What is this article about

I believe everyone will FWT and FMT.

If you don't , I recommend you to have a look at VFK Of 2015 National training team .

Let the complete set be \(U=\{1,2,\ldots,n\}\), Suppose we care about \(f_S\) The set in \(S\) yes \(U\) Subset .

Here you are. \(c_i,d_i\), Make

The product of two set power series is set merging convolution (or)/ Set symmetric difference convolution (xor) One of the .

Might as well set \(d_S\) Different from each other ( Otherwise you can use DP/ Combination number or something ).



Do it again for each set power series FMT/FWT, And then we'll just ride them together , Switch back .

Time complexity :\(O(n4^n)\)

It's too slow , Because it doesn't use the special condition of this problem .

Set or convolution

For a set power series \(f\), Definition \(f\) The Mobius transformation of is a set power series \(\hat f\), among
\hat f_S=\sum_{T\subseteq S}f_T
In turn, , Definition \(\hat f\) The Mobius inversion of is \(f\), From the inclusion exclusion principle, we can get
f_S=\sum_{T\subseteq S}{(-1)}^{|S|-|T|}\hat f_T
I believe everyone is familiar with the above contents .

Back to the formula we asked for :
\hat g_T&=\prod_S \hat{f_{S}}_T\\
&=\prod_S \sum_{K\subseteq T}f_{S,K}\\
&=\prod_{S\subseteq T}{(1+a_S)}
It's similar to the ordinary Mobius transformation ?

Put all the \(a_S\) add \(1\), Change the addition of Mobius transformation to multiplication , You can get \(\hat g_T\) 了 .

Time complexity :\(O(n2^n)\)

Set symmetric difference convolution

We still need to use those formulas .
\hat g_T&=\prod_S\hat {f_S}_T\\
&=\prod_S\sum_{K}f_{S,K}{(-1)}^{|K\cap T|}\\
&=\prod_S(1+a_S{(-1)}^{|S\cap T|})
This is also very similar to Walsh transformation , however \({(-1)}^{|S\cap T|}\) Only in \(a_S\) above , So we can't put \(a_S\) Add \(1\) After doing the variant Walsh transformation .

But we can maintain another \(\hat h_T=\prod_S(1-a_S{(-1)}^{|S\cap T|})\), Take the Walsh transformation of \(-\hat g_T\) Replacing the \(\hat h_T\), You can do it .

Time complexity :\(O(n2^n)\)


First pit


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