The WITH clause allows you to specify common table expressions (CTEs). Regular (non-recursive) common-table-expressions are essentially views that are limited in scope to a particular query. CTEs can reference each-other and can be nested. Recursive CTEs can reference themselves.

Basic CTE Examples

Create a CTE called cte and use it in the main query:

WITH cte AS (SELECT 42 AS x)
SELECT * FROM cte;

Create two CTEs cte1 and cte2, where the second CTE references the first CTE:

WITH
    cte1 AS (SELECT 42 AS i),
    cte2 AS (SELECT i * 100 AS x FROM cte1)
SELECT * FROM cte2;

CTE Materialization

DuckDB can employ CTE materialization, i.e., inlining CTEs into the main query. This is performed using heuristics: if the CTE performs a grouped aggregation and is queried more than once, it is materialized. Materialization can be explicitly activated by defining the CTE using AS MATERIALIZED and disabled by using AS NOT MATERIALIZED.

Take the following query for example, which invokes the same CTE three times:

WITH t(x) AS (complex_query)
SELECT *
FROM
    t AS t1,
    t AS t2,
    t AS t3;

Inlining duplicates the definition of t for each reference which results in the following query:

SELECT *
FROM
    (complex_query) AS t1(x),
    (complex_query) AS t2(x),
    (complex_query) AS t3(x);

If complex_query is expensive, materializing it with the MATERIALIZED keyword can improve performance. In this case, complex_query is evaluated only once.

WITH t(x) AS MATERIALIZED (complex_query)
SELECT *
FROM
    t AS t1,
    t AS t2,
    t AS t3;

If one wants to disable materialization, use NOT MATERIALIZED:

WITH t(x) AS NOT MATERIALIZED (complex_query)
SELECT *
FROM
    t AS t1,
    t AS t2,
    t AS t3;

Recursive CTEs

WITH RECURSIVE allows the definition of CTEs which can refer to themselves. Note that the query must be formulated in a way that ensures termination, otherwise, it may run into an infinite loop.

Example: Fibonacci Sequence

WITH RECURSIVE can be used to make recursive calculations. For example, here is how WITH RECURSIVE could be used to calculate the first ten Fibonacci numbers:

WITH RECURSIVE FibonacciNumbers (
    RecursionDepth, FibonacciNumber, NextNumber
) AS (
        -- Base case
        SELECT
            0 AS RecursionDepth,
            0 AS FibonacciNumber,
            1 AS NextNumber
        UNION ALL
        -- Recursive step
        SELECT
            fib.RecursionDepth + 1 AS RecursionDepth,
            fib.NextNumber AS FibonacciNumber,
            fib.FibonacciNumber + fib.NextNumber AS NextNumber
        FROM
            FibonacciNumbers fib
        WHERE
            fib.RecursionDepth + 1 < 10
    )
SELECT
    fn.RecursionDepth AS FibonacciNumberIndex,
    fn.FibonacciNumber
FROM
    FibonacciNumbers fn;

Example: Tree Traversal

WITH RECURSIVE can be used to traverse trees. For example, take a hierarchy of tags:

CREATE TABLE tag (id INTEGER, name VARCHAR, subclassof INTEGER);
INSERT INTO tag VALUES
    (1, 'U2',     5),
    (2, 'Blur',   5),
    (3, 'Oasis',  5),
    (4, '2Pac',   6),
    (5, 'Rock',   7),
    (6, 'Rap',    7),
    (7, 'Music',  9),
    (8, 'Movies', 9),
    (9, 'Art', NULL);

The following query returns the path from the node Oasis to the root of the tree (Art).

WITH RECURSIVE tag_hierarchy(id, source, path) AS (
        SELECT id, name, [name] AS path
        FROM tag
        WHERE subclassof IS NULL
    UNION ALL
        SELECT tag.id, tag.name, list_prepend(tag.name, tag_hierarchy.path)
        FROM tag, tag_hierarchy
        WHERE tag.subclassof = tag_hierarchy.id
    )
SELECT path
FROM tag_hierarchy
WHERE source = 'Oasis';

Graph Traversal

The WITH RECURSIVE clause can be used to express graph traversal on arbitrary graphs. However, if the graph has cycles, the query must perform cycle detection to prevent infinite loops. One way to achieve this is to store the path of a traversal in a list and, before extending the path with a new edge, check whether its endpoint has been visited before (see the example later).

Take the following directed graph from the LDBC Graphalytics benchmark:

CREATE TABLE edge (node1id INTEGER, node2id INTEGER);
INSERT INTO edge VALUES
    (1, 3), (1, 5), (2, 4), (2, 5), (2, 10), (3, 1),
    (3, 5), (3, 8), (3, 10), (5, 3), (5, 4), (5, 8),
    (6, 3), (6, 4), (7, 4), (8, 1), (9, 4);

Note that the graph contains directed cycles, e.g., between nodes 1, 5 and 8.

Enumerate All Paths from a Node

The following query returns all paths starting in node 1:

WITH RECURSIVE paths(startNode, endNode, path) AS (
        SELECT -- Define the path as the first edge of the traversal
            node1id AS startNode,
            node2id AS endNode,
            [node1id, node2id] AS path
        FROM edge
        WHERE startNode = 1
        UNION ALL
        SELECT -- Concatenate new edge to the path
            paths.startNode AS startNode,
            node2id AS endNode,
            array_append(path, node2id) AS path
        FROM paths
        JOIN edge ON paths.endNode = node1id
        -- Prevent adding a repeated node to the path.
        -- This ensures that no cycles occur.
        WHERE list_position(paths.path, node2id) IS NULL
    )
SELECT startNode, endNode, path
FROM paths
ORDER BY length(path), path;

Note that the result of this query is not restricted to shortest paths, e.g., for node 5, the results include paths [1, 5] and [1, 3, 5].

Enumerate Unweighted Shortest Paths from a Node

In most cases, enumerating all paths is not practical or feasible. Instead, only the (unweighted) shortest paths are of interest. To find these, the second half of the WITH RECURSIVE query should be adjusted such that it only includes a node if it has not yet been visited. This is implemented by using a subquery that checks if any of the previous paths includes the node:

WITH RECURSIVE paths(startNode, endNode, path) AS (
        SELECT -- Define the path as the first edge of the traversal
            node1id AS startNode,
            node2id AS endNode,
            [node1id, node2id] AS path
        FROM edge
        WHERE startNode = 1
        UNION ALL
        SELECT -- Concatenate new edge to the path
            paths.startNode AS startNode,
            node2id AS endNode,
            array_append(path, node2id) AS path
        FROM paths
        JOIN edge ON paths.endNode = node1id
        -- Prevent adding a node that was visited previously by any path.
        -- This ensures that (1) no cycles occur and (2) only nodes that
        -- were not visited by previous (shorter) paths are added to a path.
        WHERE NOT EXISTS (
                FROM paths previous_paths
                WHERE list_contains(previous_paths.path, node2id)
              )
    )
SELECT startNode, endNode, path
FROM paths
ORDER BY length(path), path;

Enumerate Unweighted Shortest Paths between Two Nodes

WITH RECURSIVE can also be used to find all (unweighted) shortest paths between two nodes. To ensure that the recursive query is stopped as soon as we reach the end node, we use a window function which checks whether the end node is among the newly added nodes.

The following query returns all unweighted shortest paths between nodes 1 (start node) and 8 (end node):

WITH RECURSIVE paths(startNode, endNode, path, endReached) AS (
   SELECT -- Define the path as the first edge of the traversal
        node1id AS startNode,
        node2id AS endNode,
        [node1id, node2id] AS path,
        (node2id = 8) AS endReached
     FROM edge
     WHERE startNode = 1
   UNION ALL
   SELECT -- Concatenate new edge to the path
        paths.startNode AS startNode,
        node2id AS endNode,
        array_append(path, node2id) AS path,
        max(CASE WHEN node2id = 8 THEN 1 ELSE 0 END)
            OVER (ROWS BETWEEN UNBOUNDED PRECEDING
                           AND UNBOUNDED FOLLOWING) AS endReached
     FROM paths
     JOIN edge ON paths.endNode = node1id
    WHERE NOT EXISTS (
            FROM paths previous_paths
            WHERE list_contains(previous_paths.path, node2id)
          )
      AND paths.endReached = 0
)
SELECT startNode, endNode, path
FROM paths
WHERE endNode = 8
ORDER BY length(path), path;

Recursive CTEs With USING KEY

USING KEY alters the behavior of a regular recursive CTE.

In each iteration, a regular recursive CTE appends results rows to the union table which, ultimately defines the overall result of the CTE. In contrast, a CTE with USING KEY has the ability to update rows that have been placed in the union table in an earlier iteration: if the current iteration produces a row with key k, it replaces a row with the same key k in the union table (like a dictionary). If no such row exists in the union table yet, the new row is appended to the union table as usual.

This allows a CTE to exercise fine-grained control over the union table contents. Avoiding the append-only behavior can lead to significantly smaller union table sizes. This helps query runtime, memory consumption, and makes it feasible to access the union table while the iteration is still ongoing (this is impossible for regular recursive CTEs): in a CTE WITH RECURSIVE T(...) USING KEY ..., table T denotes the rows added by the last iteration (as is usual for recursive CTEs), while table recurring.T denotes the union table built so far. References to recurring.T allow for the elegant and idiomatic translation of rather complex algorithms into readable SQL code.

Example: USING KEY

This is a recursive CTE where USING KEY has a key column (a) and a payload column (b). The payload columns correspond to the columns to be overwritten. In the first iteration we have two different keys, 1 and 2. These two keys will generate two new rows, (1, 3) and (2, 4). In the next iteration we produce a new key, 3, which generates a new row. We also generate the row (2,3), where 2 is a key that already exists from the previous iteration. This will overwrite the old payload 4 with the new payload 3.

WITH RECURSIVE tbl(a,b) USING KEY (a) AS (
    SELECT a, b
    FROM (VALUES (1, 3), (2, 4)) t(a, b)
	    UNION
    SELECT a + 1, b
    FROM tbl
    WHERE a < 3)
SELECT *
FROM tbl

Example: USING KEY References Union Table

As well as using the union table as a dictionary, we can now reference it in queries. This allows us to use results from not just the previous iteration, but also earlier ones. This new feature makes certain algorithms easier to implement.

One example is the connected components algorithm. For each node, the algorithm determines the node with the lowest ID to which it is connected. To achieve this, we use the entries in the union table to track the lowest ID found for a node. If a new incoming row contains a lower ID, we update this value.

CREATE TABLE nodes (id INTEGER);
INSERT INTO nodes VALUES (1), (2), (3), (4), (5), (6), (7), (8);
CREATE TABLE edges (node1id INTEGER, node2id INTEGER);
INSERT INTO edges VALUES
    (1, 3), (2, 3), (3, 7), (7, 8), (5, 4),
    (6, 4);

WITH RECURSIVE cc(id, comp) USING KEY (id) AS (
    SELECT n.id, n.id AS comp
    FROM nodes AS n
        UNION
    (SELECT DISTINCT ON (u.id) u.id, v.comp
    FROM recurring.cc AS u, cc AS v, edges AS e
    WHERE ((e.node1id, e.node2id) = (u.id, v.id)
       OR (e.node2id, e.node1id) = (u.id, v.id))
      AND v.comp < u.comp
    ORDER BY u.id ASC, v.comp ASC)
)
TABLE cc
ORDER BY id;

Limitations

DuckDB does not support mutually recursive CTEs. See the related issue and discussion in the DuckDB repository.

Syntax