If you want to learn more about Citus on Microsoft Azure, read this post about Hyperscale (Citus) on Azure Database for PostgreSQL.

Running `SELECT COUNT(DISTINCT)`

on your database is all too common. In applications it’s typical to have some analytics dashboard highlighting the number of unique items such as unique users, unique products, unique visits. While traditional `SELECT COUNT(DISTINCT)`

queries works well in single machine setups, it is a difficult problem to solve in distributed systems. When you have this type of query, you can’t just push query to the workers and add up results, because most likely there will be overlapping records in different workers. Instead you can do:

- Pull all distinct data to one machine and count there. (Doesn’t scale)
- Do a map/reduce. (Scales but it’s very slow)

This is where approximation algorithms or sketches come in. Sketches are probabilistic algorithms which can generate approximate results efficiently within mathematically provable error bounds. There are a many of them out there, but today we’re just going to focus on one, HyperLogLog or HLL. HLL is very successfull for estimating unique number of elements in a list. First we’ll look some at the internals of the HLL to help us understand why HLL algorithm is useful to solve distict count problem in a scalable way, then how it can be applied in a distributed fashion. Then we will see some examples of HLL usage.

## What does HLL do behind the curtains?

### Hash all elements

HLL and almost all other probabilistic counting algorithms depend on uniform distribution of the data. Since in the real world, our data is generally not distributed uniformly, HLL firsts hashes each element to make the data distribution more uniform. Here, by uniform distribution, we mean that each bit of the element has 0.5 probability of being 0 or 1. We will see why this is useful in couple of minutes. Apart from uniformity, hashing allows HLL to treat all data types same. As long as you have a hash function for your data type, you can use HLL for cardinality estimation.

### Observe the data for rare patterns

After hashing all the elements, HLL looks for the binary representation of each hashed element. It mainly looks if there are bit patterns which are less likely to occur. Existence of such rare patterns means that we are dealing with large dataset.

For this purpose, HLL looks number of leading zero bits in the hash value of each element and finds maximum number of leading zero bits. Basically, to be able to observe k leading zeros, we need 2^{k+1} trials (i.e. hashed numbers). Therefore, if maximum number of leading zeros is k in a data set, HLL concludes that there are approximately 2^{k+1} distinct elements.

This is pretty straightforward and simple estimation method. However; it has some important properties, which are especially shine in distributed environment;

- HLL has very low memory footprint. For maximum number n, we need to store just
*log log n*bits. For example; if we hash our elements into 64 bit integers, we just need to store 6 bits to make an estimation. This saves a lot of memory especially compared with naive approach where we need to remember all the values. - We only need to do one pass on the data to find maximum number of leading zeros.
- We can work with streaming data. After calculating maximum number of leading zeros, if some new data arrives we can include them into calculation without going over whole data set. We only need to find number of leading zeros of each new element, compare them with maximum number of leading zeros of whole dataset and update maximum number of leading zeros if necessary.
- We can merge estimations of two separate datasets efficiently. We only need to pick bigger number of leading zeros as maximum number of leading zeros of combined dataset. This allow us to separate the data into shards, estimate their cardinality and merge the results. This is called additivity and it allow us to use HLL in distributed systems.

### Stochastic Averaging

If you think above is not that good estimation, you are right. First of all, our prediction is always in the form of 2^{k}. Secondly we may end up with pretty far estimates if the data distribution is not uniform enough.

One possible fix for these problems could be just repeating the process with different hash functions and taking the average, which would work fine but hashing all the data multiple times is expensive. HLL fixes this problem something called stochastic averaging. Basically, we divide our data into buckets and use aforementioned algorithm for each bucket separately. Then we just take the average of the results. We use first few bits of the hash value to determine which bucket a particular element belongs to and use remaining bits to calculate maximum number of leading zeros.

Moreover, we can configure precision by choosing number of buckets to divide the data. We will need to store *log log n* bits for each bucket. Since we can store each estimation in *log log n* bits, we can create lots of buckets and still end up using insignificant amount of memory. Having such small memory footprint is especially important while operating on large scale data. To merge two estimations, we will merge each bucket then take the average. Therefore, if we plan to do merge operation, we should keep each bucket’s maximum number of leading zeros.

### More?

HLL does some other things too to increase accuracy of the estimation, however observing bit patterns and stochastic averaging is the key points of HLL. After these optimizations, HLL can estimate cardinality of a dataset with typical error rate 2% error rate using 1.5 kB of memory. Of course is is possible to increase accuracy by using more memory. We will not go into details of other steps but there are tons of content on the internet about HLL.

## HLL in distributed systems

As we mentioned, HLL has additivity property. This means you can divide your dataset into several parts, operate on them with HLL algorithm to find unique element count of each part. Then you can merge intermediate HLL results efficiently to find unique element count of all data without looking back to original data.

If you work on large scale data and you keep parts of your data in different physical machines, you can use HLL to calculate unique count over all your data without pulling whole data to one place. In fact, Citus can do this operation for you. There is a HLL extension developed for PostgreSQL and it is fully compatible with Citus. If you have HLL extension installed and want to run COUNT(DISTINCT) query on a distributed table, Citus automatically uses HLL. You do not need to do anything extra once you configured it.

## Hands on with HLL

### Setup

To play with HLL we will use Citus Cloud and GitHub events data. You can see and learn more about Citus Cloud and this data set from here. Assuming you created your Citus Cloud instance and connected it via psql, you can create HLL extension by;

```
CREATE EXTENSION hll;
```

You should create the extension at master and all workers. Then enable count distinct approximations by setting the *citus.count_distinct_error_rate* configuration value. Lower values for this configuration setting are expected to give more accurate results but take more time and use more memory for computation. We recommend setting this to 0.005.

```
SET citus.count_distinct_error_rate TO 0.005;
```

Different from previous blog post, we will only use github_events table and we will use large_events.csv data set;

```
CREATE TABLE github_events
(
event_id bigint,
event_type text,
event_public boolean,
repo_id bigint,
payload jsonb,
repo jsonb,
user_id bigint,
org jsonb,
created_at timestamp
);
SELECT create_distributed_table('github_events', 'user_id');
\COPY github_events FROM large_events.csv CSV
```

### Examples

After distributing the table, we can use regular COUNT(DISTINCT) query to find out how many unique users created an event;

```
SELECT
COUNT(DISTINCT user_id)
FROM
github_events;
```

It should return something like this;

```
count
--------
264227
(1 row)
```

It looks like this query does not have anything with HLL. However if you set *citus.count_distinct_error_rate* to something bigger than 0 and issue COUNT(DISTINCT) query; Citus automatically uses HLL. For simple use-cases like this, you don’t even need to change your queries. Exact distinct count of users who created an event is 264198, so our error rate is little bigger than 0.0001.

We can also use constraints to filter out some results. For example we can query number of unique users who created a PushEvent;

```
SELECT
COUNT(DISTINCT user_id)
FROM
github_events
WHERE
event_type = 'PushEvent'::text;
```

It would return;

```
count
--------
157471
(1 row)
```

Similarly exact distinct count for this query is 157154 and our error rate is little bigger than 0.002.

### Conclusion

If you’re having trouble scaling `count (distinct)`

in Postgres give HLL a look it may be useful if close enough counts ares feasible for you. If you have any questions about how you can scale count event further with Citus then reach out to us.