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Message #02257
Updated (by Guest): Subquery optimization: Efficient NOT IN execution with NULLs (68)

WORKLOG TASK
===================================
TASK...........: Subquery optimization: Efficient NOT IN execution with NULLs
CREATION DATE..: Fri, 27 Nov 2009, 13:22
SUPERVISOR.....: Monty
IMPLEMENTOR....: Timour
COPIES TO......:
CATEGORY.......: ServerSprint
TASK ID........: 68 (http://askmonty.org/worklog/?tid=68)
VERSION........: Server9.x
STATUS.........: InProgress
PRIORITY.......: 60
WORKED HOURS...: 0
ESTIMATE.......: 0 (hours remain)
ORIG. ESTIMATE.: 0
PROGRESS NOTES:
==(Guest  Sat, 27 Feb 2010, 10:11)==
Status updated.
No change.
==(Guest  Sat, 27 Feb 2010, 10:11)==
Status updated.
 /tmp/wklog.68.old.24229 20100227 10:11:57.000000000 +0000
+++ /tmp/wklog.68.new.24229 20100227 10:11:57.000000000 +0000
@@ 1 +1 @@
Assigned
+InProgress
==(Timour  Mon, 22 Feb 2010, 17:39)==
HighLevel Specification modified.
 /tmp/wklog.68.old.17116 20100222 17:39:48.000000000 +0200
+++ /tmp/wklog.68.new.17116 20100222 17:39:48.000000000 +0200
@@ 233,6 +233,7 @@
1. If columns a_j1,...,a_jm do not contain null values in the temporary
table at all and v_j1,...,v_jm cannot be null, create for these columns
only one index array (and of course do not create any bitmaps for them).
+[done]
2. Consider the ratio d(a_i)=N'(a_i)/V(a_i), where N'(a_i) is the number
of rows, where a_i is not null and V(a_i) is the number of distinct
@@ 264,6 +265,10 @@
7. If you get a row with nulls in all columns stop filling the temporary
table and return UNKNOWN for any tuple <v1,...,vn>.
+[This is wrong, because if we don't fill the whole temp table, there may
+ be some tuple(s) that would match some outer tuple. In such cases, if we
+ stop filling the temp table, we would miss a TRUE result. Having a partial
+ match doesn't preclude us from having a complete match].
8. [timour]
Consider that due to materialization, we already have a unique index
==(Timour  Tue, 19 Jan 2010, 18:44)==
HighLevel Specification modified.
 /tmp/wklog.68.old.22569 20100119 18:44:01.000000000 +0200
+++ /tmp/wklog.68.new.22569 20100119 18:44:01.000000000 +0200
@@ 132,11 +132,10 @@
if (nonull_key && ! nonull_key>lookup(outer_ref))
return FALSE
 if (nonull_key)
 pq.insert(nonull_key)
for (i = 1; i <= n; i++)
{
+ if (vkey[i] != nonull_key)
vkey[i].lookup(outer_ref)
if (! vkey[i].is_eof())
pq.insert(i)
@@ 167,7 +166,7 @@
/* There cannot be a complete match, as we already checked for one. */
assert(matching_keys.elements < n)
}
 else if (cur_min_key == nonull_key)
+ else if (vkey[cur_min_key] == nonull_key)
{
/*
The nonNULL key has no corresponding NULL index, so we know for
@@ 183,8 +182,10 @@
/*
Check if all null_keys contain a NULL at row 'min_row'. The procedure
internally checks all keys in a special precomputed order. A prior
 procedure determines an optimal order and a mapping
 idx_no > idx_order (encoded as an array).
+ procedure determines an optimal order and a mapping idx_no > idx_order
+ (encoded as an array).
+
+ This procedure makes sure not to match the nonNULL column.
*/
if (test_null_row(null_keys, min_row))
return TRUE
@@ 198,6 +199,14 @@
vkey[cur_min_key].next()
if (! vkey[cur_min_key].is_eof())
pq.insert(cur_min_key)
+ else if (vkey[cur_min_key] == nonull_key)
+ {
+ /*
+ If there can't be more matches for the nonull_key, we know for sure
+ there is no match, since there is no possible NULL match.
+ */
+ return FALSE
+ }
if (pq.is_empty())
{
@@ 216,7 +225,6 @@
}

3. Directions for improvement
========================================================================
==(Timour  Tue, 19 Jan 2010, 18:29)==
HighLevel Specification modified.
 /tmp/wklog.68.old.21045 20100119 18:29:12.000000000 +0200
+++ /tmp/wklog.68.new.21045 20100119 18:29:12.000000000 +0200
@@ 132,6 +132,8 @@
if (nonull_key && ! nonull_key>lookup(outer_ref))
return FALSE
+ if (nonull_key)
+ pq.insert(nonull_key)
for (i = 1; i <= n; i++)
{
==(Guest  Tue, 19 Jan 2010, 18:15)==
HighLevel Specification modified.
 /tmp/wklog.68.old.19825 20100119 18:15:30.000000000 +0200
+++ /tmp/wklog.68.new.19825 20100119 18:15:30.000000000 +0200
@@ 1,8 +1,16 @@
This a copy of the initial algorithm proposed by Igor:
======================================================
+Contents
+========================================================================
For each left side tuple (v_1,...,v_n) we have to find the following set
of rowids for the temp table containing N rows as the result of
+1. Initial idea as proposed by Igor
+2. Algorithm for IN execution with partial matching
+3. Directions for improvement
+
+
+1. Initial idea as proposed by Igor
+========================================================================
+
+For each left side tuple (v_1,...,v_n) we have to find the following
+set of rowids for the temp table containing N rows as the result of
materialization of the subquery:
R= INTERSECT (rowid{a_i=v_i} UNION rowid{a_i is null} where i runs
@@ 18,38 +26,198 @@
 it requires minimum memory: not more than N*n bits in total
 search of an element in a set is extremely cheap
Taken all above into account I could suggest the following algorithm to
build R:
+Taken all above into account I could suggest the following algorithm
+to build R:
 Using indexes (read about them below) for each column participating in the
 intersection,
 merge ordered sets rowid{a_i=v_i} in the following manner.
+ Using indexes (read about them below) for each column participating
+ in the intersection, merge ordered sets rowid{a_i=v_i} in the
+ following manner.
If a rowid r has been encountered maximum in k sets
rowid{a_i1=v_i1},...,rowid(a_ik=v_ik),
+ rowid{a_i1=v_i1},...,rowid(a_ik=v_ik),
then it has to be checked against all rowid{a_i=v_i} such that i is
not in {i1,...,ik}.
+ not in {i1,...,ik}.
As soon as we fail to find r in one of these sets we discard it.
If r has been found in all of them then r belongs to the set R.
Here we use the property (1): any r from rowid{a_i=v_i} UNION rowid{a_i
is null} is either
+Here we use the property (1):
+any r from rowid{a_i=v_i} UNION rowid{a_i is null} is either
belongs to rowid{a_i=v_i} or to rowid{a_i is null}. From this we can
infer that for any r from R
indexes a_i can be uniquely divided into two groups: one contains
indexes a_i where r belongs to
the sets rowid{a_i=v_i}, the other contains indexes a_j such that r
belongs to rowid{a_j is null}.

Now let's talk how to get elements from rowid{a_i=v_i} in a sorted order
needed for the merge procedure. We could use BTREE indexes for temp
table. But they are rather expensive and
take a lot of memory as the are implemented with RB trees.
+infer that for any r from R indexes a_i can be uniquely divided into
+two groups:
+ one contains indexes a_i where r belongs to the sets rowid{a_i=v_i},
+ the other contains indexes a_j such that r belongs to
+ rowid{a_j is null}.
+
+Now let's talk how to get elements from rowid{a_i=v_i} in a sorted
+order needed for the merge procedure. We could use BTREE indexes for
+temp table. But they are rather expensive and take a lot of memory as
+the are implemented with RB trees.
I would suggest creating for each column from the temporary table just
an array of rowids sorted by the value from column a.
Index lookup in such an array is cheap. It's also rather cheap to check
that the next rowid refers to a row with a different value in column a.
The array can be created on demand.
+2. Algorithm for IN execution with partial matching
+========================================================================
+
+2.1 Below is shown the toplevel algorithm to execute an IN predicate
+with partial matching. This algorithm is essentially the implementation
+of Item_subselect:exec().
+
+int lookup_with_null_semantics(outer_ref[], mat_subquery)
+{
+ if (index_lookup(outer_ref, mat_subquery)
+ return TRUE
+ else
+ {
+ /*
+ Check if there is a partial match (UNKNOWN) or no match (NULL).
+ */
+ if (this is the first partial match)
+ {
+ vkey[] = build array of value keys for each NULLable column
+ of mat_subquery.
+ nkey[] = build a bitmap NULL index for each column of mat_subquery
+ that contains NULLs
+ nonull_key = build a key over all nonNULL columns of mat_subquery
+ }
+ if (partial_match(outer_ref, vkey[], nkey[], nonull_key)
+ return UNKNOWN
+ else
+ return FALSE
+ }
+}
+
+2.2 The implementation of partial matching is as follows
+
+/*
+ Assumptions:
+  It has already been checked if there is a complete match by a
+ regular index lookup, and the test failed.
+  It has already been checked if there is a complete NULL row,
+ and if there was we wouldn't call this function. Thus we assume
+ that there is no complete NULL row.
+  Not all vidx_i are empty, but some can be empty. If all were empty,
+ then the only possibility for a match is a complete NULL row, which
+ we already checked.
+
+ @param outer_ref  the uter (left) IN argument.
+ @param vidx[]  array of value keys
+ Ordered sequences of rowids of the corresponding columns a_i, such
+ that all rowids in idx_i are the ones where column a_i contains some
+ value or NULL. Each idx_i is derived dynamically, for each different
+ left argument of an IN predicate.
+ @param nidx[]  array of NULL keys
+ Bitmpas, one per each column, where a bit is set if the corresponding
+ row has a NULL value for the corresponding column.
+ @nonull_key  the only key over all columns of the materialized subquery
+ that do not contain NULLs
+
+ @returns
+ @retval FALSE if there is no match
+ @retval TRUE if there is a partial match
+*/
+
+Boolean partial_match(outer_ref, vkey[], nkey[], nonull_key)
+{
+ /* Set of the keys (columns) that form a partial match. */
+ Set matching_keys = {}
+ /* A subset of all keys that need to be checked for NULL matches. */
+ Set null_keys = {}
+ Int min_key /* Key that contains the current minimum position. */
+ Int min_row /* Current row number of min_key. */
+ Int cur_min_key, cur_min_row
+ PriorityQueue pq
+
+ if (nonull_key && ! nonull_key>lookup(outer_ref))
+ return FALSE
+
+ for (i = 1; i <= n; i++)
+ {
+ vkey[i].lookup(outer_ref)
+ if (! vkey[i].is_eof())
+ pq.insert(i)
+ }
+ /*
+ Not all value keys are empty, thus we don't have only NULL
+ keys. If we had, the only possible match is a NULL row, and
+ we cheked there is no such row, therefore the result is known
+ to be FALSE.
+ In fact this algorithm makes sense for at least two nonNULL
+ columns.
+ */
+ assert(pq.elements > 1)
+
+ (min_key, min_row) = pq.pop()
+ matching_keys.add(min_key)
+ vkey[min_key].next()
+ if (! vkey[min_key].is_eof())
+ pq.insert(min_key)
+
+ while (TRUE)
+ {
+ (cur_min_key, cur_min_row) = pq.pop()
+
+ if (cur_min_row == min_row)
+ {
+ matching_keys.add(cur_min_key)
+ /* There cannot be a complete match, as we already checked for one. */
+ assert(matching_keys.elements < n)
+ }
+ else if (cur_min_key == nonull_key)
+ {
+ /*
+ The nonNULL key has no corresponding NULL index, so we know for
+ sure that the row 'min_row' is not a match.
+ */
+ (min_key, min_row) = (cur_min_key, cur_min_row)
+ matching_keys = {min_key}
+ }
+ else
+ {
+ assert(cur_min_row > min_row) /* Follows from the use of PQ. */
+ null_keys = set_difference(all keys vkey[], matching_keys)
+ /*
+ Check if all null_keys contain a NULL at row 'min_row'. The procedure
+ internally checks all keys in a special precomputed order. A prior
+ procedure determines an optimal order and a mapping
+ idx_no > idx_order (encoded as an array).
+ */
+ if (test_null_row(null_keys, min_row))
+ return TRUE
+ else
+ {
+ (min_key, min_row) = (cur_min_key, cur_min_row)
+ matching_keys = {min_key}
+ }
+ }
+
+ vkey[cur_min_key].next()
+ if (! vkey[cur_min_key].is_eof())
+ pq.insert(cur_min_key)
+
+ if (pq.is_empty())
+ {
+ /* Check the last row of the last column in PQ for NULL matches. */
+ null_keys = set_difference(all keys vkey[], matching_keys)
+ if (test_null_row(null_keys, min_row))
+ return TRUE
+ else
+ return FALSE
+ }
+ }
+
+ /* We should never get here. */
+ assert(FALSE)
+ return FALSE
+}
+
+
+
+3. Directions for improvement
+========================================================================
+
Other consideration that may be taken into account:
1. If columns a_j1,...,a_jm do not contain null values in the temporary
==(Timour  Sun, 06 Dec 2009, 14:36)==
HighLevel Specification modified.
 /tmp/wklog.68.old.12919 20091206 14:36:18.000000000 +0200
+++ /tmp/wklog.68.new.12919 20091206 14:36:18.000000000 +0200
@@ 87,3 +87,8 @@
7. If you get a row with nulls in all columns stop filling the temporary
table and return UNKNOWN for any tuple <v1,...,vn>.
+8. [timour]
+ Consider that due to materialization, we already have a unique index
+on all columns <a_1,..., a_n>. We can use the first key part of this index
+over column a_1, instead of the index rowid{a_i=v_i}. Thus we can avoid
+creating the index rowid{a_i=v_i}.
==(Timour  Fri, 04 Dec 2009, 14:04)==
HighLevel Specification modified.
 /tmp/wklog.68.old.16724 20091204 14:04:28.000000000 +0200
+++ /tmp/wklog.68.new.16724 20091204 14:04:28.000000000 +0200
@@ 10,7 +10,8 @@
Bear in mind the following specifics of this intersection:
(1) For each i: rowid{a_i=v_i} and rowid{a_i is null} are disjoint
 (2) For each i: rowid{a_i is null} is the same for each tuple
+ (2) For each i: rowid{a_i is null} is the same for each tuple,
+ that is, this set is independent of the leftside tuples.
Due to (2) it makes sense to build rowid{a_i is null} only once.
A good representation for such sets would be bitmaps:
==(Timour  Fri, 04 Dec 2009, 11:27)==
Version updated.
 /tmp/wklog.68.old.5257 20091204 11:27:11.000000000 +0200
+++ /tmp/wklog.68.new.5257 20091204 11:27:11.000000000 +0200
@@ 1 +1 @@
Benchmarks3.0
+Server9.x
==(Timour  Fri, 04 Dec 2009, 11:27)==
Category updated.
 /tmp/wklog.68.old.5242 20091204 11:27:02.000000000 +0200
+++ /tmp/wklog.68.new.5242 20091204 11:27:02.000000000 +0200
@@ 1 +1 @@
ServerRawIdeaBin
+ServerSprint

==(View All Progress Notes, 13 total)==
http://askmonty.org/worklog/index.pl?tid=68&nolimit=1
DESCRIPTION:
The goal of this task is to implement efficient execution of NOT IN
subquery predicates of the form:
<oe_1,...,oe_n> NOT IN <non_correlated subquery>
when either some oe_i, or some subqury result column contains NULLs.
The problem with such predicates is that it is possible to use index
lookups only when neither argument of the predicate contains NULLs.
If some argument contains a NULL, then due to NULL semantics, it
plays the role of a wildcard. If we were to use regular index lookups,
then we would get 'no match' for some outer tuple (thus the predicate
evaluates to FALSE), while the SQL semantics means 'partial match', and
the predicate should evaluate to NULL.
This task implements an efficient algorithm to compute such 'parial
matches', where a NULL matches any value.
HIGHLEVEL SPECIFICATION:
Contents
========================================================================
1. Initial idea as proposed by Igor
2. Algorithm for IN execution with partial matching
3. Directions for improvement
1. Initial idea as proposed by Igor
========================================================================
For each left side tuple (v_1,...,v_n) we have to find the following
set of rowids for the temp table containing N rows as the result of
materialization of the subquery:
R= INTERSECT (rowid{a_i=v_i} UNION rowid{a_i is null} where i runs
trough all indexes from [1..n] such that v_i is not null.
Bear in mind the following specifics of this intersection:
(1) For each i: rowid{a_i=v_i} and rowid{a_i is null} are disjoint
(2) For each i: rowid{a_i is null} is the same for each tuple,
that is, this set is independent of the leftside tuples.
Due to (2) it makes sense to build rowid{a_i is null} only once.
A good representation for such sets would be bitmaps:
 it requires minimum memory: not more than N*n bits in total
 search of an element in a set is extremely cheap
Taken all above into account I could suggest the following algorithm
to build R:
Using indexes (read about them below) for each column participating
in the intersection, merge ordered sets rowid{a_i=v_i} in the
following manner.
If a rowid r has been encountered maximum in k sets
rowid{a_i1=v_i1},...,rowid(a_ik=v_ik),
then it has to be checked against all rowid{a_i=v_i} such that i is
not in {i1,...,ik}.
As soon as we fail to find r in one of these sets we discard it.
If r has been found in all of them then r belongs to the set R.
Here we use the property (1):
any r from rowid{a_i=v_i} UNION rowid{a_i is null} is either
belongs to rowid{a_i=v_i} or to rowid{a_i is null}. From this we can
infer that for any r from R indexes a_i can be uniquely divided into
two groups:
 one contains indexes a_i where r belongs to the sets rowid{a_i=v_i},
 the other contains indexes a_j such that r belongs to
rowid{a_j is null}.
Now let's talk how to get elements from rowid{a_i=v_i} in a sorted
order needed for the merge procedure. We could use BTREE indexes for
temp table. But they are rather expensive and take a lot of memory as
the are implemented with RB trees.
I would suggest creating for each column from the temporary table just
an array of rowids sorted by the value from column a.
Index lookup in such an array is cheap. It's also rather cheap to check
that the next rowid refers to a row with a different value in column a.
The array can be created on demand.
2. Algorithm for IN execution with partial matching
========================================================================
2.1 Below is shown the toplevel algorithm to execute an IN predicate
with partial matching. This algorithm is essentially the implementation
of Item_subselect:exec().
int lookup_with_null_semantics(outer_ref[], mat_subquery)
{
if (index_lookup(outer_ref, mat_subquery)
return TRUE
else
{
/*
Check if there is a partial match (UNKNOWN) or no match (NULL).
*/
if (this is the first partial match)
{
vkey[] = build array of value keys for each NULLable column
of mat_subquery.
nkey[] = build a bitmap NULL index for each column of mat_subquery
that contains NULLs
nonull_key = build a key over all nonNULL columns of mat_subquery
}
if (partial_match(outer_ref, vkey[], nkey[], nonull_key)
return UNKNOWN
else
return FALSE
}
}
2.2 The implementation of partial matching is as follows
/*
Assumptions:
 It has already been checked if there is a complete match by a
regular index lookup, and the test failed.
 It has already been checked if there is a complete NULL row,
and if there was we wouldn't call this function. Thus we assume
that there is no complete NULL row.
 Not all vidx_i are empty, but some can be empty. If all were empty,
then the only possibility for a match is a complete NULL row, which
we already checked.
@param outer_ref  the uter (left) IN argument.
@param vidx[]  array of value keys
Ordered sequences of rowids of the corresponding columns a_i, such
that all rowids in idx_i are the ones where column a_i contains some
value or NULL. Each idx_i is derived dynamically, for each different
left argument of an IN predicate.
@param nidx[]  array of NULL keys
Bitmpas, one per each column, where a bit is set if the corresponding
row has a NULL value for the corresponding column.
@nonull_key  the only key over all columns of the materialized subquery
that do not contain NULLs
@returns
@retval FALSE if there is no match
@retval TRUE if there is a partial match
*/
Boolean partial_match(outer_ref, vkey[], nkey[], nonull_key)
{
/* Set of the keys (columns) that form a partial match. */
Set matching_keys = {}
/* A subset of all keys that need to be checked for NULL matches. */
Set null_keys = {}
Int min_key /* Key that contains the current minimum position. */
Int min_row /* Current row number of min_key. */
Int cur_min_key, cur_min_row
PriorityQueue pq
if (nonull_key && ! nonull_key>lookup(outer_ref))
return FALSE
for (i = 1; i <= n; i++)
{
if (vkey[i] != nonull_key)
vkey[i].lookup(outer_ref)
if (! vkey[i].is_eof())
pq.insert(i)
}
/*
Not all value keys are empty, thus we don't have only NULL
keys. If we had, the only possible match is a NULL row, and
we cheked there is no such row, therefore the result is known
to be FALSE.
In fact this algorithm makes sense for at least two nonNULL
columns.
*/
assert(pq.elements > 1)
(min_key, min_row) = pq.pop()
matching_keys.add(min_key)
vkey[min_key].next()
if (! vkey[min_key].is_eof())
pq.insert(min_key)
while (TRUE)
{
(cur_min_key, cur_min_row) = pq.pop()
if (cur_min_row == min_row)
{
matching_keys.add(cur_min_key)
/* There cannot be a complete match, as we already checked for one. */
assert(matching_keys.elements < n)
}
else if (vkey[cur_min_key] == nonull_key)
{
/*
The nonNULL key has no corresponding NULL index, so we know for
sure that the row 'min_row' is not a match.
*/
(min_key, min_row) = (cur_min_key, cur_min_row)
matching_keys = {min_key}
}
else
{
assert(cur_min_row > min_row) /* Follows from the use of PQ. */
null_keys = set_difference(all keys vkey[], matching_keys)
/*
Check if all null_keys contain a NULL at row 'min_row'. The procedure
internally checks all keys in a special precomputed order. A prior
procedure determines an optimal order and a mapping idx_no > idx_order
(encoded as an array).
This procedure makes sure not to match the nonNULL column.
*/
if (test_null_row(null_keys, min_row))
return TRUE
else
{
(min_key, min_row) = (cur_min_key, cur_min_row)
matching_keys = {min_key}
}
}
vkey[cur_min_key].next()
if (! vkey[cur_min_key].is_eof())
pq.insert(cur_min_key)
else if (vkey[cur_min_key] == nonull_key)
{
/*
If there can't be more matches for the nonull_key, we know for sure
there is no match, since there is no possible NULL match.
*/
return FALSE
}
if (pq.is_empty())
{
/* Check the last row of the last column in PQ for NULL matches. */
null_keys = set_difference(all keys vkey[], matching_keys)
if (test_null_row(null_keys, min_row))
return TRUE
else
return FALSE
}
}
/* We should never get here. */
assert(FALSE)
return FALSE
}
3. Directions for improvement
========================================================================
Other consideration that may be taken into account:
1. If columns a_j1,...,a_jm do not contain null values in the temporary
table at all and v_j1,...,v_jm cannot be null, create for these columns
only one index array (and of course do not create any bitmaps for them).
[done]
2. Consider the ratio d(a_i)=N'(a_i)/V(a_i), where N'(a_i) is the number
of rows, where a_i is not null and V(a_i) is the number of distinct
values for a_i excluding nulls.
If d(a_i) is close to N'(a_i) then do not create any index array: check
whether there is a match running through the records that have been
filtered in. Anyway if d(a_i) is close to N'(a_i) then the intersection
with rowid{a_i=v_i} will not reduce the number of remaining rowids
significantly.
In other words is V(a_i) exceeds some threshold there is no sense to
create an index for a_i.
If additionally NN'(a_i) is small do not create a bitmap for this
column either.
3. If for a column a_i d(a_i) is not close to N'(a_i), but NN'(a_i) is
small a sorted array of rowids from the set rowid{a_i is null} can be
used instead of a bitmap.
4. We always have a match if R0= INTERSECT rowid{a_i is null} is not
empty. Here i runs through all indexes from [1..n] such that v_i is not
null. For a given subset of columns this fact has to be checked only
once. It can be easily done with bitmap intersection.
5. If v1,...,vn never can be a null, then indexes (sorted arrays) can be
created only for rows with nulls.
6. If v1,...,vn never can be a null and number of rows with nulls is
small do not create indexes and do not create bitmaps.
7. If you get a row with nulls in all columns stop filling the temporary
table and return UNKNOWN for any tuple <v1,...,vn>.
[This is wrong, because if we don't fill the whole temp table, there may
be some tuple(s) that would match some outer tuple. In such cases, if we
stop filling the temp table, we would miss a TRUE result. Having a partial
match doesn't preclude us from having a complete match].
8. [timour]
Consider that due to materialization, we already have a unique index
on all columns <a_1,..., a_n>. We can use the first key part of this index
over column a_1, instead of the index rowid{a_i=v_i}. Thus we can avoid
creating the index rowid{a_i=v_i}.
ESTIMATED WORK TIME
ESTIMATED COMPLETION DATE

WorkLog (v3.5.9)