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] available on github.
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.
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[])
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.
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.
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[])
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.
Computes a requested percentile from the data, using a t-digest with the specified accuracy.
SELECT tdigest_percentile(t.c, 100, 0.95) FROM t
value- values to aggregateaccuracy- accuracy of the t-digestpercentile- value in [0, 1] specifying the percentile
Computes requested percentiles from the data, using a t-digest with the specified accuracy.
SELECT tdigest_percentile(t.c, 100, ARRAY[0.95, 0.99]) FROM t
value- values to aggregateaccuracy- accuracy of the t-digestpercentile[]- array of values in [0, 1] specifying the percentiles
Computes relative rank of a hypothetical value, using a t-digest with the specified accuracy.
SELECT tdigest_percentile_of(t.c, 100, 139832.3) FROM t
value- values to aggregateaccuracy- accuracy of the t-digesthypothetical_value- hypothetical value
Computes relative ranks of a hypothetical values, using a t-digest with the specified accuracy.
SELECT tdigest_percentile_of(t.c, 100, ARRAY[6343.43, 139832.3]) FROM t
value- values to aggregateaccuracy- accuracy of the t-digesthypothetical_value- hypothetical values
Computes t-digest with the specified accuracy.
SELECT tdigest(t.c, 100) FROM t
value- values to aggregateaccuracy- accuracy of the t-digest
Returns number of items represented by the t-digest.
SELECT tdigest(d, 100) FROM (
SELECT tdigest(t.c, 100) FROM t
) foo
Computes requested percentile from the pre-computed t-digests.
SELECT tdigest_percentile(d, 0.99) FROM (
SELECT tdigest(t.c, 100) FROM t
) foo
tdigest- t-digest to aggregate and processpercentile- value in [0, 1] specifying the percentile
Computes requested percentiles from the pre-computed t-digests.
SELECT tdigest_percentile(d, ARRAY[0.95, 0.99]) FROM (
SELECT tdigest(t.c, 100) FROM t
) foo
tdigest- t-digest to aggregate and processpercentile- values in [0, 1] specifying the percentiles
Computes relative rank of a hypothetical value, using a pre-computed t-digest.
SELECT tdigest_percentile_of(d, 349834.1) FROM (
SELECT tdigest(t.c, 100) FROM t
) foo
tdigest- t-digest to aggregate and processhypothetical_value- hypothetical value
Computes relative ranks of hypothetical values, using a pre-computed t-digest.
SELECT tdigest_percentile_of(d, ARRAY[438.256, 349834.1]) FROM (
SELECT tdigest(t.c, 100) FROM t
) foo
tdigest- t-digest to aggregate and processhypothetical_value- hypothetical values
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).
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
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