Before you get too excited, this is a probabilistic algorithm, not a deterministic one. Feels weird to call it an "extension" when you lose an absolute guarantee, but still cool nonetheless.
A rough sketch for a more direct way to extend the XOR trick to finding more than two differences:
For e.g. 3 differences: instead of a binary xor (i.e. binary-digit-wise sum mod 2), do a binary-digit-wise sum mod 3 (negating one input); a 0 (mod 3) sum result for a given bit means that the bit is the same in all entries, and 1 (mod 3) or 2 (mod 3) mean that you can partition on the bit, resulting in partitions with sizes `sum` and `input_different_element_count - sum`; then you repeat this recursively until they reach containing just 1 difference. (rounding the modulo up to the next power of two instead of odd modulos for the summing is perfectly fine, the final infinite-precision sum is in the range of [0; diffcount] anyway)
Extends trivially to more than 3 differences, and collapses to the basic trick for 2 differences. The accumulator size is O(log(diffcount) * element_size), but the recursive partitioning takes O(n) space or O(diffcount * n) time (plus some logarithm something maybe). Tradeoffs are probably reasonably possible, but the basic hashset approach can reduce its O(n) space requirement at the cost of taking >O(n) time too by partitioning on a hash.
iblt has low space efficiency for small sets, small elements, and low failure rates (and on that note is only probabilistic in its success).
We implemented https://github.com/bitcoin-core/minisketch which has optimal size efficiency-- N bits of state will always correctly recover when there are N or fewer bits of set difference, even when the set elements are small (like 32 bits, for example).
So for example you can you and I can each have sets of, say, ten thousand 32-bit elements which are identical except for 10 entries, and I can send you a 320 bit (32*10) sketch of my set and from that you can always determine the 10 (or fewer) differences. The same element and difference size with IBLT would likely take thousands of bits to have a low failure rate.
The downside is that the minisketch approach has quadratic decode complexity in the size of the set difference, but this is not a big deal when the number of differences is small by construction or thanks to recursive subdivision.
For cases where the differences are large iBLT eventually wins out-- the two ideas can also be hybridized in a variety of ways. E.g. using minisketch to make multi-element buckets in an iblt analogous to blocked bloom filter or the normal practice with cuckoo filters.
Another related scheme is cpisync which was used for many years by SKS key servers. It has communications efficiency like minisketch, but cubic decode costs.
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[ 0.23 ms ] story [ 25.0 ms ] threadFor e.g. 3 differences: instead of a binary xor (i.e. binary-digit-wise sum mod 2), do a binary-digit-wise sum mod 3 (negating one input); a 0 (mod 3) sum result for a given bit means that the bit is the same in all entries, and 1 (mod 3) or 2 (mod 3) mean that you can partition on the bit, resulting in partitions with sizes `sum` and `input_different_element_count - sum`; then you repeat this recursively until they reach containing just 1 difference. (rounding the modulo up to the next power of two instead of odd modulos for the summing is perfectly fine, the final infinite-precision sum is in the range of [0; diffcount] anyway)
Extends trivially to more than 3 differences, and collapses to the basic trick for 2 differences. The accumulator size is O(log(diffcount) * element_size), but the recursive partitioning takes O(n) space or O(diffcount * n) time (plus some logarithm something maybe). Tradeoffs are probably reasonably possible, but the basic hashset approach can reduce its O(n) space requirement at the cost of taking >O(n) time too by partitioning on a hash.
We implemented https://github.com/bitcoin-core/minisketch which has optimal size efficiency-- N bits of state will always correctly recover when there are N or fewer bits of set difference, even when the set elements are small (like 32 bits, for example).
So for example you can you and I can each have sets of, say, ten thousand 32-bit elements which are identical except for 10 entries, and I can send you a 320 bit (32*10) sketch of my set and from that you can always determine the 10 (or fewer) differences. The same element and difference size with IBLT would likely take thousands of bits to have a low failure rate.
The downside is that the minisketch approach has quadratic decode complexity in the size of the set difference, but this is not a big deal when the number of differences is small by construction or thanks to recursive subdivision.
For cases where the differences are large iBLT eventually wins out-- the two ideas can also be hybridized in a variety of ways. E.g. using minisketch to make multi-element buckets in an iblt analogous to blocked bloom filter or the normal practice with cuckoo filters.
Another related scheme is cpisync which was used for many years by SKS key servers. It has communications efficiency like minisketch, but cubic decode costs.
Can't we use this again? I mean:
1. Partition the data so that some batches have up to 1 missing element.
2. Recover the elements where possible with the XOR trick.
3. Pick another hash function, then repeat finding more missing elements.
4. Repeat until no more missing elements.