Fast, Cheap, and 98% Right: Cardinality Estimation for Big Data

The nascent era of big data brings new challenges, which in turn require new tools and algorithms. At Metamarkets, one such challenge focuses on cardinality estimation: efficiently determining the number of distinct elements within a dimension of a large-scale data set. Cardinality estimations have a wide range of applications from monitoring network traffic to data mining. If leveraged correctly, these algorithms can also be used to provide insights into user engagement and growth, via metrics such as “daily active users.”

The HyperLogLog Algorithm:  Every Bit is Great
It is well known that the cardinality of a large data set can be precisely calculated if the storage complexity is proportional to the number of elements in the data set. However, given the scale and complexity of some Druid data sets (with record counts routinely in the billions), the data ensemble is often far too large to be kept in core memory. Furthermore, because Druid data sets can be arbitrarily queried with varying time granularities and filter sets, we needed the ability to estimate dimension cardinalities on the fly across multiple granular buckets. To address our requirements, we opted to implement the HyperLogLog algorithm, originally described by Flajolet and colleagues in 2007. The HyperLogLog algorithm can estimate cardinalities well beyond 10^9 with a relative accuracy (standard error) of 2% while only using 1.5kb of memory. Other companies have also leveraged variations of this algorithm in their cardinality estimations.

HyperLogLog takes advantage of the randomized distribution of bits from hashing functions in order to estimate how many things you would’ve needed to see in order to experience a specific phenomenon.  But as that sentence probably made little sense to any reader, let’s try a simple example to explain what it does.

An Example:  Making a Hash of Things
First, there’s a fundamental mental model shift that is important to realize.  A hash function is generally understood as a function that maps a value from one (larger) space onto another (smaller) space.  In order to randomly hash on a computer system, which is binary at its core, you can view the input value as a series of bits. The hash function acts to contort the input value in some meaningful way such that an output value that is N bits long is produced. A good hash function should assure that the bits of the output value are independent and each have an equal probability (50%) of occurring.

Given a random uniform distribution for likelihoods of N 0s and 1s, you can extract a probability distribution for the likelihood of a specific phenomenon.  The phenomenon we care about is the maximum index of a 1 bit.  Specifically, we expect the following to be true:

50% of hashed values will look like this: 1xxxxxxx…x
25% of hashed values will look like this: 01xxxxxx…x
12.5% of hashed values will look like this: 001xxxxxxxx…x
6.25% of hashed values will look like this: 0001xxxxxxxx…x

So, naively speaking, we expect that if we were to hash 8 unique things, one of them will start with 001.  If we were to hash 4 unique things, we would expect one to start with 01.  This expectation can also be inverted: if the “highest” index of a 1 is 2 (we start counting with index 1 as the leftmost bit location), then we probably saw ~4 unique values.  If the highest index is 4, we probably saw ~16 unique values.  This level of approximation is pretty coarse and it is pretty easy to see that it is only approximate at best, but it is the basic idea behind HyperLogLog.

Buckets and Bits:  Tuning Precision and Scale
The adjustment HyperLogLog makes is that it essentially takes the above algorithm and introduces multiple “buckets”.  That is, you can take the first k bits of the hashed value and use that as a bucket index, then you keep track of the max(index of 1) for the remaining bits in that bucket.  The authors then provide some math for converting the values in all of the buckets back into an approximate cardinality.

Another interesting thing about this algorithm is that it introduces two parameters to adjust the accuracy of the approximation:

1)   Increasing the number of buckets (the k) increases the accuracy of the approximation
2)   Increasing the number of bits of your hash increases the highest possible number you can accurately approximate

Now, Do it in Parallel
So how exactly is all of this useful?  When working with large data sets, it is common to maintain a summarization of the data set inside of a data warehouse and run analytical queries against that summarization.  Often, including information like user ids, user cookies or IP addresses (things that are used to compute unique users) in these summarizations results in a tradeoff with the potential reduction of data volume seen in the summarization and the ability to compute cardinalities.  We wanted to be able to take advantage of the space savings and row reduction of summarization while still being able to compute cardinalities:  this is where HyperLogLog comes in.

In Druid, our summarization process applies the hash function (Murmur 128) and computes the intermediate HyperLogLog format (i.e. the list of buckets of max(index of 1)) and stores that in a column.  Thus, for every row in our summarized dataset, we have a HyperLogLog “sketch” of the unique users that were seen in the original event rows comprising that summarized line.  These sketches are combinable in an additive/commutative way, just like sum, max, and min.  In other words, this intermediate format fits in perfectly with the hierarchical scatter/gather query distribution and processing paradigm employed by Druid, allowing us to provide granular time-series and top lists of unique users, with the full arbitrary slicing and dicing power of Druid.

We don’t just end there though.  We also further optimize the storage format of the intermediate data structure depending on whether the set of buckets is sparse or dense. Stored densely, the data structure is just n buckets of 1 byte (or an array of n bytes, generally, k is less than 256, so it can be represented in one byte).  However, in the sparse case, we only need to store buckets with valid index values in them.  This means that instead of storing n buckets of 1 byte apiece, we can just store the (index, value) pairs.

We Are the 99% (ok, the 98.5%)
Given our implementation of the algorithm, the theoretical average amount of error is 1.5% (i.e. the values will be off by an average of 1.5%). The graph below shows the benchmark results for a loop that ran from 0 to Integer.MAX_VALUE and added the result of a Random.nextLong() to the HyperLogLog.  For this particular benchmark, the average error rate was found to be 1.202526%.



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