All hash table sizes you will ever need

When picking a hash table size we usually have two choices: Either, we pick a prime number or a power of 2. Powers of 2 are easy to use, as a modulo by a power of 2 is just a bit-wise and, but 1) they waste quite a bit of space, as we have to round up to the next power of 2, and 2) they require "good" hash functions, where looking at just a subset of bits is ok.

Prime numbers are more forgiving concerning the hash function, and we have more choices concerning the size, which leads to less overhead. But using a prime number requires a modulo computation, which is expensive. And we have to find a suitable prime number at runtime, which is not that simple either.

Fortunately we can solve both problems simultaneously, which is what this blog post is about. We can tackle the problem of finding prime numbers by pre-computing suitable numbers with a given maximum distance. For example when when only considering prime numbers that are at least 5% away from each other we can cover the whole space from 0 to 2^64 with just 841 prime numbers. We can solve the performance problem by pre-computing the magic numbers from Hacker's Delight for each prime number in our list, which allows us to use multiplications instead of expensive modulo computations. And we can skip prime numbers with unpleasant magic numbers (i.e., the ones that require an additional add fixup), preferring the next cheap prime number instead.

The resulting code can be found here. It contains every prime number you will ever need for hash tables, covering the whole 64bit address space. Usage is very simple, we just ask for a prime number and then perform modulo operations as needed:

class HashTable {
   primes::Prime prime;
   vector table;
public:
   HashTable(uint64_t size) { 

      prime = prime::Prime::pick(size);

      table.resize(prime.get());

   }
   ...
   Entry* getEntry(uint64_t hash) { return table[prime.mod(hash)]; }
   ...
};

The performance is quite good. On an AMD 1950X, computing the modulo for 10M values (and computing the sum of the results) takes about 4.7ns per value when using a plain (x%p), but only 0.63ns per value when using p.mod(x).

Getting this into unordered_map would be useful, it would probably improve the performance quite significantly when we have few cache misses.

Cuckoo Filters with arbitrarily sized tables

Cuckoo Filters are an interesting alternative to Bloom filters. Instead of maintaining a filter bitmap, they maintain a small (cuckoo-)hash table of key signatures, which has several good properties. For example is stores just the signature of a key instead of the key itself, but is nevertheless able to move an element to a different position in the case of conflicts.

This conflict resolution mechanism is quite interesting: Just like regular cuckoo hash tables each element has two potential positions where is be placed, a primary position i1 and a secondary position i2. These can be computed as follows:

i1 = hash(x)

i2 = i1 xor hash(signature(x))

Remember that the cuckoo filter stores only the (small) signature(x), not x itself. Thus, when we encounter a value, we cannot know if it is at its position i1 or position i2. However, we can nevertheless alternate between positions because the following holds

i1 = i2 xor hash(signature(x))

and we have the signature stored in the table. Thus, we can just use the self-inverse xor hash(signature(x)) to switch between i1 and i2, regardless of whether are currently at i1 or i2. Which is a neat little trick. This allows is to switch between positions, which is used in the cuckoo filter conflict resolution logic.

However all this hold only because the original cuckoo filters use power-of-two hash tables. If our hash table size is not a power of 2, the xor can place the alternative position beyond the size of the hash table, which breaks the filter. Thus cuckoo filter tables always had to be powers of two, even if that wasted a lot of memory.

In more recent work Lang et al. proposed using cuckoo filters with size C, where C did not have to be a power of two, offering much better space utilization. They achieved this by using a different self-inverse function:

i1 = hash(x) mod C

i2 = -(i1 + hash(signature(x)) mod C

Note that the modulo computation can be made reasonable efficient by using magic numbers, which can be precomputed when allocating the filter.

A slightly different way to formulate this is to introduce a switch function f, which switches between positions:

f(i,sig,C) = -(i + hash(sig)) mod C

i1 = hash(x) mod C

i2 = f(i1, signature(x), C)

i1 = f(i2, signature(x), C)

All this works because f is self-inverse, i.e.,

i = f(f(i, signature(x), C), signature(x), C)

for all C>0, i between 0 and C-1, and signature(x)>0.

The only problem is: Is this true? In a purely mathematical sense it is, as you can convince yourself by expanding the formula, but the cuckoo filters are not executed on abstract machines but on real CPUs. And there something unpleasant happens: We can get numerical overflows of our integer registers, which implicitly introduces a modulo 2^32 into our computation. Which breaks the self-inverseness of f in some cases, except when C is power of two itself.

Andreas Kipf noticed this problem when using the cuckoo filters with real world data. Which teaches us not to trust in formulas without additional extensive empirical validation... Fortunately we can repair the function f by using proper modular arithmetic. In pseudo-code this looks like this

f(i,sig,C)

    x=(C-1)-(hash(sig) mod C)

    if (x>=i)

       return (x-i);

    // The natural formula would be C-(i-x), but we prefer this one...

    return C+(x-i);

This computes the correct wrap-around module C, at the cost of one additional if. We can avoid the if by using predication, as shown below

f(i,sig,C)

    x=(C-1)-(hash(sig) mod C)

    m = (x<i)*(~0)

    return (m&C)+(x-i);

which can be attractive for SSE implementations where the comparison produces a bit mask anyway.

We have validated that this new f function is now self-inverse for all possible values of i, sig, and C. And we did this by not just looking at the formula, but by trying out all values programmatically. Which is a good way to get confidence in your approach; there is only a finite number of combinations, and we can test them all.

With this small fix, we can now enjoy Cuckoo Filters with arbitrarily sized tables.

Edit:  The original post did not mirror the hash space correctly (using C-... instead of (C-1)-...), thanks to Andreas Kipf for pointing this out.

Why use learning when you can fit?

We recently had a talk by Tim Kraska in our group, and he spoke among other things about learned indexes. As I had mentioned before, I am more in favor of using suitably implemented b-trees, for reasons like update friendliness and distribution independence. But nevertheless, the talk made me curious: The model they are learning is in the end very primitive. It is a two-level linear model, i.e., they are using a linear function to select another linear function. But if that is enough, why do we need machine learning? A simple function fit should work just as well.

Thus, I tried the following:
1) we sort all data and keep it in an array, just like with learned indexes
2) we build the CDF
3) we fit a linear spline to the CDF minimizing the Chebyshev norm
4) we fit a polynomial function to the spline nodes
5) now we can lookup a value by evaluating first the polynomial function, then the spline, and then retrieving the values from the array. The previous step is always the seed to a local search in the next step.

As we bound the Chebyshev norm in each step, the lookup is in O(1), without any need for machine learning or other magic boxes.

Now admittedly there was some weasel wording in the previous paragraph: The lookup is in O(1), but the "constant" here is the Chebyshev norm of the fit, which means this only works well if we can find the good fit. But just the same is true for the learned indexes, of course.

Now do we find a good fit? In theory we know how to construct the optimal fit in O(n log n), but that paper is beyond me. I am not aware of any implementation, and the paper is much too vague to allow for one. But constructing a good fit is much easier, and can also be done in O(n log n). Using that algorithm, we can construct a linear spline that maximum error efficiently, and we know what the maximum is over the whole domain. Thus, we can probe the spline to get an estimate for the real value position, and we then can perform an efficient local search on a small, known, window of the data.

The only problem is evaluating the spline itself. Evaluating a linear spline is pretty cheap, but we have to find the appropriate knot points to evaluate. Traditionally, we find these with binary search again. Note that the spline is much smaller than the original data, but still we want to avoid the binary search. Thus, we construct a polynomial function to predict the spline knot, again minimizing the Chebyshev norm, which allows us to consider only a small subset of spline nodes, leading to the before mentioned time bound.

How well does this work in practice? On the map data set from the learned indexes paper and a log normal data set we get the following. (The learned indexes numbers are from the paper, the b-tree numbers are from here, and the spline numbers from this experiments. I still do not really know what the averages mean for the learned indexes, but probably the average errors averaged over all models).

Map data	size (MB)	avg error
Learned Index (10,000)	0.15	8 ± 45
Learned Index (100,000)	1.53	2 ± 36
B-tree (10,000)	0.15	225
B-tree (100,000)	1.53	22
Spline (10,000)	0.15	193
Spline (100,000)	1.53	22

Log normal data	size (MB)	avg error
Learned Index (10,000)	0.15	17,060 ± 61,072
Learned Index (100,000)	1.53	17,005 ± 60,959
B-tree (10,000)	0.15	1330
B-tree (100,000)	1.53	3
Spline (10,000)	0.15	153
Spline (100,000)	1.53	1

Somewhat surprising the accuracy the accuracy of the spline is nearly identical to the interpolating b-tree for the real-world map data, which suggests that the separators span the domain reasonably well there. For the log normal data the spline is significantly better, and leads to nearly perfect predictions. Note that the original data sets contains many millions of data points in both cases, thus the prediction accuracy is really high.

For practical applications I still recommend the B-tree, of course, even though the polynomial+spline solution is in "O(1)" while the B-tree is in O(log n). I can update a B-tree just fine, including concurrency, recovery, etc., while I do not know how to do that with the polynomial+spline solution.
But if one wants to go the read-only/read-mostly route, the function functions could be attractive alternative the machine learning. The advantage of using fits is that the algorithms are reasonably fast, we understand how they work, and they give strong guarantees for the results.

Honest asymptotic complexity for search trees and tries

Fun fact that was pointed out to me by Viktor: All traditional books on algorithms and data structures that we could find gave the lookup costs of balanced search trees as O(log n) (i.e., the depth of the search tree), and the lookup costs of tries as O(k) (i.e., the length of the key). At a first glance this is a logarithmic time lookup against a linear time lookup, which makes people nervous when thinking about long keys.

But that argument is very unfair: Either we consider a key comparison a O(1) operation, then a tree lookup is indeed in O(log n), but then a trie lookup is in O(1)! As the length of the key has to be bounded by a constant to get O(1) comparisons, the depth of the trie is bounded, too. Or the length of the key matters, then a trie lookup is indeed in O(k), but then a tree lookup is in O(k log n). We have to compare with the key on every level, and if we are unlucky we have to look at the whole key, which gives the factor k.

Which of course makes tries much more attractive asymptotic wise. Note that we ignore wall clock times here, which are much more difficult to predict, but in many if not most cases tries are indeed much faster than search trees.

I believe the reason why text books get away with this unfair comparison is that they all present balanced search trees with integer keys:


While tries are traditionally introduced with string examples. If they had used string keys for balanced search trees instead it would have been clear that the key length matters:


The trie examines every key byte at most once, while the search tree can examine every key byte log n times. Thus, the asymptotic complexity of tries is actually better than that of balanced search tries.

Propagation of Mistakes in Papers

While reading papers on cardinality estimation I noticed something odd: The seminal paper by Flajolet and Martin on probabilistic counting gives a bias correction constant as 0.77351, while a more recent (and very useful) paper by Scheuermann and Mauve gives the constant as 0.775351. Was this a mistake? Or did they correct a mistake in the original paper?

I started searching, and there is a large number of papers that uses the value 0.775351, but there is also a number of papers that uses the value 0.77351. Judging by the number of Google hits for "Flajolet 0.77351" vs. "Flajolet 0.775351" the 0.77351 group seems to be somewhat larger, but both camps have a significant number of publications. Interestingly, not a single paper mentions both constants, and thus no paper explains what the correct constant should be.

In the end I repeated the constant computation as explained by Flajolet, and the correct value is 0.77351. We can even derive one digit more when using double arithmetic (i.e., 0.773516), but that makes no difference in practice. Thus, the original paper was correct.

But why do so many paper use the incorrect value 0.775351 then? My guess is that at some point somebody made a typo while writing a paper, introducing the superfluous digit 5, and that all other authors copied the constant from that paper without re-checking its value. I am not 100% sure what the origin of the mistake is. The incorrect value seems to appear first in the year 2007, showing up in multiple publications from that year. Judging by publication date the source seems to be this paper (also it did not cite any other papers with the incorrect value, as far as I know). And everybody else just copied the constant from somewhere else, propagating it from paper to paper.

If you find this web page because you are searching for the correct Flajolet/Martin bias correction constant, I can assure you that the original paper was correct, and that the value is 0.77351. But you do not have to trust me on this, you can just repeat the computation yourself.

Addendum to the MILP Optimization Times

In our upcoming SIGMOD paper on Adaptive Optimization of Very Large Join Queries we compared result quality and optimization times of over a dozen approaches for join ordering, over a wide range of query sizes. Which is quite a challenging problem, as the different algorithms often work under different assumptions, usually no reference implementation is available, and we had to unify them all into one framework that can handle joins from 10 to 5,000 relations.

One of the approaches that we included was Solving the Join Ordering Problem via Mixed Integer Linear Programming by Immanuel Trummer and Christoph Koch. Our implementation tries to follow the original paper faithfully, implementing the mapping from query graph to MILP problem just as described in the original paper. For some benchmarks like TPC-DS (up to 18 relations in a join, with a median of 3) that implementation worked fine. But for some other benchmarks like the Join Order Benchmark (up to 17 relations, median 8) and the SQLite join set (up to 64 relations, median 34) we saw significant optimization times on our Xeon E7-4870 system: In total 290s for the JOB queries, and 5,100s for the SQLite queries. (Note that the JOB times do not include queries with non-inner join edges, as these currently cannot be handled by the MILP approach).

Immanuel pointed out to me that we can improve the optimization time quite a bit by initializing the start position for the Gurobi solver to a solution constructed by a greedy heuristic. In particular for the SQLite queries that helps a lot, as the greedy solution works very well there and thus the start position is already very good. The optimization times for his implementation (on a weaker hardware, a 2.2 GHz Intel Core i7 laptop) are 52s for the Join Order Benchmark and 44s for the SQLite queries.

I am glad for the hint, and thus amend the numbers here. As far as a I can see it this initialization trick was not mentioned in the original SIGMOD17 paper. But of course a carefully tuned implementation will often have tricks that are unfortunately not described in detail in the corresponding publication.

If there are any more comments about any of the approaches we measured I am happy to hear them.

The Case for B-Tree Index Structures

Recently a very interesting paper made a Case for Learned Index Structures. It argued that we could, and perhaps should, replace traditional index structures with machine learning, using the following reasoning: If we consider the leaf pages of an index as a sorted array, the inner pages of the index point towards a (bucketized) position within that array. Which means that it essentially describes the cummulative distribution function (CDF), mapping from keys to array positions.

And the argument of that paper was that using machine learning we can do that mapping much better because a) the learned model (in this case neuronal network) is much smaller than a traditional b-tree, and b) the learned model can predict the CDF value much more accurately than a simple b-tree, which improves performance.

Now I am all in favor of trying out new ideas, and adapting to the data distribution is clearly a good idea, but do we really need a neural network for that? Because, after all, the neuronal network is just an approximation of the CDF function. There are many other ways to approximate a function, for example spline interpolation: We define a few knots of the spline, and then interpolate between the knots. For example (picture by D.J. Graham)

Thus, what we need for a spline are a sequence of knots where we can interpolate between, i.e., a sequence of (x,y) values.

Now, if we think back about traditional index structures, in particular B-trees, we see that they have something similar:

The inner pages consist of separator values, and offsets to the next lower level. Well, we can interpret that as a spline. Instead of just going down to the next level and then doing binary search in the next node, we can interpret our search key as position between the two separators, and then interpolate the position of our search key one the next level. This estimate will be slightly off, of course, but the same is true for the machine learning approach, and we can use the same binary search strategy starting from our estimated position. We can use that interpolation strategy on all levels, both when navigating the inner pages and then going down to the leaf nodes.

How well does that work in practice? The learned indexes paper gives accuracy results and performance results for different sizes of neuronal network models. In the paper the b-trees are depicted as being very large, but in reality that is a parameter, of course. We can get arbitrarily sized b-trees by modifying the page size of the b-tree. For comparisons we chose the b-trees to have the same size (in KB) as the neuronal networks reported in the paper. The source code of the learned indexes approach is not available, thus we only report the original numbers for the neuronal networks. Our own proof of concept code is available upon request. As data sets we used the map set and the lognormal set mentioned in the paper, as we could not obtain the other data sets.

If we just look at the accuracy of the prediction of the final tuple we get as average error the number shown below. For the b-trees we report distance between the estimated position and the real tuple position, averaged over all elements in the data set. For the neuronal networks the wording in the paper is a bit unclear, we think the numbers are the average of the average errors of the second level models, which might be slightly different.

Map data	size (MB)	avg error
Learned Index (10,000)	0.15	8 ± 45
Learned Index (100,000)	1.53	2 ± 36
Complex Learned Index	1.53	2 ± 30
B-tree (10,000)	0.15	225
B-tree (100,000)	1.53	22

Log normal data	size (MB)	avg error
Learned Index (10,000)	0.15	17,060 ± 61,072
Learned Index (100,000)	1.53	17,005 ± 60,959
Complex Learned Index	1.53	8 ± 33
B-tree (10,000)	0.15	1,330
B-tree (100,000)	1.53	3

If we look at the numbers, the interpolating b-tree doesn't perform that bad. For the map data the learned index is a bit more accurate, but the difference is small. For the log normal data the interpolating b-tree is in fact much more accurate than the learned index, being able to predict the final position very accurately.

What does that mean for index performance? That is a complicated topic, as we do not have the source code of the learned index and we do not even know precisely on which hardware the experiments were run. We thus only give some indicative numbers, being fully aware that we might be comparing apples with oranges due to various differences in hardware and implementation. If we compare the reported numbers from the paper for lognormal with our proof of
concept implementation (running on a i7-5820K @ 3.30GHz, searching for every element in the data set in shuffled order) we get

Log normal data	Total (ns)	Model (ns)	Search (ns)
Learned Index (10,000)	178	26	152
Learned Index (100,000)	152	36	127
Complex Learned Index	178	110	67
B-tree (10,000)	156	101	54
B-tree (100,000)	171	159	12

Again, the b-tree does not perform that bad, being virtually identical to the reported learned index performance (remember the caveat about hardware differences!). And the b-tree is a very well understood data structure, well tested, with efficient update support etc., while the machine learning model will have great difficulties if the data is updated later on. Thus, I would argue that traditional index structures, in particular b-trees, are still the method of choice, and will probably remain so in the foreseeable future.

Does this mean we should not consider machine learning for indexing? No, we should consider everything that helps. It is just that "everything that helps" does not just include fashionable trends like machine learning, but also efficient implementations of well known data structures.