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 by \(O(g(n,m))\) the set of functions

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

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

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

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