2
$\begingroup$

How can I assess and derive the time complexity of the BBP formula? $$ BBP(n)=4S(1,n) - 2S(4,n) - S(5,n) - S(6,n) $$ where

$$ S(j,n) = \sum_{k=0}^n{\frac{16^{n-k}mod(8k+j)}{8k+j}}+\sum_{k=n+1}^{\infty}{\frac{16^{n-k}}{8k+j}} $$

Note that the modulo is evaluated using the binary modular exponentiation algorithm (which is supposedly O(log N)) and the second sum iterates only over a few elements, i.e. up to a certain $\epsilon$.

I got O(N log N), but even if it is correct, I can't derive it.

$\endgroup$
4
$\begingroup$

A good place to start is the BBP paper. In Section 3 (1st paragraph of page 8) they state that the running time for $n$ digits is $O(n\log nM(\log n))$, where $M(m)$ is the complexity of integer multiplication; currently the best known algorithms give $M(m) = O(m\log m 2^{O(\log^* m)})$ bit complexity, though Fürer himself (inventor of the first algorithm with that complexity) argues that this is not the correct complexity measure.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.