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19:49 < maaku> I have a CS theory question for this group. A Huffman code is optimal if the probabilities are known and fixed. However is there a method for constructing trees that try to minimize the path for accessing multiple elements?
19:49 < maaku> Obviously the frequency of each A,B pairing (for accessing 2 items) or A,B,C (3, etc.) would be given.
19:50 < maaku> And by "minimize path" I mean that shared path prefixes are not double-( or triple-)counted
19:52 < kanzure> if you allow for stepwise increase in path length cost, your method could be just put the frequently-accessed stuff at the top of the tree, then for double cost the less popular stuff at the next level, and so on.
19:53 < kanzure> (assuming binary tree)
19:57 < kanzure> H(root) -> (H(tree(H(tree(...)), H(C)), H(B)), H(A)) so H(A) is cheapest to lookup, B is more expensive, C is even more expensive.
19:57 < sipa> maaku: what do you mean with 'multiple elements' ?
19:58 < gmaxwell> maaku: huffman tress also require that the probablities are all 1/2^n for optimality. (though not your question)  ... I know how to construct ones that have a depth limit, which seems relevant to your interests.
20:01 < sipa> maaku: oh, i see what you're asking
20:03 < gmaxwell> I think this shared prefix property might also just be obeyed by huffman codes, if you assume that the probablities are independant.
20:03 < maaku> gmaxwell: is there anything better than Huffman codes for non-binary probabilities?
20:04 < gmaxwell> absolutely, but nothing that maps into a binary decision tree, which is what I assume you want. (because I assume you're using this for hashtree layouts)
20:05 < maaku> gmaxwell: yes, and yes. with those restrictions I think Huffman is optimal, but would like to be corrected if wrong
20:06 < maaku> kanzure: that's what I'm trying to do. my question is more, "for given probability distributions, what's the optimal tree shape?"
20:07 < maaku> gmaxwell: constructing with a depth limit would be interesting. not what I'm working on right now, but I'd be good to know how to do that
20:08 < kanzure> uhrm for the tree sizes you are considering, is it practical to just run some quick iterative emperical method to figure that out?   and then you cna look at the results to answer your question (or if it's fast enough, just include that in your implementation anyway).
20:16 < maaku> Specifically I'm trying to find tree shapes that I should be benchmarking serialization choices against. "Maximally-unbalanced (a list) at the top, then a fully-balanced tree at the bottom" is a good intuition for what this *should* look like, based on real world use cases
20:16 < maaku> Something like this: https://imgur.com/7oL0LGq
20:17 < maaku> But I'm looking for a theoretical justification for that -- constructing a justifiable probability model, then making an optimal tree.
20:28 < gmaxwell> https://en.wikipedia.org/wiki/Package-merge_algorithm  is how you solve the capacited depth version; this https://arxiv.org/pdf/cs/0701012  also looks relevant
20:33 < maaku> thank you
20:35 < maaku> Oh, and here's a practical justification: key tree threshold signatures where 'individual' signers use their own internal thresholds. for example, a 4-way script has an 'everybody signs' top-level conditional that must actually be the combinatorial possibilities of each individual signer
20:37 < maaku> and in many cases each actor would specify their own probability distribution for their sub-key combinations. e.g. if it is 2-of-3 there is 84% confidence one set of 2 keys will be used, 15% confidence the other set, and 1% for the worst-case paper key recovery
20:39 < maaku> that could easily chop up high level reasoning about what the tree structure "should" be, and at some point it becomes better to add probability semantics to the key-tree-like signing condition language and have it auto-generate tree structures via a standard method
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