[+] A4 P3.2

assignments/a4/a4.tex

@@ -268,6 +268,20 @@ Since there are only constant-time operations outside the loop, the runtime of t
 \item[2.] 
 Running-time analysis of \texttt{starting\_coprime\_numbers}.
 
+Let $P$ be the size of the input set \code{primes}, and let $m$ be the product of the numbers in \code{primes}.
+
+Let $c_0, \dots, c_n$ be constants that doesn't depend on any variables.
+
+The inner loop \code{for p in primes} runs $P$ iterations, and since each iteration contains only a constant number of constant-time operations, taking $c_0P$ steps where $c_0$ is a constant. The outer loop iterates $m - 1$ times, taking $c_0P + c_1$ steps each time, with a total of $(m - 1)(c_0P + c_1)$ steps.
+
+Since there are only constant-time operations outisde the outer loop, the entire function will take $(m - 1)(c_0P + c_1) + c_2$ steps.
+
+Thus, the total number of basic operations is:
+\begin{align}
+RT_{\code{starting\_coprime\_numbers}}(P, m) &= (m - 1)(c_0P + c_1) + c_2 \\
+& = c_0mP + c_1m - c_0P + c_3 \\
+& \in \Theta(mP)
+\end{align}
 \item[3.]
 TODO: Running-time analysis of \texttt{coprime\_to\_all}.
 \end{enumerate}