Finding Maximum Shared Key Combinations in SQLite

Understanding the Problem: Identifying Common Key Combinations Across Rounds

The core issue revolves around identifying the maximum number of shared pkey values across different rnd (round) values in a SQLite table. The table structure is as follows:

CREATE TABLE t1(rnd integer, line integer, pkey TEXT, primary key(rnd, line));

The goal is to determine the most frequently occurring combinations of pkey values within the same rnd. For example, if pkey values AAA, CCC, and DDD appear together in multiple rnd values, we want to count how many times this combination occurs and identify the rnd values where this happens.

The challenge lies in efficiently generating and counting these combinations, especially when the number of pkey values per rnd is large (e.g., 12 pkey values per rnd). The problem becomes computationally intensive due to the exponential growth of possible combinations as the number of pkey values increases.

Potential Causes of Complexity: Exponential Growth and Self-Joins

The primary cause of complexity in this problem is the exponential growth of possible combinations of pkey values. For each rnd, the number of possible combinations of pkey values is given by the formula (2^n – 1), where (n) is the number of pkey values per rnd. For example, with 12 pkey values per rnd, there are (2^{12} – 1 = 4095) possible combinations. This exponential growth makes it computationally expensive to generate and count all possible combinations, especially when dealing with large datasets.

Another source of complexity is the need for self-joins to generate these combinations. In SQL, generating combinations typically requires joining a table with itself multiple times. For example, to generate combinations of 3 pkey values, you would need to join the table with itself twice. This approach becomes unwieldy as the number of pkey values increases, leading to a large number of self-joins and a significant increase in query execution time.

Efficient Solutions: Recursive CTEs and Subset Generation

To address the problem of exponential growth and self-joins, we can leverage recursive Common Table Expressions (CTEs) in SQLite. Recursive CTEs allow us to generate all possible subsets of pkey values in a more efficient manner. The idea is to generate all possible combinations of pkey values lexicographically and then join these combinations back to the original table to count their occurrences.

Step 1: Generate All Possible Subsets of pkey Values

The first step is to generate all possible subsets of pkey values. This can be done using a recursive CTE that builds subsets incrementally. The following SQL code demonstrates how to generate these subsets:

-- Create a temporary table to store ordered pkeys
DROP TABLE IF EXISTS ordered_pkeys;
CREATE TEMPORARY TABLE ordered_pkeys AS
SELECT pkey, row_number() OVER (ORDER BY pkey) AS id
FROM (SELECT DISTINCT pkey FROM t1);

-- Create a recursive CTE to generate all subsets
DROP VIEW IF EXISTS calc_all_subsets;
CREATE TEMPORARY VIEW calc_all_subsets AS
WITH recurse (curidx, nelems, subset) AS (
    SELECT 0, 0, ''
    UNION ALL
    SELECT r.curidx + 1, r.nelems + sq.incl,
           CASE
               WHEN length(r.subset) = 0 OR length(sq.pkey) = 0 THEN r.subset || sq.pkey
               ELSE printf('%s-%s', r.subset, sq.pkey)
           END
    FROM recurse AS r
    JOIN (SELECT id, 0 AS incl, '' AS pkey FROM ordered_pkeys
          UNION ALL
          SELECT id, 1, pkey FROM ordered_pkeys) AS sq
    ON r.curidx + 1 = sq.id
    WHERE r.curidx < (SELECT max(id) FROM ordered_pkeys)
)
SELECT nelems, subset FROM recurse;

-- Store the generated subsets in a table
DROP TABLE IF EXISTS pkey_subsets;
CREATE TABLE pkey_subsets AS
SELECT * FROM calc_all_subsets;

-- Clean up temporary tables and views
DROP TABLE IF EXISTS ordered_pkeys;
DROP VIEW IF EXISTS calc_all_subsets;

This code generates all possible subsets of pkey values and stores them in the pkey_subsets table. Each subset is represented as a string of pkey values separated by hyphens, and the nelems column indicates the number of pkey values in the subset.

Step 2: Join Subsets with Original Data and Count Occurrences

Once we have generated all possible subsets, the next step is to join these subsets back to the original table and count how many times each subset occurs within the same rnd. This can be done using the following SQL code:

-- Join subsets with original data and count occurrences
SELECT ps.nelems, ps.subset, COUNT(*) AS occur_cnt, GROUP_CONCAT(t1.rnd) AS rnds
FROM pkey_subsets AS ps
JOIN t1 ON INSTR(ps.subset, t1.pkey) > 0
GROUP BY ps.subset, t1.rnd
HAVING COUNT(*) = ps.nelems
ORDER BY ps.nelems DESC, occur_cnt DESC;

This query joins the pkey_subsets table with the original t1 table, ensuring that each pkey value in the subset is present in the same rnd. The HAVING clause ensures that only subsets where all pkey values are present in the same rnd are counted. The results are then ordered by the number of pkey values in the subset (nelems) and the number of occurrences (occur_cnt).

Step 3: Optimize for Large Datasets

While the above approach works well for small datasets, it may become computationally expensive for larger datasets with many pkey values per rnd. To optimize for larger datasets, we can limit the number of combinations generated by focusing on specific pkey values or by restricting the maximum size of the subsets.

For example, if we are only interested in combinations of up to 4 pkey values, we can modify the recursive CTE to stop generating subsets once the maximum size is reached:

-- Modify the recursive CTE to limit subset size
WITH recurse (curidx, nelems, subset) AS (
    SELECT 0, 0, ''
    UNION ALL
    SELECT r.curidx + 1, r.nelems + sq.incl,
           CASE
               WHEN length(r.subset) = 0 OR length(sq.pkey) = 0 THEN r.subset || sq.pkey
               ELSE printf('%s-%s', r.subset, sq.pkey)
           END
    FROM recurse AS r
    JOIN (SELECT id, 0 AS incl, '' AS pkey FROM ordered_pkeys
          UNION ALL
          SELECT id, 1, pkey FROM ordered_pkeys) AS sq
    ON r.curidx + 1 = sq.id
    WHERE r.curidx < (SELECT max(id) FROM ordered_pkeys)
      AND r.nelems + sq.incl <= 4  -- Limit subset size to 4
)
SELECT nelems, subset FROM recurse;

By limiting the subset size, we can significantly reduce the number of combinations generated and improve query performance.

Conclusion: Balancing Complexity and Performance

In conclusion, the problem of finding the maximum number of shared pkey values across different rnd values in SQLite involves generating and counting all possible combinations of pkey values. The exponential growth of combinations and the need for self-joins make this problem computationally intensive, especially for large datasets.

To address this, we can use recursive CTEs to generate all possible subsets of pkey values and then join these subsets back to the original data to count their occurrences. By limiting the size of the subsets or focusing on specific pkey values, we can optimize the query for larger datasets and improve performance.

This approach provides a flexible and efficient solution to the problem, allowing us to identify the most frequently occurring combinations of pkey values and the rnd values where they occur.