#### Approximation is not evil

#### One of the top requirements for database systems is reliability. Whether you run a big bank or a small retail business, you don't want to lose a cent here or there or charge your customer twice for the Pink Fluffy Unicorn he just bought, just because the DBMS gave a wrong answer. Classic OLTP operations have to be always 100% correct.

However, for the case of analytics, things become different. In some cases, it can be desirable to trade a bit of accuracy for a lot of speed. This is called

*approximation*and to many database people (and users), the concept of accepting results with less than 100% accuracy seems strange at first.

But if you know - and can control - the error introduced by approximation, it is not. It can even be very useful, if a 95% accurate answer now is worth more than a 100% accurate answer tomorrow.

Welcome to approximation in PostgreSQL 9.5.

#### Approximating queries

Approximate queries work on subsets, called samples, of the whole data set, called the population.

If the sampling is done statistically correct, a sample much smaller than the whole population gives answers close to the real answer within a known error range.

If the sampling is done statistically correct, a sample much smaller than the whole population gives answers close to the real answer within a known error range.

A possible application for the hypothetical retail business would be to find which product is currently trending.

Instead of knowing that exactly 500, 1000 and 2000 Pink Fluffy Unicorns were sold in the last three weeks, knowing that 498, 1001 and 1999 Pink Fluffy Unicorns were sold in the last three weeks with let's say 5% error tells the procurement people that Pink Fluffy Unicorns are a trending product just as fine as the exact numbers. Only, they might have to wait a few seconds for the answer instead of a few hours...

PostgreSQL 9.5 has built-in support for approximate queries. Because I'm lazy and already wrote about this I just point to the corresponding post.

Still, all the population data has to be there for approximate queries to work. How about running queries without storing the underlying data at all?

Instead of knowing that exactly 500, 1000 and 2000 Pink Fluffy Unicorns were sold in the last three weeks, knowing that 498, 1001 and 1999 Pink Fluffy Unicorns were sold in the last three weeks with let's say 5% error tells the procurement people that Pink Fluffy Unicorns are a trending product just as fine as the exact numbers. Only, they might have to wait a few seconds for the answer instead of a few hours...

PostgreSQL 9.5 has built-in support for approximate queries. Because I'm lazy and already wrote about this I just point to the corresponding post.

#### Approximating data structures

Due to the lock-free multiversion concurrency design of PostgreSQL,

*count()*has to touch each row in a table to check whether it is visible in the current transaction or not. Unlike locking DBMS like Oracle, it can only use an index to count in a few cases . Full table scan.*Distinct*always has to sort the table. It can use an index, but only covering indexes, and the larger the index is compared to the table, the less likely PostgreSQL will use it. Sorting can be tuned by raising

*work_mem*, but since this is a

**per session**parameter, it is limited by available RAM.

So

*count(distinct)*is like the worst of both worlds (In the following example*distinct*alone is slower, because it has to return ten million rows to the client,*count(distinct)*returns only one value).
Like here (times are w/o Index / w Index):

create table
hmr(id serial, value real);

insert into hmr (value) select random()*10000000 from generate_series(1,10000000);

select count (value) from hmr; --826 msec. / 817 msec.

select distinct value from hmr; --33917 msec. / 32420 msec.

select count (distinct value) from hmr; -- 9665 msec. / 9439 msec.

insert into hmr (value) select random()*10000000 from generate_series(1,10000000);

select count (value) from hmr; --826 msec. / 817 msec.

select distinct value from hmr; --33917 msec. / 32420 msec.

select count (distinct value) from hmr; -- 9665 msec. / 9439 msec.

Enter the HyperLogLog cardinality estimator. Some clever people at Google observed, that the cardinality of a multiset of evenly distributed random numbers can be predicted by finding the maximum number of leading zeroes in the binary representation of those numbers: For a maximum of

*k*leading zeroes, the cardinality is

*2^k*.

HyperLogLog uses a hash
function to transform arbitrary input values into such random numbers and thus
allows to estimate the cardinality of an input multiset for cardinalities > 10^9 with a 2-3% error, using only 1280 bytes of storage

PostgreSQL has a HyperLogLog extension, hll.

create
extension hll;

CREATE TABLE cardinality (

id integer,

set hll

);

INSERT INTO cardinality(id, set)

SELECT 1, (select hll_add_agg(hll_hash_any(value))

FROM hmr); -- 2267 msec.

SELECT hll_cardinality(set)::int FROM cardinality WHERE id = 1; -- 11 msec.

CREATE TABLE cardinality (

id integer,

set hll

);

INSERT INTO cardinality(id, set)

SELECT 1, (select hll_add_agg(hll_hash_any(value))

FROM hmr); -- 2267 msec.

SELECT hll_cardinality(set)::int FROM cardinality WHERE id = 1; -- 11 msec.

Since

Another, not so PostgreSQL specific example would be a database that has a stream table, e.g. holding only one hour worth of events at any given point in time. I showed how to do this with stock PostgreSQL and a bit of Java here and here.

If you also want to know, how many distinct events that stream has seen in total, it's impossible, unless you store all distinct values and update their counts every time a new event arrives. But then, you might end up in storing all events - which is not what you wanted in the first place if you chose to use a stream table.

With HyperLogLog it's easy. Update your HyperLogLog estimator on every new event and you get a good approximation how many distinct values the stream has seen in total.

*count distinct(value)*= 8187749 and*hll_cardinality*= 8470057, the error is ~3%Another, not so PostgreSQL specific example would be a database that has a stream table, e.g. holding only one hour worth of events at any given point in time. I showed how to do this with stock PostgreSQL and a bit of Java here and here.

If you also want to know, how many distinct events that stream has seen in total, it's impossible, unless you store all distinct values and update their counts every time a new event arrives. But then, you might end up in storing all events - which is not what you wanted in the first place if you chose to use a stream table.

With HyperLogLog it's easy. Update your HyperLogLog estimator on every new event and you get a good approximation how many distinct values the stream has seen in total.

#### Approximating indexes

#### 9.5 introduced BRIN indexes for very large tables. Unlike e.g. a btree, BRIN stores only ranges of values and points to the physical pages where a value that falls into that range

*could*possibly be found.
A BRIN index thus only gives precise answers to the question where a certain value could

*not*be found on disk.
9.6 will have Bloom-Filter indexes as an extension. Bloom filters can tell you that a value does

So, as BRIN and Bloom indexes both are approximating indexes, every index hit has to be rechecked by the DBMS against the underlying data. But if you know their limitations and use them accordingly, they too can speed up your queries quite a bit.

*not exist*in a set with perfect accuracy. But the question if a value*exists*in the set can only be answered with a probability that increases with the collision resilience of the underlying hash.So, as BRIN and Bloom indexes both are approximating indexes, every index hit has to be rechecked by the DBMS against the underlying data. But if you know their limitations and use them accordingly, they too can speed up your queries quite a bit.