boyd

Throwing cores at the problem with `pytest-xdist` is typically the lowest hanging fruit, but you still hit all the paper cuts the authors mention -- collection, DB fixtures, import time, etc.

And, further optimization is really hard when the CI plumbing starts to dominate. For example, the last Warehouse `test` job I checked has 43s of Github Actions overhead for 51s of pytest execution time (half the test action time and approaching 100% overhead).

Disclosure: Have been tinkering on a side project trying to provide 90% of these pytest optimizations automatically, but also get "time-to-first-test-failure" down to ~10 seconds (via warm runners, container snapshotting, etc.). Email in profile if anyone would like to swap notes.

I think we're somewhat talking past one another -- in any case, we'll add more in the README on minimal perfect hash functions and the differences. In short, you'd need to also have a data structure (e.g., a Bloom filter) for checking if the key is in your MPHF and then also a mapping of 1..n MPHF values to your actual values.

> PTHash and other minimum perfect hash functions return an arbitrary value if the query key did not exist when building the MPHF, so they can be a lot smaller. B-field can identify query keys that don't exist in the set (with high probability?).

Yes, exactly.

> What I'm wondering is why the Kraken2 probabilistic hash table doesn't work.

I just skimmed the paper again (has been a while since a close reading), but my refreshed understanding is:

* Like the B-field, there are also false positives.

* When multiple hashed keys (k-mers) collide in the Kraken2 hash table, it has to store a "reduced" value for those key-value pairs. While there's an elegant solution for this issue for the problem of taxonomic classification (lowest common ancestor), it still results in a loss of specificity. There's a similar issue with "indeterminate" results in the B-field, but this rate can be reduced to ~0 with secondary arrays.

* The original Kraken2 paper describes using 17 bits for taxonomic IDs (~131K unique values). I don't know how many tax IDs current Kraken2 DB builds use offhand, but the error rate climbs significantly as you use additional bits for the value vs. key (e.g., to represent >=2^20 values, see Fig S4). I don't have a good sense for the performance and other engineering tradeoffs of just extending the hash code >32 bits. I also don't know what the data structure overhead is beyond those >32 bits/pair.

So, for a metagenomics classifier specifically, some subtle tradeoffs but honestly database quality and the classification algorithm likely matters a lot more than the marginal FP rates with either data structure -- we just happen to have come to this solution.

For other applications, my sense is a B-field is generally going to be much more flexible (e.g., supporting arbitrary keys vs. a specific fixed-length encoding) but of course it depends on the specifics.

... meaning it is an "injective" function that maps unique key-value pairs, correct? Genuinely asking, I have glancing familiarity via their use in assembly algorithms but (a) don't have a formal math/CS background; and (b) haven't read any of the papers recently.

My understanding is that a perfect hash function maps elements elements to a unique integer (i.e., it's a one-to-one mapping). I think PHF data structures will also always return a value. So if you look up an element not in the constructed PHF, you'll always get a "false positive" value.

In contrast, a B-field lets you map a key to an arbitrary number of (typically non-unique) values. So I could map a million elements to "1", another million to "2", etc.

I'm not especially current (or fluent!) in that literature though, so would love pointers to anything that doesn't have the above constraints.

So a bitmap index requires a bit per unique value IIRC (plus the key and some amount of overhead). So for ~32 unique values you're already at 4 bytes, 40 bytes per key-value pair for 320 values, etc.

In comparison, a B-field will let you store 32 distinct values at ~3.4 bytes (27 bits) per key-value pair at a 0.1% FP rate.

Thank you! The "Space Requirements" section in the README has a few examples, and your comment has made me realize our (micro-)benchmark link in the README is broken.

We'll get that fixed and maybe find the time to do a larger post with some benchmarks on both space/time tradeoffs and overall performance vs. other data structures.

Ah, we need to clarify the language! The B-field will always return the correct value for an inserted key.

False positives are only returned for keys that have not been inserted. This is akin to a Bloom filter falsely returning that a key is in the set).

I think those are both good examples of where you can manage the cost of a false positive.

In genomics, we're using this to map a DNA substring (or "k-mer") to a value. We can tolerate a very low error rate for those individual substrings, especially since any erroneous values will be random (vs. having the same or correlated values). So, with some simple threshold-based filtering, our false positive problem goes away.

Again, you'll never get the incorrect value for a key in the B-field, only for a key not in the B-field (which can return a false positive with a low, tunable error rate).