Exercise 3.1.8

We can extend our notation to the case of two parameters $n$ and $m$ that can go to infinity independently at different rates. For a given function $g(n, m)$ we denote $O(g(n, m))$ the set of functions:

$$ \begin{align} O(g(n, m) = \lbrace f(n, m): &\text{there exist positive constants } c, n_0, \text{ and } m_0 \\ &\text{such that } 0 \leq f(n, m) \leq cg(n, m) \\ &\text{for all } n \geq n_0 \text{ or } m \geq m_0. \end{align} $$

Give corresponding definitions for $\Omega(g(n, m))$ and $\Theta(g(n, m))$.

In the University of Sofia, we woud have writen that tersely.

$$ \begin{align} \Omega(g(n, m) = \lbrace f(n, m): &\text{there exist positive constants } c, n_0, \text{ and } m_0 \\ &\text{such that } 0 \leq cg(n, m) \leq f(n, m) \\ &\text{for all } n \geq n_0 \text{ or } m \geq m_0. \end{align} $$

$$ \begin{align} \Theta(g(n, m) = \lbrace f(n, m): &\text{there exist positive constants } c_1, c_2, n_0, \text{ and } m_0 \\ &\text{such that } 0 \leq c_1g(n, m) \leq f(n, m) \leq c_2g(n, m) \\ &\text{for all } n \geq n_0 \text{ or } m \geq m_0. \end{align} $$