# Background

Typically when talking about product combinations, you would think of products that are closely related in terms of form and function. For instance, in China, stuffed buns, soybean milk, and tea eggs are common breakfast items. So it wouldn’t come as a surprise if these products are promoted together in a supermarket.

But then, you’ve probably heard of the infamous beer and diapers correlation. Why do these seemingly disparate items are the best product combination?

For these questions, we can find answers based on accumulated order data. The question “how to use PostgreSQL to find the best production combination” was raised by a friend in the PostgreSQL community during the PostgreSQL community activity on April 8.

In fact, this problem can also be solved by using a professional recommendation database that supports a variety of recommendation algorithms.

The statistical method covered in this article can only be used to calculate the best combination of directly related items (data included in the same order).

To calculate the combination of indirectly associated items (for example, if user A bought item 1 and item 2 and user B bought item 2 and item 3, item 1 and item 3 have an indirect relationship), you need to use the recommendation algorithm in RecDB or use a similar graph search method.

# Scenario Simulation

Assume that there is a total of 100,000 item IDs and simulate a batch of users’ order or shopping cart records with each order or shopping cart record containing 5–10 items. Create a total of around 11 million of these records.

Create a table

`postgres=# create unlogged table buy (pay_id int8, item_id int[]);  CREATE TABLE`

# Construct Data

Create a function that inserts data into the buy table (array of 5–10 items)

`create or replace function f() returns void as \$\$    declare    begin      for i in 5..15 loop        insert into buy (item_id) select array_agg((100000*random())::int8) from generate_series(1,i);      end loop;    end;    \$\$ language plpgsql strict;`

Use pgbench to generate 11 million records

`vi test.sql    select f();        pgbench -M prepared -n -r -P 1 -f ./test.sql -c 100 -j 100 -t 10000      transaction type: ./test.sql  scaling factor: 1  query mode: prepared  number of clients: 100  number of threads: 100  number of transactions per client: 10000  number of transactions actually processed: 1000000/1000000  latency average = 1.155 ms  latency stddev = 1.814 ms  tps = 85204.625725 (including connections establishing)  tps = 85411.351807 (excluding connections establishing)  script statistics:   - statement latencies in milliseconds:           1.158  select f();`

Confirm that the data has been successfully written

`postgres=# select count(*) from buy;    count     ----------   11000000  (1 row)    postgres=# select * from buy limit 10;   pay_id |                           item_id                              --------+--------------------------------------------------------------          | {6537,76804,33612,75580,8021}          | {72437,66015,2939,56128,7056}          | {40983,79581,15954,21039,6702,90279}          | {93626,8337,13416,69371,4366,75868}          | {84611,56893,25201,74038,59337,62045,59178}          | {97422,48801,69714,77056,17059,79714,21598}          | {42997,50834,57214,52866,83656,76342,5639,93416}          | {53543,24369,31552,28654,38516,63657,86564,11483}          | {58873,23162,23369,55091,32046,29907,31895,65658,5487}          | {39916,6641,85068,55870,27679,91770,46150,12290,48662,71350}  (10 rows)`

# GIN Indexes

`postgres=# create index idx_buy_item on buy using gin(item_id);`

# Split Functions

The purpose of splitting is to split the array in an order into several sets. For example, an order containing five items can be split into 10 (4+3+2+1) sets of 2 items.

`{6537,76804,33612,75580,8021}`

Split the order into the following sets

`{6537,76804}    {6537,33612}    {6537,75580}    {6537,8021}    {76804,33612}    {76804,75580}    {76804,8021}    {33612,75580}    {33612,8021}    {75580,8021}`

Create a function to complete the split work

Use a recursive query for recombination

Example

`WITH RECURSIVE   t(i) AS (    SELECT * FROM unnest('{A,B,C}'::char[])  ),   cte AS (       SELECT i AS combo, i, 1 AS ct        FROM t      UNION ALL        SELECT cte.combo || t.i, t.i, ct + 1        FROM cte, t        WHERE ct <= 3 -- Combine 4 (3+1) times         AND position(t.i in cte.combo) = 0 -- Newly added characters are not included in existing characters  )   SELECT ARRAY(SELECT combo FROM cte ORDER BY ct, combo) AS result;                          result                         ---------------------------------------------------   {A,B,C,AB,AC,BA,BC,CA,CB,ABC,ACB,BAC,BCA,CAB,CBA}  (1 row)`

# Function 1 That Returns a Specified Number of Sets

Assume that the array doesn’t include replicate elements

`create or replace function array_regroup(    i_arr int[], -- Enter an array    i_elems int -- Scramble into sets of fixed length  ) returns setof int[] as \$\$  declare    v_arr_len int := array_length(i_arr, 1); -- The length of the array that has been entered  begin    -- Protection    if i_elems > v_arr_len then      raise notice 'you cann''t return group len % more then %', i_elems, v_arr_len;      return;    elsif i_elems = v_arr_len then      return next i_arr;      return;    elsif i_elems = 1 then      return query select array(select i) from unnest(i_arr) t(i);      return;    end if;      return query    WITH RECURSIVE     t(i) AS (        select array(select i) from unnest(i_arr) t(i)    ),     cte AS (       SELECT i AS combo, i, 1 AS ct        FROM t      UNION ALL        SELECT array(select i from (select unnest(array_cat(cte.combo, t.i)) order by 1) t(i)), t.i, ct + 1        FROM cte, t        WHERE cte.ct <= i_elems-1 -- Combine any times         AND (not cte.combo @> t.i) -- Newly added values are not included in existing value sets    )     SELECT combo FROM cte where array_length(combo,1)=i_elems group by combo;       return;  end;  \$\$ language plpgsql strict;  postgres=# select array_regroup(array[1,2,3],2);   array_regroup   ---------------   {2,3}   {1,2}   {1,3}  (3 rows)`

# Function 2 That Returns All Sets

`create or replace function array_regroup(    i_arr int[], -- Enter an array  ) returns setof int[] as \$\$  declare    v_arr_len int := array_length(i_arr, 1); -- The length of the array that has been entered  begin      return query    WITH RECURSIVE     t(i) AS (        select array(select i) from unnest(i_arr) t(i)    ),     cte AS (       SELECT i AS combo, i, 1 AS ct        FROM t      UNION ALL        SELECT array(select i from (select unnest(array_cat(cte.combo, t.i)) order by 1) t(i)), t.i, ct + 1        FROM cte, t        WHERE cte.ct <= v_arr_len-1 -- Combine any times         AND (not cte.combo @> t.i) -- Newly added values are not included in existing value sets    )     SELECT combo FROM cte group by combo;       return;  end;  \$\$ language plpgsql strict;  postgres=# select array_regroup(array[1,2,3]);   array_regroup   ---------------   {2}   {2,3}   {1,2}   {1}   {1,2,3}   {3}   {1,3}  (7 rows)`

# Function 3 That Returns a Specified Number of Sets and That Only Displays Sets Containing Certain Elements (For Example, Arrays That Contain the Bread ID)

`create or replace function array_regroup(    i_arr int[], -- Enter an array    i_elems int -- Scramble into sets of fixed length    i_arr_contain int[] -- Arrays that contain specified item IDs  ) returns setof int[] as \$\$  declare    v_arr_len int := array_length(i_arr, 1); -- The length of the array that has been entered  begin    -- Protection    if i_elems > v_arr_len then      raise notice 'you cann''t return group len % more then %', i_elems, v_arr_len;      return;    elsif i_elems = v_arr_len then      return next i_arr;      return;    elsif i_elems = 1 then      return query select array(select i) from unnest(i_arr) t(i);      return;    end if;      return query    WITH RECURSIVE     t(i) AS (        select array(select i) from unnest(i_arr) t(i)    ),     cte AS (       SELECT i AS combo, i, 1 AS ct        FROM t      UNION ALL        SELECT array(select i from (select unnest(array_cat(cte.combo, t.i)) order by 1) t(i)), t.i, ct + 1        FROM cte, t        WHERE cte.ct <= i_elems-1 -- Combine any times         AND (not cte.combo @> t.i) -- Newly added values are not included in existing value sets         AND (cte.combo @> i_arr_contain)    )     SELECT combo FROM cte where array_length(combo,1)=i_elems group by combo;       return;  end;  \$\$ language plpgsql strict;  postgres=# select array_regroup(array[1,2,3,4,5],2,array[1]);   array_regroup   ---------------   {1,2}   {1,3}   {1,4}   {1,5}  (4 rows)    Time: 1.150 ms`

# Find the One Best Combination Item for a Single Item

For example, find the best combination item for bread.

Assume that the item ID of bread is 6537.

`postgres=# select item_id from buy where item_id @> array[6537];  ......   {60573,17248,6537,77857,43349,66208,13656}   {97564,50031,79924,24255,6537,21174,39117}   {24026,78667,99115,87856,64782,8344,73169,41478,63091,29609,6537,71982,75382}   {53094,97465,26156,54181,6537}  (1101 rows)  Time: 5.791 ms    postgres=# explain select item_id from buy where item_id @> array[6537];                                     QUERY PLAN                                      ---------------------------------------------------------------------------------   Bitmap Heap Scan on buy  (cost=457.45..51909.51 rows=55000 width=60)     Recheck Cond: (item_id @> '{6537}'::integer[])     ->  Bitmap Index Scan on idx_buy_item  (cost=0.00..443.70 rows=55000 width=0)           Index Cond: (item_id @> '{6537}'::integer[])  (4 rows)`

Split order data into sets and find the sets that have the highest occurrence of this item ID.

`postgres=# select count(*), array_regroup(item_id,2,array[6537]) from buy where item_id @> array[6537] group by 2 order by 1 desc;   count | array_regroup   -------+---------------       3 | {6537,55286}       3 | {6537,48661}       3 | {6537,78337}       3 | {6537,72623}       3 | {6537,81442}       3 | {6537,66414}       3 | {6537,35346}       3 | {6537,79565}       3 | {3949,6537}  ......    Time: 286.859 ms`

# Find the Two Best Combination Items for a Single Item

For example, find the two best combination items for bread.

`postgres=# select count(*), array_regroup(item_id,3,array[6537]) from buy where item_id @> array[6537] group by 2 order by 1 desc;   count |   array_regroup      -------+--------------------       1 | {32,999,6537}       1 | {6537,49957,91533}       1 | {6537,49957,88377}       1 | {6537,49957,57887}       1 | {6537,49957,55192}       1 | {6537,49952,95266}       1 | {6537,49952,56916}       1 | {6537,49945,60492}       1 | {6537,49940,92888}  ......    Time: 1055.414 ms`

# Find the One Best Combination Item throughout the Network

This may take a long time.

`select count(*), array_regroup(item_id,2) from buy group by 2 order by 1 desc limit 10;`

# Find the N Best Combination Items throughout the Network

This may take a long time.

`select count(*), array_regroup(item_id, n) from buy group by 2 order by 1 desc limit 10;`

# Summary

1. This case doesn’t require highly technical methods, and only splits arrays by recommendation level to count the number of occurrences.

The following database features are used in this case:

1.1. Support for the array type

1.2. PL/pgSQL programming on the service side

1.3. Retrieval of array elements by index (containing a specific element)

1.4. The MPP distributed database architecture for computation performance enhancement For more information, refer to Alibaba Cloud HybridDB for PostgreSQL.

2. Note that the statistical method in this article has a limitation.

The statistical method covered in this article can only be used to calculate the best combination of directly related items (data included in the same order).

To calculate the combination of indirectly associated items (for example, if user A bought item 1 and item 2 and user B bought item 2 and item 3, item 1 and item 3 have an indirect relationship), you need to use the recommendation algorithm in RecDB or use a similar graph search method.

3. Alibaba Cloud HybridDB for PostgreSQL provides the MPP feature to support horizontal scaling, which is very suitable for OLAP scenarios. For example, the “group by” operation which is used many times in this case can result in significant performance improvement.

4. The CPU-based multi-core parallel computing is added in PostgreSQL 9.6 and can significantly improve performance in OLAP scenarios. For example, the “group by” operation which is used many times in this case can have significant performance improvement.

# References

https://github.com/DataSystemsLab/recdb-postgresql

https://www.ibm.com/developerworks/cn/web/1103_zhaoct_recommstudy1/index.html

# Original Source

https://www.alibabacloud.com/blog/using-hybriddbpostgresql-to-find-the-best-product-combination-for-marketing_594994?spm=a2c41.13103612.0.0

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