Figure out the cost of each individual operator.
Only count the number of IOs added by each operator.
| Operation | RA | IOs Added (#pages) | Memory (#tuples) |
|---|---|---|---|
| Table Scan | $R$ | $\frac{|R|}{\mathcal P}$ | $O(1)$ |
| Projection | $\pi(R)$ | $0$ | $O(1)$ |
| Selection | $\sigma(R)$ | $0$ | $O(1)$ |
| Union | $R \cup S$ | $0$ | $O(1)$ |
| Sort (In-Mem) | $\tau(R)$ | $0$ | $O(|R|)$ |
| Sort (On-Disk) | $\tau(R)$ | $\frac{2 \cdot \lfloor log_{\mathcal B}(|R|) \rfloor}{\mathcal P}$ | $O(\mathcal B)$ |
| (B+Tree) Index Scan | $Index(R, c)$ | $\log_{\mathcal I}(|R|) + \frac{|\sigma_c(R)|}{\mathcal P}$ | $O(1)$ |
| (Hash) Index Scan | $Index(R, c)$ | $1$ | $O(1)$ |
| Operation | RA | IOs Added (#pages) | Memory (#tuples) |
|---|---|---|---|
| Nested Loop Join (Buffer $S$ in mem) | $R \times S$ | $0$ | $O(|S|)$ |
| Nested Loop Join (Buffer $S$ on disk) | $R \times_{disk} S$ | $(1+ |R|) \cdot \frac{|S|}{\mathcal P}$ | $O(1)$ |
| 1-Pass Hash Join | $R \bowtie_{1PH, c} S$ | $0$ | $O(|S|)$ |
| 2-Pass Hash Join | $R \bowtie_{2PH, c} S$ | $\frac{2|R| + 2|S|}{\mathcal P}$ | $O(1)$ |
| Sort-Merge Join | $R \bowtie_{SM, c} S$ | [Sort] | [Sort] |
| (Tree) Index NLJ | $R \bowtie_{INL, c}$ | $|R| \cdot (\log_{\mathcal I}(|S|) + \frac{|\sigma_c(S)|}{\mathcal P})$ | $O(1)$ |
| (Hash) Index NLJ | $R \bowtie_{INL, c}$ | $|R| \cdot 1$ | $O(1)$ |
| (In-Mem) Aggregate | $\gamma_A(R)$ | $0$ | $adom(A)$ |
| (Sort/Merge) Aggregate | $\gamma_A(R)$ | [Sort] | [Sort] |
Estimating IOs requires Estimating $|Q(R)|$
Unlike estimating IOs, cardinality estimation doesn't care about the algorithm, so we'll just be working with raw RA.
Also unlike estimating IOs, we care about the cardinality of $|Q(R)|$ as a whole, rather than the contribution of each individual operator.
| Operator | RA | Estimated Size |
|---|---|---|
| Table | $R$ | $|R|$ |
| Projection | $\pi(Q)$ | $|Q|$ |
| Union | $Q_1 \uplus Q_2$ | $|Q_1| + |Q_2|$ |
| Cross Product | $Q_1 \times Q_2$ | $|Q_1| \times |Q_2|$ |
| Sort | $\tau(Q)$ | $|Q|$ |
| Limit | $\texttt{LIMIT}_N(Q)$ | $N$ |
| Selection | $\sigma_c(Q)$ | $|Q| \times \texttt{SEL}(c, Q)$ |
| Join | $Q_1 \bowtie_c Q_2$ | $|Q_1| \times |Q_2| \times \texttt{SEL}(c, Q_1\times Q_2)$ |
| Distinct | $\delta_A(Q)$ | $\texttt{UNIQ}(A, Q)$ |
| Aggregate | $\gamma_{A, B \leftarrow \Sigma}(Q)$ | $\texttt{UNIQ}(A, Q)$ |
Every select or distinct operator passes 10% of all rows.
(Queries are typically standardized first)
(The specific % varies by DBMS. E.g., Teradata uses 10% for the first AND clause, and 75% for every subsequent clause)
The 10% rule is a fallback when everything else fails.
Usually, databases collect statistics...
We assume that for $\sigma_c(Q)$ or $\delta_A(Q)$...
COUNT(*)COUNT(DISTINCT A) (for each A)MIN(A), MAX(A) (for each numeric A)If (1) fails, fall back to the 10% rule.
If (2) or (3) fails, it'll often still be a good enough estimate.
Estimating $\delta_A(Q)$ requires only COUNT(DISTINCT A)
Selectivity is a probability ($\texttt{SEL}(c, Q) = P(c)$)
| $P(A = x_1)$ | $=$ | $\frac{1}{\texttt{COUNT(DISTINCT A)}}$ |
| $P(A \in (x_1, x_2, \ldots, x_N))$ | $=$ | $\frac{N}{\texttt{COUNT(DISTINCT A)}}$ |
| $P(A \leq x_1)$ | $=$ | $\frac{x_1 - \texttt{MIN(A)}}{\texttt{MAX(A)} - \texttt{MIN(A)}}$ |
| $P(x_1 \leq A \leq x_2)$ | $=$ | $\frac{x_2 - x_1}{\texttt{MAX(A)} - \texttt{MIN(A)}}$ |
| $P(A = B)$ | $=$ | $\textbf{min}\left( \frac{1}{\texttt{COUNT(DISTINCT A)}}, \frac{1}{\texttt{COUNT(DISTINCT B)}} \right)$ |
| $P(c_1 \wedge c_2)$ | $=$ | $P(c_1) \cdot P(c_2)$ |
| $P(c_1 \vee c_2)$ | $=$ | $1 - (1 - P(c_1)) \cdot (1 - P(c_2))$ |
(With constants $x_1$, $x_2$, ...)
Idea 1: Pick 100 tuples at random from each input table.
Assume: $\texttt{UNIQ}(A, R) = \texttt{UNIQ}(A, S) = N$
It takes $O(\sqrt{N})$ samples from both $R$ and $S$
to get even one match.
To be resumed later in the term when we talk about AQP
How DBs Do It: Instrument queries while running them.
Ideal Case: You have some $$f(x) = \left(\texttt{SELECT COUNT(*) WHERE A = x}\right)$$ (and similarly for the other aggregates)
Slightly Less Ideal Case: You have some $$f(x) \approx \left(\texttt{SELECT COUNT(*) WHERE A = x}\right)$$
If this sounds like CDF-based indexing... you're right!
... but we're not going to talk about NNs today
Simpler/Faster Idea: Break $f(x)$ into chunks
| Name | YearsEmployed | Role |
|---|---|---|
| 'Alice' | 3 | 1 |
| 'Bob' | 2 | 2 |
| 'Carol' | 3 | 1 |
| 'Dave' | 1 | 3 |
| 'Eve' | 2 | 2 |
| 'Fred' | 2 | 3 |
| 'Gwen' | 4 | 1 |
| 'Harry' | 2 | 3 |
| YearsEmployed | COUNT |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 2 |
| 4 | 1 |
COUNT(DISTINCT YearsEmployed) | $= 4$ |
MIN(YearsEmployed) | $= 1$ |
MAX(YearsEmplyed) | $= 4$ |
COUNT(*) YearsEmployed = 2 | $= 4$ |
| YearsEmployed | COUNT |
|---|---|
| 1-2 | 5 |
| 3-4 | 3 |
COUNT(DISTINCT YearsEmployed) | $= 4$ |
MIN(YearsEmployed) | $= 1$ |
MAX(YearsEmplyed) | $= 4$ |
COUNT(*) YearsEmployed = 2 | $= \frac{5}{2}$ |
| YearsEmployed | COUNT |
|---|---|
| 1-4 | 8 |
COUNT(DISTINCT YearsEmployed) | $= 4$ |
MIN(YearsEmployed) | $= 1$ |
MAX(YearsEmplyed) | $= 4$ |
COUNT(*) YearsEmployed = 2 | $= \frac{8}{4}$ |
| Value | COUNT |
|---|---|
| 1-10 | 20 |
| 11-20 | 0 |
| 21-30 | 15 |
| 31-40 | 30 |
| 41-50 | 22 |
| 51-60 | 63 |
| 61-70 | 10 |
| 71-80 | 10 |
SELECT … WHERE A = 33 |
$= \frac{1}{40-30}\cdot 30 = 3$ |
SELECT … WHERE A > 33 |
$= \frac{40-33}{40-30}\cdot 30+22$ $\;\;\;+63+10+10$ $= 126$ |
CREATE TABLE R (
A int,
B int UNIQUE
...
PRIMARY KEY A
);
No duplicate values in the column. $$\texttt{COUNT(DISTINCT A)} = \texttt{COUNT(*)}$$
CREATE TABLE S (
B int,
...
FOREIGN KEY B REFERENCES R.B
);
All values in the column appear in another table. $$\pi_{attrs(S)}\left(S \bowtie_B R\right) \subseteq S$$
Not expressible in SQL
One set of columns uniquely determines another.
$\pi_{A}(\delta(\pi_{A, B}(R)))$ has no duplicates and...
$$\pi_{attrs(R)-A}(R) \bowtie_A \delta(\pi_{A, B}(R)) = R$$
Foreign keys are like pointers. What happens with broken pointers?
Foreign keys are defined with update triggers ON INSERT [X], ON UPDATE [X], ON DELETE [X]. Depending on what [X] is, the constraint is enforced differently:
CASCADENO ACTIONSET NULL
CASCADE and NO ACTION ensure that the data never has broken pointers, so
A generalization of keys: One set of attributes that uniquely identify another.
Two rows with the same As must have the same Bs
(but can still have identical Bs for two different As)
"All functional dependencies should be keys."
(Otherwise you want two separate relations)
(for more details, see CSE 560)
$$R \bowtie_{R.A = S.B} S = \sigma_{R.A = S.B}(R \times S)$$ (and $S.B$ is a foreign key referencing $R.A$)
The (foreign) key constraint gives us two things... $$\texttt{COUNT}(\texttt{DISTINCT } A) \approx \texttt{COUNT}(\texttt{DISTINCT } B)$$ and $$\texttt{COUNT}(\texttt{DISTINCT } A) = |R|$$
Based on the first property the total number of rows is roughly... $$|R| \times |S| \times \frac{1}{\texttt{COUNT}(\texttt{DISTINCT } A)}$$
Then based on the second property... $$ = |R| \times |S| \times \frac{1}{|R|} = |S|$$
(Statistics/Histograms will give you the same outcome... but constraints can be easier to propagate)
Next class: Exam Review