What does tilde mean, in big-O notation?
It's a variant of the big-O that “ignores” logarithmic factors:

$$f(n) \in \tilde O(h(n))$$

is equivalent to:

$$ \exists k : f(n) \in O \!\left( h(n)\log^k(h(n)) \right) $$

From Wikipedia:


  Essentially, it is big-$O$ notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the “nitpicking” within growth-rates that are stated as too tightly bounded for the matters at hand (since $\log^k n$ is always $o(n^\varepsilon)$ for any constant $k$ and any $\varepsilon &gt; 0$).