How can repeated addition/multiplication be done in polynomial time?

Question

I can see how adding 2 unsigned four-bit values is $O(n)$. We just go from the rightmost digits to the leftmost digits and add the digits up sequentially. We can also perform multiplication in polynomial time ($O(n^2)$) via the algorithm we all learned in grade school.

However, how can we add up or multiply say $i$ numbers together in polynomial time? After we add up 2 numbers together, we get a bigger number that will require more bits to represent. Same with multiplication.

How can we ensure that these extra bits do not produce exponential blowup?

Answer 1

From what you said, I suppose that you consider complexity in terms of number of bits of the input.

Say you add up two numbers $a$ and $b$, with respectively $n_1$ and $n_2$ bits, then the result is at most $\max(n_1, n_2) + 1$ bits since $a + b \leq 2 \times \max(a, b)$.

For the multiplication, the result is at most $2 \times \max(n_1, n_2)$ bits since $a \times b \leq \max(a, b)^2$.

The important thing is that the number of bits necessary to represent the sum / product of two numbers is polynomial in the number of bits to represent those numbers. Hence, for $i$ numbers, the number of bits necessary to represent the answer is still polynomial in the number of bits of the input (since composition of polynomial functions yields polynomial functions).

Edit: actually, the really important thing is that $i$ is not part of the input, it is fixed. Otherwise, if you take for instance $2 \times 2 \times \ldots \times 2 = 2^i$, the answer takes $2^{i-1}$ bits, while the input takes $i + \log i$ bits. But if $i$ is fixed, then $2^{i-1}$ is a constant. This kind of distinction happens quite often in computer science, especially in the field of fixed parameter tractability.