Z)= Z_2^21

for the 2328-th group G in GAP's library of small groups. The penulimate command shows that the parallel computation achieves a speedup of 10.4 .

    gap> Processes:=List([1..15],i->ChildProcess());;
    gap> fn:=function(i);return GroupHomology(SmallGroup(128,i),2);end;;
    gap> for p in Processes do
    > ChildPut(fn,"fn",p);
    > od;
    gap> Exec("date +%s");L:=ParallelList([1..2328],"fn",Processes);;Exec("date +%s");
    1716105545
    1716105554
    gap> Exec("date +%s");L1:=List([1..2328],fn);;Exec("date +%s");
    1716105586
    1716105680

    gap> speedup:=1.0*(680-586)/(554-545);
    10.4444

    gap> L[2328];
    [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

The function `ParallelList()` is built from HAP's six core functions for parallel computation.

#### 15.2 A non-embarassingly parallel computation

The following commands use core functions to compute the product A=M× N of two random matrices by distributing the work over two processors.

    gap> M:=RandomMat(10000,10000);;
    gap> N:=RandomMat(10000,10000);;
    gap>
    gap> s:=ChildProcess();;
    gap>
    gap> Exec("date +%s");
    1716109418
    gap> Mtop:=M{[1..5000]};;
    gap> Mbottom:=M{[5001..10000]};;
    gap> ChildPut(Mtop,"Mtop",s);
    gap> ChildPut(N,"N",s);
    gap> NextAvailableChild([s]);;
    gap> ChildCommand("Atop:=Mtop*N;;",s);;
    gap> Abottom:=Mbottom*N;;
    gap> A:=ChildGet("Atop",s);;
    gap> Append(A,Abottom);;
    gap> Exec("date +%s");
    1716110143

    gap> AA:=M*N;;Exec("date +%s");
    1716111389

    gap> speedup:=1.0*(111389-110143)/(110143-109418);
    1.71862

The next commands compute the product A=M× N of two random matrices by distributing the work over fifteen processors. The parallelization is very naive (the entire matrices M and N are communicated to all processes) and the computation achieves a speedup of 7.6.

    gap> M:=RandomMat(15000,15000);;
    gap> N:=RandomMat(15000,15000);;
    gap> S:=List([1..15],i->ChildCreate());;

    gap> Exec("date +%s");
    1716156583
    gap> ChildPutObj(M,"M",S);
    gap> ChildPutObj(N,"N",S);
    gap> for i in [1..15] do
    > cmd:=Concatenation("A:=M{[1..1000]+(",String(i),"-1)*1000}*N;");
    > ChildCommand(cmd,S[i]);
    > od;
    gap> A:=[];;
    gap> for i in [1..15] do
    >  C:=ChildGet("A",S[i]);
    >  Append(A,C);
    > od;
    gap> Exec("date +%s");
    1716157489

    gap> AA:=M*N;;Exec("date +%s");
    1716164405

    gap> speedup:=1.0*(64405-57489)/(57489-56583);
    7.63355

#### 15.3 Parallel persistent homology

Section 5.8 illustrates an alternative method of computing the persitent Betti numbers of a filtered pure cubical complex. The method lends itself to parallelisation. However, the following parallel computation of persistent Betti numbers achieves only a speedup of 1.5 due to a significant time spent transferring data structures between processes. On the other hand, the persistent Betti function could be used to distribute computations over several computers. This might be useful for larger computations that require significant memory resources.

    gap> file:=HapFile("data247.txt");;
    gap> Read(file);;
    gap> F:=ThickeningFiltration(T,25);;
    gap> S:=List([1..15],i->ChildCreate());;
    gap> N:=[0,1,2];;
    gap> Exec("date +%s");P:=ParallelPersistentBettiNumbers(F,N,S);;Exec("date +%s");
    1717160785
    1717161285

    gap> Exec("date +%s");Q:=PersistentBettiNumbersAlt(F,N);;Exec("date +%s");
    1717161528
    1717162276
    gap> speedup:=1.0*(1717162276-1717161528)/(1717161285-1717160785);
    1.496

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