What does $\omega{(1)}\cdot O(logN)$ mean?
It presumably means a product of the form $f(N)\,g(N)$, where $f(N)\in\omega(1)$ and $g(N) \in O(\log n)$.

However, that's a very strange thing to write, since every well-behaved function&nbsp;$h$ can be written in this way. If $h\in\omega(1)$, write it as $h(N)\cdot1$, which is valid, since $1\in O(\log N)$; if $h\in O(1)$, write it was $N\cdot (h(N)/N)$.