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---
template: templates/cse4562_2021_slides.erb
title: "Cost-Based Optimization"
date: March 11, 2021
textbook: Ch. 16
---
<section>
<section>
<h3>Remember the Real Goals</h3>
<ol>
<li>Accurately <b>rank</b> the plans.</li>
<li>Don't spend more time optimizing than you get back.</li>
<li>Don't pick a plan that uses more memory than you have.</li>
</ol>
</section>
<section>
<table style="font-size: 70%">
<tr><th>Operation</th><th>RA</th><th>Total IOs (#pages)</th><th>Memory (#tuples)</th></tr>
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<tr >
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<td>Table Scan</td>
<td>$R$</td>
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<td >$\frac{|R|}{\mathcal P}$</td>
<td >$O(1)$</td>
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</tr>
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<tr >
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<td>Projection</td>
<td>$\pi(R)$</td>
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<td >$\textbf{io}(R)$</td>
<td >$O(1)$</td>
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</tr>
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<tr >
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<td>Selection</td>
<td>$\sigma(R)$</td>
<td>$\textbf{io}(R)$</td>
<td>$O(1)$</td>
</tr>
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<tr >
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<td>Union</td>
<td>$R \uplus S$</td>
<td>$\textbf{io}(R) + \textbf{io}(S)$</td>
<td>$O(1)$</td>
</tr>
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<tr >
<td style="vertical-align: middle;">Sort <span >(In-Mem)</span></td>
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<td style="vertical-align: middle;">$\tau(R)$</td>
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<td >$\textbf{io}(R)$</td>
<td >$O(|R|)$</td>
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</tr>
<tr>
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<td >Sort (On-Disk)</td>
<td >$\tau(R)$</td>
<td >$\frac{2 \cdot \lfloor log_{\mathcal B}(|R|) \rfloor}{\mathcal P} + \textbf{io}(R)$</td>
<td >$O(\mathcal B)$</td>
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</tr>
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<tr >
<td><span >(B+Tree)</span> Index Scan</td>
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<td>$Index(R, c)$</td>
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<td >$\log_{\mathcal I}(|R|) + \frac{|\sigma_c(R)|}{\mathcal P}$</td>
<td >$O(1)$</td>
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</tr>
<tr>
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<td span >(Hash) Index Scan</td>
<td span >$Index(R, c)$</td>
<td >$1$</td>
<td >$O(1)$</td>
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</tr>
</table>
<ol style="font-size: 50%; margin-top: 50px;">
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<li >Tuples per Page ($\mathcal P$) <span>–
<li >Size of $R$ ($|R|$)</li>
<li >Pages of Buffer ($\mathcal B$)</li>
<li >Keys per Index Page ($\mathcal I$)</li>
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</ol>
</section>
<section>
<table style="font-size: 70%">
<tr><th width="300px">Operation</th><th>RA</th><th>Total IOs (#pages)</th><th style="font-size: 80%;">Mem (#tuples)</th></tr>
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<tr >
<td style="font-size: 60%">Nested Loop Join <span >(Buffer $S$ in mem)</span></td>
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<td>$R \times_{mem} S$</td>
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<td >$\textbf{io}(R)+\textbf{io}(S)$</td>
<td >$O(|S|)$</td>
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</tr>
<tr>
<td style="font-size: 60%">Block NLJ (Buffer $S$ on disk)</td>
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<td >$R \times_{disk} S$</td>
<td >$\frac{|R|}{\mathcal B} \cdot \frac{|S|}{\mathcal P} + \textbf{io}(R) + \textbf{io}(S)$</td>
<td >$O(1)$</td>
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</tr>
<tr>
<td style="font-size: 60%">Block NLJ (Recompute $S$)</td>
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<td >$R \times_{redo} S$</td>
<td >$\textbf{io}(R) + \frac{|R|}{\mathcal B} \cdot \textbf{io}(S)$</td>
<td >$O(1)$</td>
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</tr>
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<tr >
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<td>1-Pass Hash Join</td>
<td>$R \bowtie_{1PH, c} S$</td>
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<td >$\textbf{io}(R) + \textbf{io}(S)$</td>
<td >$O(|S|)$</td>
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</tr>
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<tr >
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<td>2-Pass Hash Join</td>
<td>$R \bowtie_{2PH, c} S$</td>
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<td >$\frac{2|R| + 2|S|}{\mathcal P} + \textbf{io}(R) + \textbf{io}(S)$</td>
<td >$O(1)$</td>
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</tr>
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<tr >
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<td>Sort-Merge Join </td>
<td>$R \bowtie_{SM, c} S$</td>
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<td >[Sort]</td>
<td >[Sort]</td>
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</tr>
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<tr >
<td><span >(Tree)</span> Index NLJ</td>
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<td>$R \bowtie_{INL, c}$</td>
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<td >$|R| \cdot (\log_{\mathcal I}(|S|) + \frac{|\sigma_c(S)|}{\mathcal P})$</td>
<td >$O(1)$</td>
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</tr>
<tr>
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<td >(Hash) Index NLJ</td>
<td >$R \bowtie_{INL, c}$</td>
<td >$|R| \cdot 1$</td>
<td >$O(1)$</td>
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</tr>
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<tr >
<td><span >(In-Mem)</span> Aggregate</td>
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<td>$\gamma_A(R)$</td>
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<td >$\textbf{io}(R)$</td>
<td >$adom(A)$</td>
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</tr>
<tr>
<td style="font-size: 90%">(Sort/Merge) Aggregate</td>
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<td >$\gamma_A(R)$</td>
<td >[Sort]</td>
<td >[Sort]</td>
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</tr>
</table>
</section>
</section>
<section>
<section>
<h3>Cardinality Estimation</h3>
<h4>(The Hard Parts)</h4>
<dl>
<dt style="margin-top: 50px;">$\sigma_c(Q)$ (Cardinality Estimation)</dt>
<dd>How many tuples will a condition $c$ allow to pass?</dd>
<dt style="margin-top: 50px;">$\delta_A(Q)$ (Distinct Values Estimation)</dt>
<dd>How many distinct values of attribute(s) $A$ exist?</dd>
</dl>
</section>
<section>
<h3>Remember the Real Goals</h3>
<ol>
<li>Accurately <b>rank</b> the plans.</li>
<li>Don't spend more time optimizing than you get back.</li>
</ol>
</section>
<section>
<h3>(Some) Estimation Techniques</h3>
<dl style="font-size: 80%">
<div class="fragment highlight-grey" data-fragment-index="1">
<dt>Guess Randomly</dt>
<dd>Rules of thumb if you have no other options...</dd>
</div>
<div class="fragment highlight-grey" data-fragment-index="1">
<dt>Uniform Prior</dt>
<dd>Use basic statistics to make a very rough guess.</dd>
</div>
<div>
<dt>Sampling / History</dt>
<dd>Small, Quick Sampling Runs (or prior executions of the query).</dd>
</div>
<div>
<dt>Histograms</dt>
<dd>Using more detailed statistics for improved guesses.</dd>
</div>
<div>
<dt>Constraints</dt>
<dd>Using rules about the data for improved guesses.</dd>
</div>
</dl>
</section>
</section>
<section>
<section>
<h3>(Some) Estimation Techniques</h3>
<dl style="font-size: 80%">
<dt style="color: grey;">Guess Randomly</dt>
<dd style="color: grey;">Rules of thumb if you have no other options...</dd>
<dt style="color: grey;">Uniform Prior</dt>
<dd style="color: grey;">Use basic statistics to make a very rough guess.</dd>
<dt style="color: blue;">Sampling / History</dt>
<dd style="color: blue;">Small, Quick Sampling Runs (or prior executions of the query).</dd>
<dt style="color: grey;">Histograms</dt>
<dd style="color: grey;">Using more detailed statistics for improved guesses.</dd>
<dt style="color: grey;">Constraints</dt>
<dd style="color: grey;">Using rules about the data for improved guesses.</dd>
</dl>
</section>
<section>
<p><b>Idea 1:</b> Pick 100 tuples at random from each input table.</p>
</section>
<section>
<svg data-src="2021-03-11/JoinIssue.svg" />
</section>
<section>
<h3>The Birthday Paradox</h3>
<p style="margin-top: 50px;">
Assume: $\texttt{UNIQ}(A, R) = \texttt{UNIQ}(A, S) = N$
</p>
<p style="margin-top: 50px;">
It takes $O(\sqrt{N})$ samples from both $R$ and $S$ <br/> to get even <b>one match.</b>
</p>
</section>
<section>
<p>To be resumed later in the term when we talk about AQP</p>
</section>
<section>
<p><b>How DBs Do It</b>: Instrument queries while running them.<ul>
<li class="fragment">The first time you run a query it <i>might</i> be slow.</li>
<li class="fragment">The second, third, fourth, etc... times it'll be fast.</li>
</ul></p>
</section>
</section>
<section>
<section>
<h3>(Some) Estimation Techniques</h3>
<dl style="font-size: 80%">
<dt style="color: grey;">Guess Randomly</dt>
<dd style="color: grey;">Rules of thumb if you have no other options...</dd>
<dt style="color: grey;">Uniform Prior</dt>
<dd style="color: grey;">Use basic statistics to make a very rough guess.</dd>
<dt style="color: grey;">Sampling / History</dt>
<dd style="color: grey;">Small, Quick Sampling Runs (or prior executions of the query).</dd>
<dt style="color: blue;">Histograms</dt>
<dd style="color: blue;">Using more detailed statistics for improved guesses.</dd>
<dt style="color: grey;">Constraints</dt>
<dd style="color: grey;">Using rules about the data for improved guesses.</dd>
</dl>
</section>
<section>
<h3>Limitations of Uniform Prior</h3>
<dl>
<div class="fragment highlight-grey" data-fragment-index="1">
<dt>Don't always have statistics for $Q$</dt>
<dd>For example, $\pi_{A \leftarrow (B \times C)}(R)$</dd>
</div>
<div class="fragment highlight-grey" data-fragment-index="1">
<dt>Don't always have clear rules for $c$</dt>
<dd>For example, $\sigma_{\texttt{FitsModel}(A, B, C)}(R)$</dd>
</div>
<div class="fragment highlight-blue" data-fragment-index="1">
<dt>Attribute values are not always uniformly distributed.</dt>
<dd>For example, <span style="font-size: 60%"> $|\sigma_{SPC\_COMMON = 'pin\ oak'}(T)|$ vs $|\sigma_{SPC\_COMMON = 'honeylocust'}(T)|$</span></dd>
</div>
<div class="fragment highlight-grey" data-fragment-index="1">
<dt>Attribute values are sometimes correlated.</dt>
<dd>For example, $\sigma_{(stump < 5) \wedge (diam > 3)}(T)$</dd>
</div>
</dl>
</section>
<section>
<p class="fragment highlight-grey" data-fragment-index="1">
<b>Ideal Case:</b> You have some
$$f(x) = \left(\texttt{SELECT COUNT(*) WHERE A = x}\right)$$
(and similarly for the other aggregates)
</p>
<p class="fragment" data-fragment-index="1">
<b>Slightly Less Ideal Case:</b> You have some
$$f(x) \approx \left(\texttt{SELECT COUNT(*) WHERE A = x}\right)$$
</p>
</section>
<section>
<p>If this sounds like CDF-based indexing... you're right!</p>
<p class="fragment">... but we're not going to talk about NNs today</p>
</section>
</section>
<section>
<section>
<p>
<b>Simpler/Faster Idea: </b> Break $f(x)$ into chunks
</p>
</section>
<section>
<h3>Example Data</h3>
<table style="font-size: 80%">
<tr><th>Name</th> <th>YearsEmployed</th> <th>Role</th></tr>
<tr><td>'Alice'</td> <td>3</td> <td>1</td></tr>
<tr><td>'Bob'</td> <td>2</td> <td>2</td></tr>
<tr><td>'Carol'</td> <td>3</td> <td>1</td></tr>
<tr><td>'Dave'</td> <td>1</td> <td>3</td></tr>
<tr><td>'Eve'</td> <td>2</td> <td>2</td></tr>
<tr><td>'Fred'</td> <td>2</td> <td>3</td></tr>
<tr><td>'Gwen'</td> <td>4</td> <td>1</td></tr>
<tr><td>'Harry'</td> <td>2</td> <td>3</td></tr>
</table>
</section>
<section>
<h3>Histograms</h3>
<table style="font-size: 70%">
<tr><th>YearsEmployed</th><th>COUNT</th></tr>
<tr><td>1</td> <td>1</td> </tr>
<tr><td>2</td> <td>4</td> </tr>
<tr><td>3</td> <td>2</td> </tr>
<tr><td>4</td> <td>1</td> </tr>
</table>
<table>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(DISTINCT YearsEmployed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MIN(YearsEmployed)</code> </td><td class="fragment">$= 1$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MAX(YearsEmplyed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(*) YearsEmployed = 2</code> </td><td class="fragment">$= 4$</td></tr>
</table>
</section>
<section>
<h3>Histograms</h3>
<table style="font-size: 70%">
<tr><th>YearsEmployed</th><th>COUNT</th></tr>
<tr><td>1-2</td> <td>5</td> </tr>
<tr><td>3-4</td> <td>3</td> </tr>
</table>
<table>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(DISTINCT YearsEmployed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MIN(YearsEmployed)</code> </td><td class="fragment">$= 1$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MAX(YearsEmplyed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(*) YearsEmployed = 2</code> </td><td class="fragment">$= \frac{5}{2}$</td></tr>
</table>
</section>
<section>
<h3>The Extreme Case</h3>
<table style="font-size: 70%">
<tr><th>YearsEmployed</th><th>COUNT</th></tr>
<tr><td>1-4</td> <td>8</td> </tr>
</table>
<table>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(DISTINCT YearsEmployed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MIN(YearsEmployed)</code> </td><td class="fragment">$= 1$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>MAX(YearsEmplyed)</code> </td><td class="fragment">$= 4$</td></tr>
<tr class="fragment"><td style="font-size: 70%"><code>COUNT(*) YearsEmployed = 2</code> </td><td class="fragment">$= \frac{8}{4}$</td></tr>
</table>
</section>
<section>
<h3>More Example Data</h3>
<table style="font-size: 80%; float: left;">
<tr><th>Value</th> <th>COUNT</th> </tr>
<tr><td> 1-10</td> <td>20</td> </tr>
<tr><td>11-20</td> <td> 0</td> </tr>
<tr><td>21-30</td> <td>15</td> </tr>
<tr><td>31-40</td> <td>30</td> </tr>
<tr><td>41-50</td> <td>22</td> </tr>
<tr><td>51-60</td> <td>63</td> </tr>
<tr><td>61-70</td> <td>10</td> </tr>
<tr><td>71-80</td> <td>10</td> </tr>
</table>
<table style="margin-top: 100px;">
<tr class="fragment">
<td style="font-size: 70%; width: 350px;"><code>SELECT …
<td class="fragment" style="font-size: 80%; text-align: left; width: 200px;">$= \frac{1}{40-30}\cdot 30 = 3$</td>
</tr>
<tr><td style="height: 70px;"></td><td></td></tr>
<tr class="fragment">
<td style="font-size: 70%; width: 350px;"><code>SELECT …
<td class="fragment" style="font-size: 80%; text-align: left; width: 200px;">$= \frac{40-33}{40-30}\cdot 30+22$ $\;\;\;+63+10+10$ $= 126$ </td>
</tr>
</table>
</section>
</section>
<section>
<section>
<h3>(Some) Estimation Techniques</h3>
<dl style="font-size: 80%">
<dt style="color: grey;">Guess Randomly</dt>
<dd style="color: grey;">Rules of thumb if you have no other options...</dd>
<dt style="color: grey;">Uniform Prior</dt>
<dd style="color: grey;">Use basic statistics to make a very rough guess.</dd>
<dt style="color: grey;">Sampling / History</dt>
<dd style="color: grey;">Small, Quick Sampling Runs (or prior executions of the query).</dd>
<dt style="color: grey;">Histograms</dt>
<dd style="color: grey;">Using more detailed statistics for improved guesses.</dd>
<dt style="color: blue;">Constraints</dt>
<dd style="color: blue;">Using rules about the data for improved guesses.</dd>
</dl>
</section>
<section>
<h3>Key / Unique Constraints</h3>
<pre style="margin-top: 50px;"><code class="sql">
CREATE TABLE R (
A int,
B int UNIQUE
...
PRIMARY KEY A
);
</code></pre>
<p style="margin-top: 50px;">
No duplicate values in the column.
$$\texttt{COUNT(DISTINCT A)} = \texttt{COUNT(*)}$$
</p>
</section>
<section>
<h3>Foreign Key Constraints</h3>
<pre style="margin-top: 50px;"><code class="sql">
CREATE TABLE S (
B int,
...
FOREIGN KEY B REFERENCES R.B
);
</code></pre>
<p style="margin-top: 50px;">
All values in the column appear in another table.
$$\pi_{attrs(S)}\left(S \bowtie_B R\right) \subseteq S$$
</p>
</section>
<section>
<h3>Functional Dependencies</h3>
<pre style="margin-top: 50px;"><code class="sql">
Not expressible in SQL
</code></pre>
<p style="margin-top: 50px;">
One set of columns uniquely determines another.<br/>
$\pi_{A}(\delta(\pi_{A, B}(R)))$ has no duplicates and...
$$\pi_{attrs(R)-A}(R) \bowtie_A \delta(\pi_{A, B}(R)) = R$$
</p>
</section>
<section>
<h3>Constraints</h3>
<h4>The Good</h4>
<ul>
<li style="font-size: 70%" class="fragment">Sanity check on your data: Inconsistent data triggers failures.</li>
<li style="font-size: 70%" class="fragment">More opportunities for query optimization.</li>
</ul>
<h4 style="margin-top: 50px;" class="fragment">The Not-So Good</h4>
<ul>
<li style="font-size: 70%" class="fragment">Validating constraints whenever data changes is (usually) expensive.</li>
<li style="font-size: 70%" class="fragment">Inconsistent data triggers failures.</li>
</ul>
</section>
<section>
<h3>Foreign Key Constraints</h3>
<p style="margin-top: 50px;">Foreign keys are like pointers. What happens with broken pointers?</p>
</section>
<section>
<h3>Foreign Key Enforcement</h3>
<p>Foreign keys are defined with update triggers <code>ON INSERT [X]</code>, <code>ON UPDATE [X]</code>, <code>ON DELETE [X]</code>. Depending on what [X] is, the constraint is enforced differently:</p>
<dl style="font-size: 80%">
<dt><code>CASCADE</code></dt>
<dd>Create/delete rows as needed to avoid invalid foreign keys.</dd>
<dt><code>NO ACTION</code></dt>
<dd>Abort any transaction that ends with an invalid foreign key reference.</dd>
<dt><code>SET NULL</code></dt>
<dd>Automatically replace any invalid foreign key references with NULL</dd>
</dl>
</section>
<section>
<p style="font-weight: bold;">
<code>CASCADE</code> and <code>NO ACTION</code> ensure that the data never has broken pointers, so
</p>
$$\pi_{attrs(S)}\left(S \bowtie_B R\right) = S$$
</section>
<section>
<h3>Functional Dependencies</h3>
<p style="margin-top: 50px;"><b>A generalization of keys:</b> One set of attributes that uniquely identify another.</p>
<ul>
<li>SS# uniquely identifies Name.</li>
<li>Employee uniquely identifies Manager.</li>
<li>Order number uniquely identifies Customer Address.</li>
</ul>
<p class="fragment">Two rows with the same As must have the same Bs</p>
<p class="fragment" style="font-size: 80%">(but can still have identical Bs for two different As)</p>
</section>
<section>
<h3>Normal Forms</h3>
<p style="margin-top: 50px;">"All functional dependencies should be keys."</p>
<p class="fragment">(Otherwise you want two separate relations)</p>
<p class="fragment">(for more details, see CSE 560)</p>
</section>
<section>
<p style="font-size: 70%">
$$P(A = B) = min\left(\frac{1}{\texttt{COUNT}(\texttt{DISTINCT } A)}, \frac{1}{\texttt{COUNT}(\texttt{DISTINCT } B)}\right)$$
</p>
</section>
<section>
<p>
$$R \bowtie_{R.A = S.B} S = \sigma_{R.A = S.B}(R \times S)$$
(and $S.B$ is a foreign key referencing $R.A$)
</p>
<p class="fragment" style="margin-top: 30px; font-size: 80%">
The (foreign) key constraint gives us two things...
$$\texttt{COUNT}(\texttt{DISTINCT } A) \approx \texttt{COUNT}(\texttt{DISTINCT } B)$$
<span style="font-size: 60%; font-weight: bold; margin: 0px;">and</span>
$$\texttt{COUNT}(\texttt{DISTINCT } A) = |R|$$
</p>
<p class="fragment" style="margin-top: 30px; font-size: 80%">
Based on the first property the total number of rows is roughly...
$$|R| \times |S| \times \frac{1}{\texttt{COUNT}(\texttt{DISTINCT } A)}$$
</p>
<p class="fragment" style="margin-top: 30px; font-size: 80%">
Then based on the second property...
$$ = |R| \times |S| \times \frac{1}{|R|} = |S|$$
</p>
<p class="fragment" style="margin-top: 30px; font-size: 50%">(Statistics/Histograms will give you the same outcome... but constraints can be easier to propagate)</p>
</section>
</section>
