Question

How can you compute a shortest addition chain (sac) for an arbitrary n <= 600 within one second?

Notes

This is the programming competition on codility for this month.

Addition chains are numerically very important, since they are the most economical way to compute x^n (by consecutive multiplications).

Knuth’s Art of Computer Programming, Volume 2, Seminumerical Algorithms has a nice introduction to addition chains and some interesting properties, but I didn’t find anything that enabled me to fulfill the strict performance requirements.

What I’ve tried (spoiler alert)

Firstly, I constructed a (highly branching) tree (with the start 1-> 2 -> ( 3 -> …, 4 -> …)) such that for each node n, the path from the root to n is a sac for n. But for values >400, the runtime is about the same as for making a coffee.

Then I used that program to find some useful properties for reducing the search space. With that, I’m able to build all solutions up to 600 while making a coffee. But for n, I need to compute all solutions up to n. Unfortunately, codility measures the class initialization’s runtime, too…

Since the problem is probably NP-hard, I ended up hard-coding a lookup table. But since codility asked to construct the sac, I don’t know if they had a lookup table in mind, so I feel dirty and like a cheater. Hence this question.

Update

If you think a hard-coded, full lookup table is the way to go, can you give an argument why you think a full computation/partly computed solutions/heuristics won’t work?

Answer 1

I have just got my Golden Certificate for this problem. I will not provide a full solution because the problem is still available on the site.I will instead give you some hints:

1. You might consider doing a deep-first search.
2. There exists a minimal star-chain for each n < 12509
3. You need to know how prune your search space.
4. You need a good lower bound for the length of the chain you are looking for.
5. Remember that you need just one solution, not all.

Good luck.