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t-digest extension

This PostgreSQL extension implements t-digest, a data structure for on-line accumulation of rank-based statistics such as quantiles and trimmed means. The algorithm is also very friendly to parallel programs.

The t-digest data structure was introduced by Ted Dunning in 2013, and more detailed description and example implementation is available in his github repository [1]. In particular, see the paper [2] explaining the idea. Some of the code was inspired by tdigestc [3] and tdigest [4] by ajwerner.

The accuracy of estimates produced by t-digests can be orders of magnitude more accurate than those produced by previous digest algorithms in spite of the fact that t-digests are much more compact when stored on disk.

Basic usage

The extension provides two functions, which you can see as a replacement of percentile_cont aggregate:

  • tdigest_percentile(value double precision, compression int, quantile double precision)

  • tdigest_percentile(value double precision, compression int, quantiles double precision[])

  • tdigest_percentile_of(value double precision, compression int, value double precision)

  • tdigest_percentile_of(value double precision, compression int, values double precision[])

That is, instead of running

SELECT percentile_cont(0.95) WITHIN GROUP (ORDER BY a) FROM t

you might now run

SELECT tdigest_percentile(a, 100, 0.95) FROM t

and similarly for the variants with array of percentiles. This should run much faster, as the t-digest does not require sort of all the data and can be parallelized. Also, the memory usage is very limited, depending on the compression parameter.

Accuracy

All functions building the t-digest summaries accept accuracy parameter that determines how detailed the histogram approximating the CDF is. The value essentially limits the number of "buckets" in the t-digest, so the higher the value the larger the digest.

Each bucket is represented by two double precision values (i.e. 16B per bucket), so 10000 buckets means the largest possible t-digest is ~160kB. That is however before the transparent compression all varlena types go through, so the on-disk footprint may be much smaller.

It's hard to say what is a good accuracy value, as it very much depends on the data set (how non-uniform the data distribution is, etc.), but given a t-digest with N buckets, the error is roughly 1/N. So t-digests build with accuracy set to 100 have roughly 1% error (with respect to the total range of data), which is more than enough for most use cases.

This however ignores that t-digests don't have uniform bucket size. Buckets close to 0.0 and 1.0 are much smaller (thus providing more accurate results) while buckets close to the median are much bigger. That's consistent with the purpose of the t-digest, i.e. estimating percentiles close to extremes.

Advanced usage

The extension also provides a tdigest data type, which makes it possible to precompute digests for subsets of data, and then quickly combine those "partial" digest into a digest representing the whole data set. The prebuilt digests should be much smaller compared to the original data set, allowing significantly faster response times.

To compute the t-digest use tdigest aggregate function. The digests can then be stored on disk and later summarized using the tdigest_percentile functions (with tdigest as the first argument).

  • tdigest(value double precision, compression int)

  • tdigest_percentile(digest tdigest, compression int, quantile double precision)

  • tdigest_percentile(digest tdigest, compression int, quantiles double precision[])

  • tdigest_percentile_of(digest tdigest, compression int, value double precision)

  • tdigest_percentile_of(digest tdigest, compression int, values double precision[])

So for example you may do this:

-- table with some random source data
CREATE TABLE t (a int, b int, c double precision);

INSERT INTO t SELECT 10 * random(), 10 * random(), random()
                FROM generate_series(1,10000000);

-- table with pre-aggregated digests into table "p"
CREATE TABLE p AS SELECT a, b, tdigest(c, 100) AS d FROM t GROUP BY a, b;

-- summarize the data from "p" (compute the 95-th percentile)
SELECT a, tdigest_percentile(d, 0.95) FROM p GROUP BY a ORDER BY a;

The pre-aggregated table is indeed much smaller:

db=# \d+
                         List of relations
 Schema | Name | Type  | Owner | Persistence |  Size  | Description 
--------+------+-------+-------+-------------+--------+-------------
 public | p    | table | user  | permanent   | 120 kB | 
 public | t    | table | user  | permanent   | 422 MB | 
(2 rows)

And on my machine the last query takes ~1.5ms. Compare that to queries on the source data:

\timing on

-- exact results
SELECT a, percentile_cont(0.95) WITHIN GROUP (ORDER BY c)
  FROM t GROUP BY a ORDER BY a;
  ...
Time: 6956.566 ms (00:06.957)

-- tdigest estimate (no parallelism)
SET max_parallel_workers_per_gather = 0;
SELECT a, tdigest_percentile(c, 100, 0.95) FROM t GROUP BY a ORDER BY a;
  ...
Time: 2873.116 ms (00:02.873)

-- tdigest estimate (4 workers)
SET max_parallel_workers_per_gather = 4;
SELECT a, tdigest_percentile(c, 100, 0.95) FROM t GROUP BY a ORDER BY a;
  ...
Time: 893.538 ms

This shows how much more efficient the t-digest estimate is compared to the exact query with percentile_cont (the difference would increase for larger data sets, due to increased overhead for spilling to disk).

It also shows how effective the pre-aggregation can be. There are 121 rows in table p so with 120kB disk space that's ~1kB per row, each representing about 80k values. With 8B per value, that's ~640kB, i.e. a compression ratio of 640:1. As the digest size is not tied to the number of items, this will only improve for larger data set.

Pre-aggregated data

When dealing with data sets with a lot of redundancy (values repeating many times), it may be more efficient to partially pre-aggregate the data and use functions that allow specifying the number of occurrences for each value. This reduces the number of SQL-function calls.

There are five such aggregate functions:

  • tdigest_percentile(value double precision, count bigint, compression int, quantile double precision)

  • tdigest_percentile(value double precision, count bigint, compression int, quantiles double precision[])

  • tdigest_percentile_of(value double precision, count bigint, compression int, value double precision)

  • tdigest_percentile_of(value double precision, count bigint, compression int, values double precision[])

  • tdigest(value double precision, count bigint, compression int)

Incremental updates

An existing t-digest may be updated incrementally, either by adding a single value, or by merging-in a whole t-digest. For example, it's possible to add 1000 random values to the t-digest like this:

DO LANGUAGE plpgsql $$
DECLARE
  r record;
BEGIN
  FOR r IN (SELECT random() AS v FROM generate_series(1,1000)) LOOP
    UPDATE t SET d = tdigest_add(d, r.v);
  END LOOP;
END $$;

The overhead of doing this is fairly high, though - the t-digest has to be deserialized and serialized over and over, for each value we're adding. That overhead may be reduced by pre-aggregating data, either into an array or a t-digest.

DO LANGUAGE plpgsql $$
DECLARE
  a double precision[];
BEGIN
  SELECT array_agg(random()) INTO a FROM generate_series(1,1000);
  UPDATE t SET d = tdigest_add(d, a);
END $$;

Alternatively, it's possible to use pre-aggregated t-digest values instead of the arrays:

DO LANGUAGE plpgsql $$
DECLARE
  r record;
BEGIN
  FOR r IN (SELECT mod(i,3) AS a, tdigest(random(),100) AS d FROM generate_series(1,1000) s(i) GROUP BY mod(i,3)) LOOP
    UPDATE t SET d = tdigest_union(d, r.d);
  END LOOP;
END $$;

It may be undesirable to perform compaction after every incremental update (esp. when adding the values one by one). All functions in the incremental API allow disabling compaction by setting the compact parameter to false. The disadvantage is that without the compaction, the resulting digests may be somewhat larger (by a factor of 10). It's advisable to use either the multi-value functions (with compaction after each batch) if possible, or force compaction, e.g. by doing something like this:

UPDATE t SET d = tdigest_union(NULL, d);

Trimmed aggregates

The extension provides two aggregate functions allowing to calculate trimmed (truncted) sum and average.

  • tdigest_sum(digest tdigest, low double precision, high double precision)

  • tdigest_avg(digest tdigest, low double precision, high double precision)

The low and high parameters specify where to truncte the data.

Functions

tdigest_percentile(value, accuracy, percentile)

Computes a requested percentile from the data, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile(t.c, 100, 0.95) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest
  • percentile - value in [0, 1] specifying the percentile

tdigest_percentile(value, count, accuracy, percentile)

Computes a requested percentile from the data, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile(t.c, t.a, 100, 0.95) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • percentile - value in [0, 1] specifying the percentile

tdigest_percentile(value, accuracy, percentile[])

Computes requested percentiles from the data, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile(t.c, 100, ARRAY[0.95, 0.99]) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest
  • percentile[] - array of values in [0, 1] specifying the percentiles

tdigest_percentile(value, count, accuracy, percentile[])

Computes requested percentiles from the data, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile(t.c, t.a, 100, ARRAY[0.95, 0.99]) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • percentile[] - array of values in [0, 1] specifying the percentiles

tdigest_percentile_of(value, accuracy, hypothetical_value)

Computes relative rank of a hypothetical value, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile_of(t.c, 100, 139832.3) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest
  • hypothetical_value - hypothetical value

tdigest_percentile_of(value, count, accuracy, hypothetical_value)

Computes relative rank of a hypothetical value, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile_of(t.c, t.a, 100, 139832.3) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • hypothetical_value - hypothetical value

tdigest_percentile_of(value, accuracy, hypothetical_value[])

Computes relative ranks of a hypothetical values, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile_of(t.c, 100, ARRAY[6343.43, 139832.3]) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest
  • hypothetical_value - hypothetical values

tdigest_percentile_of(value, count, accuracy, hypothetical_value[])

Computes relative ranks of a hypothetical values, using a t-digest with the specified accuracy.

Synopsis

SELECT tdigest_percentile_of(t.c, t.a, 100, ARRAY[6343.43, 139832.3]) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • hypothetical_value - hypothetical values

tdigest(value, accuracy)

Computes t-digest with the specified accuracy.

Synopsis

SELECT tdigest(t.c, 100) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest

tdigest(value, count, accuracy)

Computes t-digest with the specified accuracy. The values are added with as many occurrences as determined by the count parameter.

Synopsis

SELECT tdigest(t.c, t.a, 100) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences for each value
  • accuracy - accuracy of the t-digest

tdigest_count(tdigest)

Returns number of items represented by the t-digest.

Synopsis

SELECT tdigest_count(d) FROM (
    SELECT tdigest(t.c, 100) FROM t
) foo

tdigest_percentile(tdigest, percentile)

Computes requested percentile from the pre-computed t-digests.

Synopsis

SELECT tdigest_percentile(d, 0.99) FROM (
    SELECT tdigest(t.c, 100) FROM t
) foo

Parameters

  • tdigest - t-digest to aggregate and process
  • percentile - value in [0, 1] specifying the percentile

tdigest_percentile(tdigest, percentile[])

Computes requested percentiles from the pre-computed t-digests.

Synopsis

SELECT tdigest_percentile(d, ARRAY[0.95, 0.99]) FROM (
    SELECT tdigest(t.c, 100) FROM t
) foo

Parameters

  • tdigest - t-digest to aggregate and process
  • percentile - values in [0, 1] specifying the percentiles

tdigest_percentile_of(tdigest, hypothetical_value)

Computes relative rank of a hypothetical value, using a pre-computed t-digest.

Synopsis

SELECT tdigest_percentile_of(d, 349834.1) FROM (
    SELECT tdigest(t.c, 100) FROM t
) foo

Parameters

  • tdigest - t-digest to aggregate and process
  • hypothetical_value - hypothetical value

tdigest_percentile_of(tdigest, hypothetical_value[])

Computes relative ranks of hypothetical values, using a pre-computed t-digest.

Synopsis

SELECT tdigest_percentile_of(d, ARRAY[438.256, 349834.1]) FROM (
    SELECT tdigest(t.c, 100) FROM t
) foo

Parameters

  • tdigest - t-digest to aggregate and process
  • hypothetical_value - hypothetical values

tdigest_add(tdigest, double precision)

Performs incremental update of the t-digest by adding a single value.

Synopsis

UPDATE t SET d = tdigest_add(d, random());

Parameters

  • tdigest - t-digest to update
  • element - value to add to the digest
  • compression - compression t (used when t-digest is NULL)
  • compact - force compaction (default: true)

tdigest_add(tdigest, double precision[])

Performs incremental update of the t-digest by adding values from an array.

Synopsis

UPDATE t SET d = tdigest_add(d, ARRAY[random(), random(), random()]);

Parameters

  • tdigest - t-digest to update
  • elements - array of values to add to the digest
  • compression - compression t (used when t-digest is NULL)
  • compact - force compaction (default: true)

tdigest_union(tdigest, tdigest)

Performs incremental update of the t-digest by merging-in another digest.

Synopsis

WITH x AS (SELECT tdigest(random(), 100) AS d FROM generate_series(1,1000))
UPDATE t SET d = tdigest_union(t.d, x.d) FROM x;

Parameters

  • tdigest - t-digest to update
  • tdigest_add - t-digest to merge into tdigest
  • compression - compression t (used when t-digest is NULL)
  • compact - force compaction (default: true)

tdigest_json(tdigest)

Returns the t-digest as a JSON value. The function is also exposed as a cast from tdigest to json.

Synopsis

SELECT tdigest_json(d) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

SELECT CAST(d AS json) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

Parameters

  • tdigest - t-digest to cast to a json value

tdigest_double_array(tdigest)

Returns the t-digest as a double precision[] array. The function is also exposed as a cast from tdigest to double precision[].

Synopsis

SELECT tdigest_double_array(d) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

SELECT CAST(d AS double precision[]) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

Parameters

  • tdigest - t-digest to cast to a double precision[] value

tdigest_avg(value, count, accuracy, low, high)

Computes trimmed mean of values, discarding values at the low and high end. The low and high values specify which part of the sample should be included in the mean, so e.g. low = 0.1 and high = 0.9 means 10% low and high values will be discarded.

Synopsis

SELECT tdigest_avg(t.v, t.c, 100, 0.1, 0.9) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • low - low threshold percentile (values below are discarded)
  • high - high threshold percentile (values above are discarded)v

tdigest_avg(tdigest, low, high)

Computes trimmed mean of values, discarding values at the low and high end. The low and high values specify which part of the sample should be included in the mean, so e.g. low = 0.1 and high = 0.9 means 10% low and high values will be discarded.

Synopsis

SELECT tdigest_avg(d, 0.05, 0.95) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

Parameters

  • tdigest - tdigest to calculate mean from
  • low - low threshold percentile (values below are discarded)
  • high - high threshold percentile (values above are discarded)

tdigest_sum(value, accuracy, low, high)

Computes trimmed sum of values, discarding values at the low and high end. The low and high values specify which part of the sample should be included in the sum, so e.g. low = 0.1 and high = 0.9 means 10% low and high values will be discarded.

Synopsis

SELECT tdigest_sum(t.v, 100, 0.1, 0.9) FROM t

Parameters

  • value - values to aggregate
  • accuracy - accuracy of the t-digest
  • low - low threshold percentile (values below are discarded)
  • high - high threshold percentile (values above are discarded)

tdigest_sum(value, count, accuracy, low, high)

Computes trimmed sum of values, discarding values at the low and high end. The low and high values specify which part of the sample should be included in the sum, so e.g. low = 0.1 and high = 0.9 means 10% low and high values will be discarded.

Synopsis

SELECT tdigest_sum(t.v, t.c, 100, 0.1, 0.9) FROM t

Parameters

  • value - values to aggregate
  • count - number of occurrences of the value
  • accuracy - accuracy of the t-digest
  • low - low threshold percentile (values below are discarded)
  • high - high threshold percentile (values above are discarded)

tdigest_sum(tdigest, low, high)

Computes trimmed sum of values, discarding values at the low and high end. The low and high values specify which part of the sample should be included in the sum, so e.g. low = 0.1 and high = 0.9 means 10% low and high values will be discarded.

Synopsis

SELECT tdigest_sum(d, 0.05, 0.95) FROM (
    SELECT tdigest(t.c, 100) AS d FROM t
) foo;

Parameters

  • tdigest - tdigest to calculate sum from
  • low - low threshold percentile (values below are discarded)
  • high - high threshold percentile (values above are discarded)

tdigest_avg(tdigest, double precision, double precision)

Calculates average of values between the low and high threshold.

Synopsis

SELECT tdigest_avg(tdigest(v, 100), 0.25, 0.75) FROM generate_series(1,10000)

Parameters

  • tdigest - t-digest to calculate average for
  • low - low threshold (truncate values below)
  • high - high threshold (truncate values above)

tdigest_sum(tdigest, double precision, double precision)

Calculates sum of values between the low and high threshold.

Synopsis

SELECT tdigest_sum(tdigest(v, 100), 0.25, 0.75) FROM generate_series(1,10000)

Parameters

  • tdigest - t-digest to calculate sum for
  • low - low threshold (truncate values below)
  • high - high threshold (truncate values above)

Notes

At the moment, the extension only supports double precision values, but it should not be very difficult to extend it to other numeric types (both integer and/or floating point, including numeric). Ultimately, it could support any data type with a concept of ordering and mean.

The estimates do depend on the order of incoming data, and so may differ between runs. This applies especially to parallel queries, for which the workers generally see different subsets of data for each run (and build different digests, which are then combined together).

License

This software is distributed under the terms of PostgreSQL license. See LICENSE or http://www.opensource.org/licenses/bsd-license.php for more details.

[1] https://github.com/tdunning/t-digest

[2] https://github.com/tdunning/t-digest/blob/master/docs/t-digest-paper/histo.pdf

[3] https://github.com/ajwerner/tdigestc

[4] https://github.com/ajwerner/tdigest