opus

onstraint that the they span a range of <=N. https://github.com/sipa/bitcoin/blob/202006_cuckoo_filter/sr... If you're lucky there will be closed form expressions for the encoding and decoding equations. (There are for both of the above, at least for some parameters, but in both those examples the implementations use small tables because for the ranges involved the tables end up being faster than sqrts). Another simple example of these is converting subsets of N out of a collection of M elements, where the underlying counting function is just the binomial function. I often do this when constructing tests for trying many combinations of choices without a whole bunch of nested loops or uglier constructs. Though if you don't need indexing iterating can of often be more simply done in other ways. For a particularly extreme example, to iterate the subsets define your membership as the bits in an unsigned integer. Start with N least significant bits set then update your state each step like so (stopping when the N most significant bits are set): t=(state&-state)+state; state=t|((1<<(popcount(state^t)-2))-1);

coding 4 sorted generation numbers with the requirement that all 4 are within a window of half the total range[2]. In prior published work on cuckoo filters used a large table of all possible values of combinations with replacement (to efficiently pack small sorted numbers). We found the algebraic code to be faster than a big dumb table, presumably due to cache locality, even though our fastest encoder/decoder still use tables but only tiny ones (at least for the sizes we were considering). A challenge for implementing these sorts of functions is that their inverses often require operations like integer square or cube roots which are not particularly fast unless the ranges are small enough to implement them with tables. [1] https://github.com/xiph/opus/blob/master/celt/cwrs.c#L74 [2] https://github.com/sipa/bitcoin/blob/202006_cuckoo_filter/sr...