Lattices

A lattice is a framework for creating and populating materialized views, and for recognizing that a materialized view can be used to solve a particular query.

A lattice represents a star (or snowflake) schema, not a general schema. In particular, all relationships must be many-to-one, heading from a fact table at the center of the star.

The name derives from the mathematics: a lattice is a partially ordered set where any two elements have a unique greatest lower bound and least upper bound.

[HRU96] observed that the set of possible materializations of a data cube forms a lattice, and presented an algorithm to choose a good set of materializations. Calcite’s recommendation algorithm is derived from this.

The lattice definition uses a SQL statement to represent the star. SQL is a useful short-hand to represent several tables joined together, and assigning aliases to the column names (it more convenient than inventing a new language to represent relationships, join conditions and cardinalities).

Unlike regular SQL, order is important. If you put A before B in the FROM clause, and make a join between A and B, you are saying that there is a many-to-one foreign key relationship from A to B. (E.g. in the example lattice, the Sales fact table occurs before the Time dimension table, and before the Product dimension table. The Product dimension table occurs before the ProductClass outer dimension table, further down an arm of a snowflake.)

A lattice implies constraints. In the A to B relationship, there is a foreign key on A (i.e. every value of A’s foreign key has a corresponding value in B’s key), and a unique key on B (i.e. no key value occurs more than once). These constraints are really important, because it allows the planner to remove joins to tables whose columns are not being used, and know that the query results will not change.

Calcite does not check these constraints. If they are violated, Calcite will return wrong results.

A lattice is a big, virtual join view. It is not materialized (it would be several times larger than the star schema, because of denormalization) and you probably wouldn’t want to query it (far too many columns). So what is it useful for? As we said above, (a) the lattice declares some very useful primary and foreign key constraints, (b) it helps the query planner map user queries onto filter-join-aggregate materialized views (the most useful kind of materialized view for DW queries), (c) gives Calcite a framework within which to gather stats about data volumes and user queries, (d) allows Calcite to automatically design and populate materialized views.

Most star schema models force you to choose whether a column is a dimension or a measure. In a lattice, every column is a dimension column. (That is, it can become one of the columns in the GROUP BY clause to query the star schema at a particular dimensionality). Any column can also be used in a measure; you define measures by giving the column and an aggregate function.

If “unit_sales” tends to be used much more often as a measure rather than a dimension, that’s fine. Calcite’s algorithm should notice that it is rarely aggregated, and not be inclined to create tiles that aggregate on it. (By “should” I mean “could and one day will”. The algorithm does not currently take query history into account when designing tiles.)

But someone might want to know whether orders with fewer than 5 items were more or less profitable than orders with more than 100. All of a sudden, “unit_sales” has become a dimension. If there’s virtually zero cost to declaring a column a dimension column, I figured let’s make them all dimension columns.

The model allows for a particular table to be used more than once, with a different table alias. You could use this to model say OrderDate and ShipDate, with two uses to the Time dimension table.

Most SQL systems require that the column names in a view are unique. This is hard to achieve in a lattice, because you often include primary and foreign key columns in a join. So Calcite lets you refer to columns in two ways. If the column is unique, you can use its name, [“unit_sales”]. Whether or not it is unique in the lattice, it will be unique in its table, so you can use it qualified by its table alias. Examples:

  • [“sales”, “unit_sales”]
  • [“ship_date”, “time_id”]
  • [“order_date”, “time_id”]

A “tile” is a materialized table in a lattice, with a particular dimensionality. (What Kylin calls a “cuboid”.) The “tiles” attribute of the lattice JSON element defines an initial set of tiles to materialize.

If you run the algorithm, you can omit the tiles attribute. Calcite will choose an initial set. If you include the tiles attribute, the algorithm will start with that list and then start finding other tiles that are complementary (i.e. “fill in the gaps” left by the initial tiles).

References