Tag Archives: algorithms

Redis HyperLogLog and KMinHash performance

The last few blog posts explored the topic of counting unique items efficiently using two specific sketching techniques – HyperLogLogs (HLLs) and KMinHash. The underlying motivation of these techniques was to use probabilistic data structures for counting high cardinality data sets, with a focus on being efficient in both time and space, trading off some accuracy in the counts. For high cardinality data sets, this is a reasonable tradeoff in some domains. We saw how HLLs in Redis provided unique counts with an error of about 0.18% with a bounded 12KB memory size per key. We also saw how well additional operations like unions and intersections fared with HLLs, and how KMinHash provided a more accurate measure over HLLs for intersection operations.

One of the strong advantages of sketching techniques is their efficiency vis-a-vis time and space system measures. Therefore, while we concluded that KMinHash provided more accurate results over HLLs for intersections, it would be good to set it in context alongside a performance comparison so that tradeoffs can be made between accuracy and performance. The purpose of this blog post is to cover the system performance measures of the two methods, using Redis as a store for the counts.

Context for the performance tests

All tests involved two sets: Set A: 175,000 elements, Set B: 10,000 elements, and their intersection Set A n B: 7,500 elements. The elements were added to Redis data structures using Python client code.

The test code operated in two phases. The first added the elements to keys representing the HLL and KMinHash sets. Once all additions were completed, the second phase computed the intersection cardinality using HLL Inclusion/Exclusion principle and the KMinHash method, respectively. The details of the HLL based implementation have been covered in the first and second posts. The KMinHash algorithm has been covered in the third post. Readers can review those posts to familiarise themselves with the details.

The tests were performed on a MacBook Pro 1.6 GHz Intel Core i5 processor, 4 GB 1600 MHz DDR3 RAM. Redis version was 3.0.3 compiled from source, and started with default configuration (at least, as far as the performance related configuration goes). The test code used Python Redis client 2.10.3.

Implementation details

Counting with HLLs

For HLLs, the add phase used PFADD with Redis pipeline mechanism, and a pipeline batch size of 10,000. The compute phase merged the HLL keys using PFMERGE to compute A u B and then computed the intersection count using the Inclusion/Exclusion principle.

Here’s how the add phase looks like:

Counting with KMinHash

For KMinHash, recall that the algorithm was implemented using Redis sorted sets storing the IDs as items in the set sorted according to their hashes (which acted as scores). The add phase added/updated elements in the Redis sorted sets. The compute phase computed the Jaccard coefficient estimate using the algorithm described in post 3, and from there computed A n B cardinality.

At the time of computing the Jaccard coefficient, we should only consider the ‘k’ minimum values of the MinHash sets. However, the sets may have more than ‘k’ elements during the add phase. This gives us a knob to tradeoff between time and memory. For example, we could either keep exactly ‘k’ elements at all times thereby making sure that memory is bounded. This does require more operations in the add phase to ensure the cardinality is maintained (via a ZREM operation, for instance). The motivation to keep the memory bounded might come if we need to maintain a lot of such MinHash sets (say, for different dimensions being measured) and cumulatively, the amount of memory might shoot up very high. On the other hand, we could allow the memory to be slightly unbounded, but make the add operation very fast. This could be a valid strategy if the number of additions is going to happen very fast and saving on time is crucial.

Based on the above choices, I tried three different approaches for implementing KMinHash.

  • Optimise for time (time-optimised): Add multiple elements as a batch using the Redis pipeline mechanism, and truncate the batch to the size of ‘k’ once we have added the batch. Note that in this approach, the MinHash set’s size could grow beyond ‘k’ (depending on the size of the batch).

Here’s how the add phase looks using batch addition. Note the cardinality adjustment at the end of the batch.

  • Optimise for memory (mem-optimised): Bound the cardinality of the KMinHash sets to ‘k’ at add time itself. We do this by truncating the MinHash set to ‘k’ elements after any addition that potentially increases the set’s size. We maintain some state on the client – the current cardinality of the sorted set and the current max MinHash value. This state acts as a cache to help avoid some calls to the Redis server.

This is how the mem-optimised version looks. Note the cardinality adjustment after every add post ‘k’ elements. The local state is maintained in variables like elements_added and max_min_hash

  • Server side scripting (scripting): Redis has a mechanism to execute something akin to stored procedures of a database, by writing them using the LUA scripting language. In this method, define a LUA script that updates the KMinHash set keeping the cardinality bounded to ‘k’. Call this script in pipeline mode during the add phase using the Redis command EVAL or EVALSHA.

Here’s the Lua script that is loaded and executed in the Redis server process. Note how the cardinality is adjusted after every addition post ‘k’ elements. The difference with the mem-optimised approach is that all state is maintained in Redis itself.

The native support for HLL in Redis acts as an advantage and it should be intuitively clear that HLLs score better than KMinHash overall. So, this is not really to show whether KMinHash is better than HLL (which it is not), but to illustrate the comparative system measures for similar cardinality sets in both approaches, as also among the various KMinHash implementation strategies.

Time comparison

Things to measure here included the time indicators between HLL and KMinHash implementations, and also across the various KMinHash implementations. To compare various KMinHash implementations, the high level ‘time’ command was used. The tests were run multiple times to see stability of the time measures across different data sets. The results of the same are as below:

  • KMinHash – time-optimised: 8.5 seconds (average real time)
  • KMinHash – mem-optimised: 13.85 seconds (average real time)
  • KMinHash – scripting: 11.2 seconds (average real time)

Note that this time includes the add phase and compute phase; however, since the HLL addition and cardinality computation is fixed, the time difference is only accounted for by the various KMinHash strategies used.

To compare times between HLL and KMinHash specifically, the Python profiler cProfile was used and the cumulative time measured across individual calls. The results are as below:

  • HLL addition: 6.1 seconds
  • KMinHash addition – time-optimised: 7.2 seconds
  • KMinHash addition – mem-optimised: 12.9 seconds
  • KMinHash addition – scripting: 10.2 seconds
  • HLL intersection: 0
  • KMinHash intersection – time-optimised: 0.03 seconds
  • KMinHash intersection – mem-optimised: 0.04 seconds
  • KMinHash intersection – scripting: 0.02 seconds

Note that the times in the profiled runs don’t add up exactly to the measurements using the ‘time’ command. I suspect this could be due to the profiler overhead.

From the above, we can draw the following conclusions:

  • As expected, HLL performance is the best among all approaches in terms of time measures.
  • The best performance among KMinHash approaches is from the time-optimised approach, followed by the Lua scripting approach and finally by the mem-optimised approach. This is as expected.
  • The time-optimised approach is slower than the HLL approach by about 18%. In comparison, the slowest KMinHash approach (mem-optimised) is almost 100% slower.
  • The time difference for intersection computation is not significant to consider and hence additions is what should be considered for selecting an approach.

Memory comparison

In terms of memory, HLL is a very efficient data structure compared to sorted sets. There are probably parameters that can be tuned for optimising set memory as well, but these will likely cause some increased load on processing time. I did not consider this in my tests.

One thing to note is that the memory is bounded in both cases after the addition of all elements: 12KB for HLL, memory for max ‘k’ elements in KMinHash. The length of the objects used can be determined using the redis command DEBUG OBJECT <key-name>. The results after adding elements from a representative dataset to both HLL and KMinHash keys are as follows:

  • HLL key 175000 size: Serialized length 10491 bytes
  • HLL Key 10000 size: Serialized length 8526 bytes
  • KMinHash key 1 size: Serialized length 187530 bytes
  • KMinHash key 2 size: Serialised length 179048 bytes

One other thing that is relevant is how the memory grows as elements are added to the sets, as these spikes can cause pressure on Redis server memory when there are a lot of such sets. Referring to the various approaches for KMinHash, we see that the time-optimised approach can add more than ‘k’ elements as they are added in batches. To measure this, I ran the redis command INFO periodically and monitored the used_memory_peak_human value as the add phase was in progress. The results are as follows:

  • KMinHash – time-optimised: 67.2 MB
  • KMinHash – mem-optimised: 62.2 MB
  • KMinHash – scripting: 62.1 MB

With the Lua scripting technique, there is also an increase in the Lua memory used by Redis (used_memory_lua) to about 50176 bytes compared to the default of 36864 bytes.

My first implementation of the time-optimised technique adjusted the cardinality of the KMinHash set to ‘k’ only when the intersection cardinality was computed (sort of a lazy approach), instead of adjusting it after every batch addition. With this approach, the used_memory_peak_human value rose as high as 125.86 MB.

From the above, we can draw the following conclusions:

  • Memory used by KMinHash is an order of magnitude more than that used by HLL.
  • The mem-optimised approach is only marginally better in used_memory_peak compared to the time-optimised approach.
  • A lazy time-optimised approach that clears memory only at the end of the add phase does significantly increase memory consumption – almost 100% more than the optimised cases.


HLL is superior to KMinHash based implementations by a reasonable margin from a performance perspective, which is expected given that it is a highly optimised implementation in the Redis server. However, for the accuracy gains of KMinHash, the penalty doesn’t seem too high. Given that the time-optimised approach gives the best time performance, with only marginally weaker memory performance, it is possibly the best implementation overall for KMinHash. So, it could well be something that is implemented along side HLLs to provide an efficient and accurate unique value counting solution in a BigData analytics system.

While I have tried to optimise the code as much as I could, I might not have got everything completely right, as my Redis knowledge isn’t too high. If anyone has suggestions to improve this implementation, or alternate ideas, I request readers to please post those in comments for the benefit of all.


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.

Some interesting points about unions:

  • 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.

The good parts about this approach are:

  • 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.


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.

HyperLogLog in Redis

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.