# How to use parallel program to select the longest element in a list?

I need to find the longest element in a list of elements of Weyl group. The way I do it is:

    winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner


Here g1 and g2 are two given subsets of the Weyl group I would like to make the above faster by using parallel computations. Is there some way in Sage or Python to do this? Thank you very much!

Edit: the full codes are:

def find_permutation2(L1, L2):
perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
assert [L2[i-1] for i in perm] == L1
r=perm.reduced_word()
return r

def LongestPermWInSBWInverseSA(W,A,B):
t1=StablizerOfTuple(A)
t2=StablizerOfTuple(sorted(B))
s=W.simple_reflections()
#print(t1,t2)
t3=[]
for i in t1:
t3.append(s[i])
t4=[]
for i in t2:
t4.append(s[i])
t5=find_permutation2(B,sorted(B))
w=W.one()
for i in t5:
w=w*s[i]
#print(t4,w,t3)
r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
r=r2

return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2):
g1=W.subgroup(S1)
g2=W.subgroup(S2)
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner

return r

def SymmetricGroupActionOnListSi(i,L): #  w=s_i,  L=[a1,a2,...,an]
r1=[]
for j in L:
r1.append(j)

t1=r1[i-1]
r1[i-1]=r1[i]
r1[i]=t1

r=r1

return r

def StablizerOfTuple(A): # A is a list, weakly increasing
k = len(A)
r=[]
for i in [1..k-1]:
t1=SymmetricGroupActionOnListSi(i,A)
if t1==A:
r.append(i)
return r

A=[1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4]
B=[3, 3, 3, 5, 5, 5, 4, 4, 4, 6, 6, 6]
kk=len(A)
#print(kk)
typ='A'
W=SymmetricGroup(kk)
w=LongestPermWInSBWInverseSA(W,A,B)
w

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At very least you can speed up things by factor of 2 or so if you take multiplication u1*w out of the inner loop.

( 2022-04-20 14:10:41 +0200 )edit

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You can try something like this:

import functools
import multiprocessing

N_CPU = multiprocessing.cpu_count()
print('CPUs:',N_CPU)

def myprod(g1,w,u2):
return max(t*w*u2 for t in g1, key=lambda t: t.length())

with multiprocessing.Pool(processes=N_CPU) as pool:
winner = max( pool.imap_unordered(functools.partial(myprod,g1,w), g2), key=lambda t: t.length() )

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( 2022-04-20 16:54:03 +0200 )edit

The first thing you can try is using max with a key function to compare the elements by their length:

sage: G = WeylGroup(['F',4])
sage: %time w_max = max(G, key=lambda w:w.length())
CPU times: user 3.09 s, sys: 18 µs, total: 3.09 s
Wall time: 3.09 s
sage: w_max.length()
24
sage: w_max
[-1  0  0  0]
[ 0 -1  0  0]
[ 0  0 -1  0]
[ 0  0  0 -1]


Is the above 3s is too slow? If yes, it would be interesting to implement a max function in Sage which would do the computation in parallel, maybe using the @parallel decorator?

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@Sebastien, thank you very much for your help. One problem is that I need to find the longest word in a subset of Weyl group (not the whole Weyl group) which satisfies certain properties. I attached the full codes. In the example in the end of the codes, it takes very long time to compute. So I would like to parallelize it and send it to a computer cluster.

( 2022-04-20 13:57:57 +0200 )edit