# Re: [HACKERS] improving GROUP BY estimation

```On 31 March 2016 at 22:02, Tom Lane <t...@sss.pgh.pa.us> wrote:
> I'm just concerned about what happens when the Dellera paper stops being
> available.  I don't mind including that URL as a backup to the written-out
> argument I just suggested.
>```
```
Here's an updated patch with references to both papers, and a more
detailed justification for the formula, along with the other changes
discussed. Note that although equation (2) in the Dell'Era paper looks
different from the Yao formula, it's actually the same.

Regards,
Dean
```
```diff --git a/src/backend/utils/adt/selfuncs.c b/src/backend/utils/adt/selfuncs.c
new file mode 100644
index a6555e9..99f5f7c
@@ -3439,9 +3439,51 @@ estimate_num_groups(PlannerInfo *root, L
reldistinct = clamp;

/*
-			 * Multiply by restriction selectivity.
+			 * Update the estimate based on the restriction selectivity,
+			 * guarding against division by zero when reldistinct is zero.
+			 * Also skip this if we know that we are returning all rows.
*/
-			reldistinct *= rel->rows / rel->tuples;
+			if (reldistinct > 0 && rel->rows < rel->tuples)
+			{
+				/*
+				 * Given a table containing N rows with n distinct values in a
+				 * uniform distribution, if we select p rows at random then
+				 * the expected number of distinct values selected is
+				 *
+				 * n * (1 - product((N-N/n-i)/(N-i), i=0..p-1))
+				 *
+				 * = n * (1 - (N-N/n)! / (N-N/n-p)! * (N-p)! / N!)
+				 *
+				 * See "Approximating block accesses in database
+				 * organizations", S. B. Yao, Communications of the ACM,
+				 * Volume 20 Issue 4, April 1977 Pages 260-261.
+				 *
+				 * Alternatively, re-arranging the terms from the factorials,
+				 * this may be written as
+				 *
+				 * n * (1 - product((N-p-i)/(N-i), i=0..N/n-1))
+				 *
+				 * This form of the formula is more efficient to compute in
+				 * the common case where p is larger than N/n.  Additionally,
+				 * as pointed out by Dell'Era, if i << N for all terms in the
+				 * product, it can be approximated by
+				 *
+				 * n * (1 - ((N-p)/N)^(N/n))
+				 *
+				 * See "Expected distinct values when selecting from a bag
+				 * without replacement", Alberto Dell'Era,
+				 *
+				 * The condition i << N is equivalent to n >> 1, so this is a
+				 * good approximation when the number of distinct values in
+				 * the table is large.  It turns out that this formula also
+				 * works well even when n is small.
+				 */
+				reldistinct *=
+					(1 - pow((rel->tuples - rel->rows) / rel->tuples,
+							 rel->tuples / reldistinct));
+			}
+			reldistinct = clamp_row_est(reldistinct);

/*
* Update estimate of total distinct groups.
diff --git a/src/test/regress/expected/subselect.out b/src/test/regress/expected/subselect.out
new file mode 100644
index de64ca7..0fc93d9
--- a/src/test/regress/expected/subselect.out
+++ b/src/test/regress/expected/subselect.out
@@ -807,27 +807,24 @@ select * from int4_tbl where
explain (verbose, costs off)
select * from int4_tbl o where (f1, f1) in
(select f1, generate_series(1,2) / 10 g from int4_tbl i group by f1);
-                              QUERY PLAN
-----------------------------------------------------------------------
- Hash Join
+                           QUERY PLAN
+----------------------------------------------------------------
+ Hash Semi Join
Output: o.f1
Hash Cond: (o.f1 = "ANY_subquery".f1)
->  Seq Scan on public.int4_tbl o
Output: o.f1
->  Hash
Output: "ANY_subquery".f1, "ANY_subquery".g
-         ->  HashAggregate
+         ->  Subquery Scan on "ANY_subquery"
Output: "ANY_subquery".f1, "ANY_subquery".g
-               Group Key: "ANY_subquery".f1, "ANY_subquery".g
-               ->  Subquery Scan on "ANY_subquery"
-                     Output: "ANY_subquery".f1, "ANY_subquery".g
-                     Filter: ("ANY_subquery".f1 = "ANY_subquery".g)
-                     ->  HashAggregate
-                           Output: i.f1, (generate_series(1, 2) / 10)
-                           Group Key: i.f1
-                           ->  Seq Scan on public.int4_tbl i
-                                 Output: i.f1
-(18 rows)
+               Filter: ("ANY_subquery".f1 = "ANY_subquery".g)
+               ->  HashAggregate
+                     Output: i.f1, (generate_series(1, 2) / 10)
+                     Group Key: i.f1
+                     ->  Seq Scan on public.int4_tbl i
+                           Output: i.f1
+(15 rows)

select * from int4_tbl o where (f1, f1) in
(select f1, generate_series(1,2) / 10 g from int4_tbl i group by f1);
```
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