Is there another way to write Θ(max(m, n)^2 * min(m, n))?

Steps to Solve

1. Identify the original expression

The original expression is given as $ \theta(\max(m, n)^2 * \min(m, n)) $.

2. Rewrite using constants

To better manipulate the expression, recognize that $\max(m, n)$ can be represented in terms of $m$ and $n$:

• Let $M = \max(m, n)$
• Let $m = \min(m, n)$

Thus, we can rewrite the expression as $ \theta(M^2 * m) $.

3. Relationship between max and min

In any case, we can express the maximum and minimum in a different way. We know that: $$ M^2 = M \cdot M $$ and since $M$ is the overall largest value, we can analyze it based on its relation with $m$.

4. Substitute back into the expression

Since $ M = \max(m, n) $, we can say: $$ \theta(M^2 * m) = \theta(\max(m, n) * \max(m, n) * \min(m, n)) $$

5. Using properties of $\theta$ notation

The complexity $\theta$ implies both upper and lower bounds, thus we can just mention these bounds but in terms of either $m$ or $n$. We can write: $$ \theta(\max(m, n)^2 * \min(m, n)) = \theta(n^2 * m) $$

6. Final alternative representation

Hence, the final representation could be simplified into: $$ \theta(m^2*n) \text{ where } m = \min(m, n), n = \max(m, n). $$