CREATE VIEW salesSinceLastMonth AS
        SELECT l.*
        FROM lineitem l, orders o
        WHERE l.orderkey = o.orderkey
        AND o.orderdate > DATE(NOW() - '1 Month')
    

      SELECT partkey FROM salesSinceLastMonth
      ORDER BY shipdate DESC LIMIT 10;
    

      SELECT suppkey, COUNT(*)
      FROM salesSinceLastMonth
      GROUP BY suppkey;
    

      SELECT partkey, COUNT(*)
      FROM salesSinceLastMonth
      GROUP BY partkey;
    

Opportunity: Views exist to be queried frequently

Idea: Pre-compute and save the view’s contents!
(like an index)

When the base data changes,
the view needs to be updated too!

Our view starts off initialized

Idea: Recompute the view from scratch when data changes.

Not quite facepalm-worthy, but not ideal.

Intuition

Intuition

$+$ is like $\uplus$

$\bowtie$ is like $\times$

Are these types of patterns common?

Rings/Semirings

Semiring: $\left< \mathbb S, +, \times, \mathbb 0, \mathbb 1\right>$

Any set of 'things' $\mathbb S$ such that...

Rings/Semirings

Ring: $\left< \mathbb S, +, \times, \mathbb 0, \mathbb 1, -\right>$

Any semiring with an additive inverse....

THIS TANGENT ENDS NOW

Basic Questions

• What does $\Delta \mathcal R$ mean?
• What is $\mathcal R + \Delta \mathcal R$?
• How do we derive $\Delta Q$?

What is $\Delta \mathcal R$?

What is $\Delta \mathcal R$?

A Set/Bag of Insertions

+

A Set/Bag of Deletions

What is $\Delta \mathcal R$?

... this feels a bit wrong

$\texttt{VIEW} \leftarrow \texttt{VIEW} + $$\Delta Q(\mathcal D, \Delta \mathcal D)$

$\sigma(\mathcal R) \rightarrow \sigma(\mathcal R \uplus \Delta \mathcal R)$

$ \equiv $ $\sigma(\mathcal R)$ $ \uplus $ $\sigma(\Delta \mathcal R)$

Set/Bag difference also commutes through selection

$\pi(\mathcal R) \rightarrow \pi(\mathcal R \uplus \Delta \mathcal R)$

$ \equiv $ $\pi(\mathcal R)$ $ \uplus $ $\pi(\Delta \mathcal R)$

Does this work under set semantics?

$\mathcal R_1 \uplus \mathcal R_2 \rightarrow \mathcal R_1 \uplus \Delta \mathcal R_1 \uplus \mathcal R_2 \uplus \Delta \mathcal R_2$

$ \equiv $ $\mathcal R_1 \uplus \mathcal R_2$ $ \uplus $ $\Delta \mathcal R_1 \uplus \Delta \mathcal R_2$

How do we derive $\Delta Q$?

So far: $$\Delta Q_{ins}(\mathcal D, \Delta \mathcal D) = Q(\Delta \mathcal Q_{ins})$$ $$\Delta Q_{del}(\mathcal D, \Delta \mathcal D) = Q(\Delta \mathcal Q_{del})$$

Multisets

$$\{1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5\}$$

(not compact)

$$\{1 \rightarrow \times 3; 2 \rightarrow \times 5; 3 \rightarrow \times 2; 4 \rightarrow \times 6; 5 \rightarrow \times 1\}$$

Multiset representation: Tuple $\rightarrow$ # of occurrences

multiplicity

Multiset Deltas

Insertions = Positive Multiplicity

Deletions = Negative Multiplicity

+ = Bag/Multiset Union

Multiset Deltas

What does Union Do?

$\{ A \rightarrow 1, B \rightarrow 2 \} \uplus \{ B \rightarrow 2, C \rightarrow 4 \} = $

$=\{$ $A \rightarrow 1,$ $B \rightarrow 5,$ $C \rightarrow 4$ $\}$

$\{ A \rightarrow 1 \} \uplus \{ A \rightarrow -1 \}$ $= \{A \rightarrow 0\}$ $ = \{\}$

Multiset Deltas

What does Cross-Product Do?

$\{ A \rightarrow 1, B \rightarrow 3 \} \times \{ C \rightarrow 4 \} $

$= \{ \left< A,C \right> \rightarrow$ $4$ $\left< B,C \right> \rightarrow$ $12$ $\}$

Multiset Deltas

What does Projection Do?

$\pi_{\text{1st attr}}\{ \left< A,X \right> \rightarrow 1, \left< A,Y \right> \rightarrow 2, \left< B,Z \right> \rightarrow 5, \}$

$= \{$ $A \rightarrow 1, A \rightarrow 2,$ $B \rightarrow 5\}$

$= \{A \rightarrow 3, B \rightarrow 5\}$

Multiset Deltas

What does Selection Do?

$\sigma_{A}\{ A \rightarrow 1, B \rightarrow 5, C \rightarrow 3 \}$

$= \{ A \rightarrow 1, B \rightarrow 0, C \rightarrow 0 \}$

$= \{ A \rightarrow 1\}$