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In the Clustal Omega paper, it says the following about mBED ($N$ is the number of input sequences, I assume):

we use a modified version of mBed (Blackshields et al , 2010), which has complexity of $O(N \log N)$

But in the mBED paper it says the following (section 2):

A number of $t$ sequences are initially sampled from the input dataset $X .$ Following the LLR algorithm, this value is set by default to $t=\left(\log _{2} N\right)^{2} .$

After the seed sequences in $R$ have been chosen, all sequences in the input data are associated with a $t$-dimensional vector. This is done simply by computing the distances from all sequences to each of the $t$ seeds.

Then my conclusion is that this would lead to a complexity of $O(N (\log N)^2)$ because for each sequence (in total $N$ of them), distances to $t = (\log N)^2$ sequences must be calculated. Where did I do wrong?

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Turns out that I was dumb: I misread.

The Clustal Omega paper (in the supplementary materials) chose $t = \log N$, which I think originates from this claim from this part of the Linial et al. paper:

In random polynomial time $X$ may be embedded in $l_{2}^{O(\log n)}$ with distortion $\leq(1+\epsilon) \cdot c_{2}(X)$ (for any $\epsilon>0)$

Several questions still remain though:

  1. The newest implementation of Clustal Omega (that I downloaded) has source code that instead chooses $t = (\log N)^2$, why changed?
  2. The original mBED paper chooses $t = (\log N)^2$, but it was not justified on how they chose this number instead of $\log N$
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