Assume that we measure the complexity of an algorithm (for some problem) by two parameters $n$ and $m$ (where $m \le n$). What is the formal interpretation of the following claim: there is no algorithm that solves the given problem in $o(m + \log{n})$?
In particular, does it mean that an $O(\log{n})$ algorithm is possible?
What this means is:
For every algorithm solving the problem, it is not the case that the worst-case running time $T(m,n)$ in terms of $m,n$ satisfies the following property: for every $c > 0$ there is $m_0(c)$ such that for all $n \geq m \geq m_0(c)$, $$ T(m,n) < c(m + \log n). $$
This doesn't rule out an $O(\log n)$ algorithm, since $\log n$ doesn't satisfy the property written above.
