Big o notation for sublinear algorithm in streaming algorithm Excuse me if this is obvious.
At 1:55 of this Coursera video on streaming algorithms (see pasted image below for relevant slide), the professor mentions sublinear storage. I get the $N^{\alpha}$ but I don't get the $log^{O(1)} N$. I take it that $O(1)$ is a constant which we could set it as $k$ for example.

Does he mean that we can have $(log N)^k$ or taking $log N$ a total of $k$ times? I have read the sublinear portion of wikipedia which makes me believe it it is more like $(log N)^k$
How does this relate to $N^{\alpha}$ mentioned on the same line? Is this an alternate notation or another possible space complexity for other streaming algorithms?
A function $f$ is $\log^{O(1)}N$ if there exists a constant $k$ such that $f(n) \leq \left(\log N\right)^k$ (and not $\underbrace{\log … \log}_{k\text{ times}} N$), so you were right on this point.
The notation $N^{\alpha}$, $\alpha &lt; 1$ is NOT another notation for $\log^{O(1)}N$ but another complexity family. If you chose $0 &lt; \alpha &lt; 1$ and $0 &lt; k$, $\log ^kN$ will always be $o(N^{\alpha})$.