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Re: Re: Permanent Computation Efficiency

Sunt wrote:
> Daniel, thank you for your suggestion!
> However, I still didn't quite know the mechanism of the function Coefficient[].
> And how to determine the computation complexities of the two permanent[]
> functions by big O notation?
> In my opinion, the complexity of permanent2[] is O(n!). Is it right?
> Thanks a lot!
> Sunt

Coefficient is taking derivatives, without expanding the expression. Not 
sure what is the complexity of this approach. It cannot be better than 
exponential, if i recall permanent complexity correctly. Probably 
factorial in dimension of the matrix.

The complexity of permanent2[] seems to be O(n!) for basic symbolic 
matrices of the sort I used. I base that on some simple timings using

mat[n_] := Array[m, {n,n}];

Timing[p9b = permanent2[mat[9]];]
Timing[p10b = permanent2[mat[10]];]
Timing[p11b = permanent2[mat[11]];]

Since the sub-permenents share sub-sub-computations, and memo-ization is 
used, I'm not sure what is the complexity in general. When I use integer 
entries, the methods behave as though they might be better than O(n!), 
and indeed permanent2 seems to get five times slower for every jump of 2 
in n. If this reflects the actual complexity, the improvement over n! is 
due to substantial use of memory.

Daniel Lichtblau
Wolfram Research

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