Evaluate $a^n$ using the binary expansion of $n$

Asked Jun 02 '21 at 21:18

Active Jun 02 '21 at 21:18

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Use the following algorithm to devise an algorithm for computing $a^n$,

Hint: use binary expansion of $n$ to evaluate $a^n$.

The following algorithm is based on the above algorithm to evaluate $a^n$, but I don't see that it does anything extra and I don't see it using the hint too,

Question: We can use both algorithms above to evaluate $a^n$. I don't see how algorithm presented above based on the first algorithm above does answer the question?

asked Jun 02 '21 at 21:18

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The first procedure calculates $a^{2^n}$ while the second one calculates $a^n$. See for example Fast exponentiation algorithm - How to arrive at it?. – dxiv Jun 02 '21 at 21:28
@dxiv. Thank you. The question is to devise algorithm based on the first algorithm to evaluate $a^n$. So you are saying the second algorithm does answer the question please? – Avv Jun 02 '21 at 21:30
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I think that it is important to understand descending recursive functions. For example, an algorithm to return $(n!) ~: n \in \Bbb{Z^+}$ is : if $n=1$ then return $(1)$. Else return $[n \times (n-1)!].$ – user2661923 Jun 02 '21 at 21:31
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@Avra No, not saying that. I suggest you read the linked question and answers. – dxiv Jun 02 '21 at 21:33
@dxiv. This is very helpful. Thank you. I see this is known as exponentiaon by squaring. – Avv Jun 02 '21 at 21:57
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This becomes crystal clear if you write the exponent in binary Horner form, e.g. see this answer in the dupe. – Bill Dubuque Jun 02 '21 at 22:04
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@Avra Or sometimes square-and-multiply. You can find several questions about this on SO as well. – dxiv Jun 02 '21 at 22:07

Evaluate $a^n$ using the binary expansion of $n$

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