### Bloom filter and cuckoo filter

The multi-level cuckoo filter (MLCF) in SlimDB builds on the cuckoo filter (CF) so I read the cuckoo filter paper. The big deal about the cuckoo filter is that it supports delete and a bloom filter does not. As far as I know the MLCF is updated when sorted runs arrive and depart a level -- so delete is required. A bloom filter in an LSM is per sorted run and delete is not required because the filter is created when the sorted run is written and dropped when the sorted run is unlinked.

I learned of the blocked bloom filter from the cuckoo filter paper (see here or here). RocksDB uses this but I didn't know it had a name. The benefit of it is to reduce the number of cache misses per probe. I was curious about the cost and while the math is complicated, the paper estimates a 10% increase on the false positive rate for a bloom filter with 8 bits/key and a 512-bit block which is similar to a typical setup for RocksDB.

I am always interested in things that use less space for filters and block indexes with an LSM so I spent time reading the paper. It is a great paper and I hope that more people read it. The cuckoo filter (CF) paper claims better space-efficiency than a bloom filter and the claim is repeated in the SlimDB paper as:

The paper has a lot of interesting math that I was able to follow. It provides formulas for the number of bits/key for a bloom filter, cuckoo filter and semisorted cuckoo filter. The semisorted filter uses 1 less bit/key than a regular cuckoo filter. The formulas assuming E is the target false positive rate, b=4, and A is the load factor:

The target load factor is 0.95 (A = 0.95) and that comes at a cost in CPU overhead when creating the CF. Assuming A=0.95 then a bloom filter uses 10 bits/key, a cuckoo filter uses 10.53 and a semisorted cuckoo filter uses 9.47. So the cuckoo filter uses either 5% more or 5% less space than a bloom filter when the target FPR is 1% which is a different perspective from the quote I listed above. Perhaps my math is wrong and I am happy for an astute reader to explain that.

When the target FPR rate is 0.1% then a bloom filter uses 15 bits/key, a cuckoo filter uses 13.7 and a semisorted cuckoo filter uses 12.7. The savings from a cuckoo filter are larger here but the common configuration for a bloom filter in an LSM has been to target a 1% FPR. I won't claim that we have proven that FPR=1% is the best rate and recent research on Monkey has shown that we can do better when allocating space to bloom filters.

The first graph shows the number of bits/key as a function of the FPR for a bloom filter (BF) and cuckoo filter (CF). The second graph shows the ratio for bits/key from BF versus bits/key from CF. The results for semisorted CF, which uses 1 less bit/key, are not included. For the second graph a CF uses less space than a BF when the value is greater than one. The graph covers FPR from 0.00001 to 0.09 which is 0.001% to 9%. R code to generate the graphs is here.

From the paper there is more detail on CPU efficiency in table 3, figure 5 and figure 7. Table 3 has the speed to create a filter, but the filter is much larger (192MB) than a typical per-run filter with an LSM and there will be more memory system stalls in that case. Regardless the blocked bloom filter has the least CPU overhead during construction.

Figure 5 shows the lookup performance as a function of the hit rate. Fortunately performance doesn't vary much with the hit rate. The cuckoo filter is faster than the blocked bloom filter and the block bloom filter is faster than the semisorted cuckoo filter.

Figure 7 shows the insert performance as a function of the cuckoo filter load factor. The CPU overhead per insert grows significantly when the load factor exceeds 80%.

I learned of the blocked bloom filter from the cuckoo filter paper (see here or here). RocksDB uses this but I didn't know it had a name. The benefit of it is to reduce the number of cache misses per probe. I was curious about the cost and while the math is complicated, the paper estimates a 10% increase on the false positive rate for a bloom filter with 8 bits/key and a 512-bit block which is similar to a typical setup for RocksDB.

**Space Efficiency**I am always interested in things that use less space for filters and block indexes with an LSM so I spent time reading the paper. It is a great paper and I hope that more people read it. The cuckoo filter (CF) paper claims better space-efficiency than a bloom filter and the claim is repeated in the SlimDB paper as:

The tl;dr for me is that the space savings from a cuckoo filter is significant when the false positive rate (FPR) is sufficiently small. But when the target FPR is 1% then a cuckoo filter uses about the same amount of space as a bloom filter.However, by selecting an appropriate fingerprint size f and bucket size b, it can be shown that the cuckoo filter is more space-efficient than the Bloom filter when the target false positive rate is smaller than 3%

The paper has a lot of interesting math that I was able to follow. It provides formulas for the number of bits/key for a bloom filter, cuckoo filter and semisorted cuckoo filter. The semisorted filter uses 1 less bit/key than a regular cuckoo filter. The formulas assuming E is the target false positive rate, b=4, and A is the load factor:

- bloom filter: ceil(1.44 * log2(1/E))
- cuckoo filter: ceil(log2(1/E) + log2(2b)) / A == (log2(1/E) + 3) / A
- semisorted cuckoo filter: ceil(log2(1/E) + 2) / A

The target load factor is 0.95 (A = 0.95) and that comes at a cost in CPU overhead when creating the CF. Assuming A=0.95 then a bloom filter uses 10 bits/key, a cuckoo filter uses 10.53 and a semisorted cuckoo filter uses 9.47. So the cuckoo filter uses either 5% more or 5% less space than a bloom filter when the target FPR is 1% which is a different perspective from the quote I listed above. Perhaps my math is wrong and I am happy for an astute reader to explain that.

When the target FPR rate is 0.1% then a bloom filter uses 15 bits/key, a cuckoo filter uses 13.7 and a semisorted cuckoo filter uses 12.7. The savings from a cuckoo filter are larger here but the common configuration for a bloom filter in an LSM has been to target a 1% FPR. I won't claim that we have proven that FPR=1% is the best rate and recent research on Monkey has shown that we can do better when allocating space to bloom filters.

The first graph shows the number of bits/key as a function of the FPR for a bloom filter (BF) and cuckoo filter (CF). The second graph shows the ratio for bits/key from BF versus bits/key from CF. The results for semisorted CF, which uses 1 less bit/key, are not included. For the second graph a CF uses less space than a BF when the value is greater than one. The graph covers FPR from 0.00001 to 0.09 which is 0.001% to 9%. R code to generate the graphs is here.

**CPU Efficiency**From the paper there is more detail on CPU efficiency in table 3, figure 5 and figure 7. Table 3 has the speed to create a filter, but the filter is much larger (192MB) than a typical per-run filter with an LSM and there will be more memory system stalls in that case. Regardless the blocked bloom filter has the least CPU overhead during construction.

Figure 5 shows the lookup performance as a function of the hit rate. Fortunately performance doesn't vary much with the hit rate. The cuckoo filter is faster than the blocked bloom filter and the block bloom filter is faster than the semisorted cuckoo filter.

Figure 7 shows the insert performance as a function of the cuckoo filter load factor. The CPU overhead per insert grows significantly when the load factor exceeds 80%.

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