# Counting unique items fast – Better intersections with MinHash

This is the third post in a series that is exploring sketching techniques to count unique items. In the first post, I explored the HyperLogLog (HLL) data structure and its implementation in Redis. In the second post, I expanded on the topic of unions and intersections of sets using HyperLogLogs.

Regarding intersections, I showed how the Inclusion/Exclusion principle could be used to compute intersection cardinalities. However, in studies conducted by others, the method has been found to produce inaccurate results in some conditions.

In this post, I explore a different sketching technique that claims to improve the accuracy of these results. I also provide the results of tests I did comparing the accuracy of the two methods. Since I will be using terminology introduced in the last post, I request readers to familiarise themselves with those first before continuing here.

## Intersection counts using the MinHash sketch

In my research to solve the problem of improving accuracy of intersection cardinalities, I found an effort by AdRoll who introduced a different approach to solve this problem using a new sketching method called k-MinHash.

The MinHash (MH) sketch is a way to estimate a quantity called the Jaccard coefficient of sets, which measures how similar two sets are. Mathematically, given two sets A and B, Jaccard coefficient is defined as | A n B | / | A u B |.  So, MH(A, B) is approximately | A n B | / | A u B |. Hence,

| A n B | is approximately equal to MH(A, B) x | A u B |.

Note that we can compute | A u B | by merging individual HLLs, as unions are lossless.

## The MinHash sketch

I found this blog to be an excellent introduction to MinHash, including a proof of how it approximates to the Jaccard coefficient. The MinHash sketch involves computing the hash of every element in the set, and maintaining k elements which have the smallest hashes from which the Jaccard coefficient is derived. Since k will be very small compared to the set’s cardinality in our cases, this is also a space-efficient sketching technique.

### Understanding MinHash

The intuitive understanding for MinHash is as follows, summarised from the blog above:

• Define hmin(S) as the element with smallest hash in S.
• Given two sets A and B, if hmin(A) = hmin(B) (say, an element ‘x’), then it can be proved that x = hmin(A u B) and x is in A n B. This can be proved by contradiction. Say x is not in hmin(A u B), there must be an element ‘y’ = hmin(A u B) that has a smaller hash than x, but that element should have been either hmin(A) or hmin(B). Hence, by contradiction, x = hmin(A u B). Now, since hmin(A) = hmin(B), assuming a good hash function, x is in A & B, i.e. x is in A n B.
• If h is a good random hashing function, x can be assumed to be a random element in A u B.
• Probability(hmin(A) = hmin(B)) = Probability of a random element of A u B that is also present in A n B. The latter quantity is | A n B | / | A u B |, which is the Jaccard coefficient. So we see that the probability of an element with the smallest hash being in two sets can be linked to the Jaccard Coefficient.

To make this argument stronger and avoid a freak hashing function accident, we look for not 1, but k min values in the MinHash set that could belong to both A and B. I find this approach somewhat similar to how we used stochastic averaging in the HLL case. There seem to be two ways of getting k values. The first is to have k different hash functions and use the same element in the set. However, since it is going to be difficult to find so many good hashing functions, we instead use the same hash function and compare not 1 but k different elements in the two MinHash sets. The blog above describes how the proof can be extended to k values, and the way to compute the probabilities in such a case. I leave that out of this blog and the interested readers can get it from there.

Although the blog mentions a value of k as small as 400, in my experiments I have found good results only with k being 4096 or 8192 (which are closer to the numbers mentioned in the AdRoll blog).

### MinHash Algorithm

The algorithm for computing MinHash is as follows:

Given 2 sets A & B, and a fixed ‘k':

• Define a hash function that maps A & B elements to hashed integer values.
• Define hmin(S, k) = elements of S with smallest k hashes.
• As elements are seen from A and B, maintain hmin(A, k) and hmin(B, k) by keeping the k smallest hashes in each set.
• Compute hmin(hmin(A, k) u hmin(B, k), k). This is the same as hmin(A u B, k). This can be deduced using similar logic to the proof shown above when k=1. Let hmin(A u B, k) = X
• Compute X n hmin(A, k) n hmin(B, k) = Y. These are elements with smallest hashes that belong to A u B and A n B.
• Jaccard coefficient (A, B) = approximately | Y | / k

## Implementing the MinHash sketch using Redis

To implement the MinHash algorithm, we can use a state store that:

• can store a set of the actual items and their hash values  together
• is able to sort these items on the hash values so as to maintain the smallest k values
• can intersect and merge these sets

As it happens, Redis has a very suitable data structure that supports these operations – the sorted set. Each MinHash structure can be a sorted set with the item as the member and the hash as its score on which Redis sorts and maintains the order of the set. This feels ideal because given the high throughput, low latency characteristics of Redis, we can use it as a shared state store and manage the MinHash sets as part of a streaming application.

Using a Redis sorted set, we can do the following:

• Add the first k items along with their hashes to the set, using ZADD
• From then, update the set by replacing the member with the highest rank with the incoming item provided the incoming item’s hash is smaller. ZRANGE and ZREM family of functions provide these capabilities.
• We can use ZUNIONSTORE and ZINTERSTORE to get the required intermediate sets X and Y mentioned above.
• ZCARD gets the cardinality of the sets required for computing the Jaccard coefficient.

Since my goal for the time being is to compare accuracy, I have not considered performance characteristics of this algorithm too much, and improvements could be possible.

## Comparing Inclusion/Exclusion principle and MinHash for intersection accuracy

### Test Setup

To evaluate how the two methods we have seen so far fare against each other in terms of accuracy, I used the same test design as what Neustar followed, that I spoke about in the last post, (although I have certainly not been as exhaustive as them). Specifically, the test parameters were driven by the two measures:

• overlap(A, B) = | A n B | / | B |, where B is the smaller set.
• cardinality_ratio(A, B) = | A | / | B |, where B is the smaller set.

I kept the value of | B | fixed and then varied | A | and | A n B |. Once the cardinalities of A, B and A n B were decided, I generated random numbers between 1 and 1B such that the required cardinality constraints were met. For each combination of A, B and A n B, I generated 100 such sets and ran the tests. The results were averaged over the 100 tests. Each run for one such combination of A, B and A n B did the following:

• Added elements of A and B to respective HyperLogLog keys in Redis
• Added elements of A and B to respective MinHash keys (backed by Redis sorted sets as described above)
• Computed A n B using HLLs and the Inclusion / Exclusion method.
• Computed A n B using KMinHash

The results for each run were logged, then compared for accuracy.

Remember that there were thresholds for overlap and cardinality within which Neustar found Inclusion/Exclusion to be satisfactory. I divided the test data into two categories – one which satisfied the thresholds (the good case) and one which didn’t (the bad case). For the HLL register size in Redis, the good case was when:

• overlap(A, B) >= 0.05, AND
• cardinality_ratio(A, B) < 20

For the good case, I fixed | B | as 10000, and created values for | A | and | A n B | that satisfied the above criteria. So for e.g. I took values of 0.05, 0.1, 0.2, 0.5, 0.75 for overlap. This gives the required | A n B | values. Similarly, I took values of 2, 5, 10 and 17.5 for cardinality ratio. This gives the required values for | A |.

For the bad case, I fixed | B | as 1000 (a smaller value). The values for B and | A n B | were created such that they violated either both the criteria or just the cardinality_ratio. I was motivated by a use case where one of the sets was very small compared to the other. Remember a use case where a publisher may be looking to see which people from a particular locality visit a particular web page. In this, the number of users visiting a popular web page would be very large, but the number of people in a locality could get very small.

Given | A |, | B | and | A n B |, I ran the tests for all combinations of these. In each case, I computed the % error of | A n B | when computed using both Inclusion / Exclusion and KMinHash.

### Test Results

The results below are aggregated and averaged by cardinality ratio and overlap individually. For example, when averaged by cardinality ratio, it computes the average of error percentages across all chosen overlap values.

The following are the results from the runs with good threshold parameters:

The following points can be made from the results:

• The error % of intersection counts through both methods seem to be significantly higher than the erro % of HLLs itself, irrespective of the sketching method used. Not shown here, but observed in my tests is that the error % of the individual HLLs are always quite small – < 0.5%.
• The accuracy of both KMinHash and Inclusion/Exclusion improves as the overlap increases (i.e. the intersection count is larger than any individual error terms) or when cardinality ratio decreases (i.e. the set sizes are comparable to each other, thereby one set doesn’t contribute a huge error term compared to the intersection count). This is as expected.
• KMinHash performs better than Inclusion / Exclusion for almost all cases except where the overlap is quite high, when Inclusion / Exclusion seems to be marginally better.
• KMinHash errors seem to be more linear in nature compared to Inclusion/Exclusion errors.

The following are the test results from runs with bad threshold parameters:

Note: The scale for the y-axis is switched to log scale for showing the results clearly.

The following points can be made from the test results:

• The error percentages for the pathological cases is quite bad for Inclusion / Exclusion (in the order of 1000s), whereas KMinHash seems to be performing quite a lot better.
• Even otherwise, KMinHash is performing significantly better than Inclusion / Exclusion in most cases.
• As seen for the good cases, with increasing overlap and decreasing cardinality ratio, the error percentages improve as usual.
• The really bad performance of the Inclusion Exclusion method for low overlaps (very small A n B) when taken in context may not appear substantially bad.We are talking about intersection counts of 50 and less sometimes and it may not be too bad to just predict these as 0. However, what is interesting to note is the KMinHash is able to perform better even for such very small cases and predict good values.
• Values of overlap higher than 0.05 are within threshold limits for the overlap, however values of the cardinality ratio are outside of the threshold limits. The results show that the error values are high even when one of the two measures is outside the threshold values.
• Again, high overlap cases seem to perform well in Inclusion / Exclusion compared to KMinHash, although again, the difference is marginal.
• Another point not shown here is that the standard deviation of the error percentages for KMinHash is much lesser than Inclusion / Exclusion method in all cases – thereby indicating more reliable performance.

## Conclusion

In conclusion, we can say that getting intersection counts through sketching techniques do carry a reasonable error percentage that needs to be considered when exposing analytics using them. KMinHash is an effective sketching technique that gives more accurate results for intersection counts compared to the Inclusion/Exclusion method of HLLs. The method I have discussed here is probably not as space or time efficient as the HLL Inclusion/Exclusion method. So, as of now, it is a tradeoff between accuracy and performance characteristics that should be considered when picking an implementation. In a future post, I will try and discuss the performance characteristics of the KMinHash method in more detail.

# Counting unique items fast – Unions and Intersections

In the last blog, I covered HyperLogLog (HLL) in Redis and spoke about how it can enable counting cardinalities of very large sets very efficiently. I concluded the blog by broadening the scope to counting the results of set operations. In this blog, I will expand on this scope.

To recap, some of the use cases of counting the results of set operations are as follows:

• Imagine we are maintaining Daily unique users in a set. Can I combine these sets to get weekly or monthly unique users? (akin to a rollup operation)
• Imagine I have a set of users who have visited a specific web page. And another who are from a particular locality. Can I combine these two sets to see which users from that locality visited the web page? (akin to a slice operation)

The first one is a set union operation and the second an intersection operation. These are the two basic operations we can expect to perform given two sets. So, how does HLL fare with these?

## Unions in HLL

Unions are said to be ‘loss-less’ in HLL. In other words, they work extremely well. So, to get the count of the union of two sets, we can ‘merge’ the HLL data structures representing the two sets to get a new HLL data structure and get the count of the result. The merge operation for two HLLs with equal number of buckets involves taking the maximum value of each pair of buckets and assigning that as the value of the same bucket in the resultant HLL.

To see the intuition behind this, remember that the HLL algorithm only maintains the maximum number of consecutive zeros seen in the hashes of the items for a given bucket. So, if two items are hashing to the same bucket, the one with the maximum number of zeros contributes the value to be stored in the bucket. Hence, our algorithm for merging HLLs described above will be equivalent to replaying the stream by merging the original items.

In Redis, there is explicit support for this operation using a command called PFMERGE

• PFMERGE <result> <key1> <key2> … – Merges the HLLs stored at key1, key2, etc into result.

One can issue PFCOUNT <result> to get the cardinality of the union.

• The merge operation is associative – hence we can merge multiple keys together into one single result, as indeed the PFMERGE command allows in Redis.
• The merge operation is parallelizable with respect to the buckets and hence can be implemented very fast.
• The merge operation assumes that the number of buckets is fixed between the sets being merged. When sizes are different, there is a way to ‘fold’ the HLL data structures with larger number of buckets into a HLL with the smallest number of buckets. This is described well here. I suppose the scenario of different bucket sizes arises if we are using some library where we could create a HLL with a specified bucket size. Of course, this is not applicable for Redis.

## Intersections in HLL

Intersections in HLL are not loss-less. Again seeking into our intuitive explanation for unions, imagine we replay a merged set where only common items in 2 sets are included. If we add the distinct elements of the sets into this HLL,  it is easy to see that the value of a bucket could be masked by a larger distinct value present in either of the original sets.

One possible way to workaround this limitation is to explore if it will be ok to maintain another key to just manage the intersection. For example, to satisfy the use case above, we could maintain one HLL for users who visited the web page, say users:<page>, and another for users from every locality, like users:<page>:<locality>. A stream processing framework will update both keys for an incoming event.

• Number of updates will be bounded by the number of combinations of dimensions we want to count for. This can be done in a streaming manner for a small number of combinations.
• Reads of intersection counts will be fast and accurate too.

The issues with this approach are:

• It is easy to see that this can become a combinatorial nightmare with many different combinations of dimensions to maintain.
• Each intersection key would be more storage space, and hence causes more load, particularly for in-memory systems like Redis.
• The method would only work if all the dimensional information came in the same event. For e.g. if we got information about users visiting pages from one source and user-locality information from another, there would no way of updating the intersection key in a straightforward manner without doing a join.

Intersection using the Inclusion / Exclusion Principle

This is yet another approach talked about for computing intersections with HLLs. Dig into the set algebra, Venn diagrams and other such topics in your child’s math textbooks and you’ll find the Inclusion/Exclusion principle there. It just says:

| A u B | = | A | + | B | – | A n B |  – where | A | means cardinality of set A.

So, can we say | A n B | = HLL(A) + HLL(B) – HLL(AuB)?

Folks at Neustar conducted experiments on this method and were kind enough to publish the results in a very detailed blog post. Again, recommended reading – for understanding how to frame an experiment for answering such a question.

To paraphrase the results, they defined two measures of sets:

• overlap(A, B) = | A n B | / | B |, where B is the smaller set.
• cardinality_ratio(A, B) = | A | / | B |, where B is the smaller set.

Using these measures, they formulated some empirical rules that determine when this method gave somewhat acceptable results. The results are within a reasonable error range when:

• overlap(A, B) >= overlap_cutoff, AND
• cardinality_ratio(A, B) < cardinality_ratio_cutoff

If one of these conditions is violated, they found the error % of the intersection could be high in the order of thousands. In their results, the cutoffs are given based on the bucket size. For the Redis bucket size (16384), this becomes:

• overlap(A, B) >= 0.05
• cardinality_ratio(A, B) < 20

I reasoned the intuition behind these results from this diagram based off the Neustar blog (hopefully, I am not too far off):

A high error value of | A n B | results from both the sets contributing to the error almost equally or one of them dominating. In the case where both sets contribute the error almost equally, this can happen if the true value of intersection itself is very small compared to the size of the sets – measured by the overlap factor. In the cases where one of the sets contributes largely to the error, this can happen when the other set is very small compared to the size of the larger set – measured by the cardinality ratio.

Cases like these can certainly happen in real world. For e.g. our own use case of people who have visited a web page can be very large compared to the people who live in a certain locality – if the locality is very small.

So, to summarise, using the Inclusion / Exclusion principle allows us to compute an estimate for the intersection cardinality with no additional state or storage. It can also be done reasonably fast due to the low latency of APIs like PFCOUNT and PFMERGE. However, for certain commonly occurring use cases, its error ratios are likely to be unusably large.

In the next post, I will explore an alternative approach to intersections using a different sketching algorithm that is said to provide better results. I will also discuss results of my own tests comparing the accuracy of the two algorithms.

# Counting unique items fast

Analytics systems count. Trivial as this may sound, implementing one is far from easy. Indeed, Nathan Marz, creator of Apache Storm, tweeted thus:
90% of analytics startups: 1. Find something new to count that no one else is counting 2. Raise \$10M

In this blog post and the next, I will try and summarise a couple of ways I have learnt to do this efficiently at scale for the specific use case of counting unique items.

Consider a standard use case for several ad-tech companies – counting the number of unique devices they get to see, commonly referred to as audience count. Or a web analytics use case – counting number of users who have seen a particular web page. One can also slice and dice these counts to provide more context. For e.g. how many unique users are viewing an article from browser vs mobile app and so on. Typically, these counts get collected from events captured by an analytics system. They then get exposed by analytics products for their customers along with context, trends, etc to form a basis for decision making. A recent example that illustrates this well is the Parse.ly blog on their new Analytics features for publishers.

Technically, the difficulty in counting uniques as opposed to counting occurrences in events is that while the latter is a simple counter, the former is, at the core, a set cardinality operation. A user visiting a web page twice should be counted as only one user, as opposed to two views. To get the set cardinality, one needs to manage the set. At today’s BigData scale:

• Sets are getting larger. Hundreds of millions of unique users and upwards does not raise eyebrows that much anymore.
• Sets are updated very fast. Sites are processing several thousands and upwards of queries per second each of which need to update the set.
• Users are demanding more. They expect to see the updates to these counts as quickly as possible.

## Technical Approaches

In technical terms, the unique counts exposed by these analytics can be treated as views of the events that drive these analytics. Data warehousing or the batch mode BigData processing solutions powered by frameworks like Hadoop have traditionally separated the collection of event streams and the generation of these views. More contemporary approaches have been proposing a change to this approach, in which the processing of the event streams results in the creation of these ‘materialised views’ directly.

At the core of these more recent approaches are streaming solutions and techniques that operate on incoming event streams and update state for views that expose this information in near real time. There are established stream processing engines that provide the framework and API to consume large event streams in a scalable fashion – such as Apache Storm, Apache Spark, Apache Samza and more recently Apache Flink. Generating the view though, is still a solution the developers need to solve themselves. And their solution needs to still meet all requirements the stream processing frameworks themselves meet.

In order to solve our uniques problem, we need to maintain a set of ’n’ identities, where n is very large and is updated very fast. Also, counting of these n identities needs to happen with very low (sub second) latencies. It is easy to see that for large ’n’s the time and/or space complexity of solving this problem conventionally is going to be large.

## HyperLogLog

One novel approach to address these constraints has been available for a good amount of time, albeit it has not been very well known. This is the concept of maintaining sketches. Informally, a ‘sketch’ is a data structure that summarises large volumes of data into very small amounts of space so as to provide approximate answers to queries about the data with extremely low latency and well-defined error percentages. There are several different sketches, and one set of them deal with counting unique or distinct values.

The specific one I discuss here is a sketch called HyperLogLog (HLL). There are several great articles on the web that describe a HLL. The one I found most intuitive to follow from a layman perspective was from Neustar, previously AggregateKnowledge.

Here is my attempt to summarise and para-phrase the black magic of HLL (although I strongly recommend reading the original article):

• Say we have a good hash function that converts the item we want to add to a set, into a binary bit stream.
• By counting the number of consecutive zero bits in the hash, we can *estimate* the size of the set. The intuition mentioned in the Neustar link above is that counting a consecutive stream of zeros is somewhat like counting the number of consecutive heads we get when tossing a coin. The larger the number of heads, the more number of times we can guess we have tossed the coin.
• We improve the estimate using a procedure called Stochastic averaging, in which we maintain not one, but many such estimates and take a harmonic mean of these. In order to maintain multiple estimates, we split the hash into two parts: a prefix that indexes into a bucket to hold an estimate and the suffix that is used to count the consecutive zero bits.
• There are also some corrections to make the estimates more accurate in cases where the buckets are too empty or too full.

There are several libraries and systems that implement a HLL algorithm (so we are spared from having to implement one ourselves). However, the one I have used is an implementation in the awesome in-memory data structure server – Redis. The blog on the Redis implementation of HyperLogLog is a classic in itself and is also a highly recommended read. The level of thought and work that went into an efficient implementation of HLL in Redis is great learning. In the blog, the Redis HLL standard error has been mentioned as 0.81%. I have actually seen lower in my tests.

Redis exposes the following APIs for manipulating the HLL set:

• PFADD <key> <item> – Adds item to HLL represented by the key
• PFCOUNT <key> – gives the estimate of the number of items added to the key.

The PF prefix stands for Philippe Flajolet who is credited with a lot of work on HLL.

Concerning the internals of HLLs in Redis, the following points are interesting:

• The length of the prefix of the binary hash in Redis is 14 bits. That means, it uses 16384 buckets.
• The hash is 64-bit, hence the remaining 50 bits is where we look for the consecutive zero bits. This means we can represent the value of the counter using at most 6 bits (2^6 = 64 > 50). Hence the total space required for a HLL is bounded by 16384 x 6 bits = 96 Kbits = 12 KB, which is amazingly small for storing a very large cardinality number. (Note that it is possible to lay out a bit array and index into the specific offset in this array representing a bucket)
• Although 12KB is the maximum amount of memory required for a HLL, for smaller sets, the sizes are much smaller. Specifically, like with many other things in Redis, the underlying data structure has a dual representation, a memory efficient one for smaller sets (called ’sparse’ representation) and the 12 KB one for larger sets (‘dense’ representation). This is particularly useful if you need to store lots of HLLs with low cardinality.
• You can use the command PFDEBUG ENCODING <key> to see what representation Redis is currently using for the key.
• The switch from sparse to dense encoding is controlled via a configuration parameter – hll-sparse-max-bytes – default 3000 bytes. The Redis documentation has more details on how to tune this parameter.

## The Good Parts with HLLs in Redis

There are some obvious benefits we can see with HLLs in Redis:

• The 12KB bounded size for a practically unbounded set (read billions of items) is extremely memory efficient.
• The operation PFCOUNT is fast enough for real time queries. Reading directly from this for front end dashboards is totally possible.

There are some subtler benefits, too:

• The operation PFADD is quite fast too, as can be expected from the low latency high throughput performance of Redis, in general. This means that updates to the set represented by the HLL can happen in a streaming fashion. I have built data pipelines using Storm that add IDs to Redis HLL keys and operate with sub second latencies (doing a lot of other work too)
• Since Redis is single threaded, adding the same element to a HLL from different threads works correctly.
• Adding an element to a HLL is idempotent. Hence, when your stream processing framework follows at least once semantics and replays can cause duplicate execution of the PFADD commands, we do not need to worry about consistency.
• The value of a Redis HLL is an encoded String. Hence, it is possible to retrieve or dump the value as a set of bytes and load it into a different Redis server to get identical results. One can imagine that this would be staggeringly fast compared to having to re-add a billion items to another server.

## Set Operations in HLLs

As mentioned above, HLLs are sketches for sets. When modelled like this, one could naturally think if set operations are possible. From a use case perspective, this certainly makes sense.

• Imagine we are maintaining Daily unique users in a set. Can I combine these sets to get weekly or monthly unique users? (akin to a rollup operation)
• Imagine I have a set of users who have visited a specific web page. And another who are from a particular locality. Can I combine these two sets to see which users from that locality visited the web page? (akin to a slice operation)

In set theoretic terms, the first of these would be a union of existing HLLs, while the second is an intersection. It turns out that unions of HLLs is possible, but intersections need more work. I will explore these operations in a following blog post.