Thursday's slides
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src/teaching/cse-562/2021sp/slide/2021-02-11-RA-Equivs.erb
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src/teaching/cse-562/2021sp/slide/2021-02-11-RA-Equivs.erb
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---
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template: templates/cse4562_2021_slides.erb
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title: "Relational Algebra Equivalences"
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date: February 11, 2021
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textbook: "Ch. 16.2"
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---
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<section>
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<section>
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<h3>The running theme</h3>
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<p>
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<span class="fragment highlight-grey" data-fragment-index="1">If X and Y are </span><u>equivalent</u><span class="fragment highlight-grey" data-fragment-index="1"> and Y is <u>better</u>,<br/>
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then replace all Xs with Ys</span>
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</p>
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<p class="fragment" data-fragment-index="1" style="font-size: 70%;"><b>Today's focus</b>: Provable Equivalence for RA Expressions</p>
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</section>
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<section>
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<h3>Equivalence</h3>
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$$Q_1 = \pi_{A}\left( \sigma_{c}( R ) \right)$$
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$$Q_2 = \sigma_{c}\left( \pi_{A}( R ) \right)$$
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<div class="fragment">
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$$Q_1 \stackrel{?}{\equiv} Q_2$$
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</div>
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</section>
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<section>
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<h3>Ground Rules</h3>
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<dl>
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<dt class="fragment" data-fragment-index="1">Only Instances Matter</dt>
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<dd class="fragment" data-fragment-index="1">Obviously $Q_1 \neq Q_2$. What we care about is whether $Q_1(R) = Q_2(R)$...</dd>
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<dt class="fragment" data-fragment-index="2">Data Independent</dt>
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<dd class="fragment" data-fragment-index="2">... for <i>all</i> valid input data $R$.</dd>
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<dd class="fragment" data-fragment-index="3" style="font-size: 70%">However, it's fair to talk about equivalence when we know the data has some properties. (more on this later)</dd>
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<dt class="fragment" data-fragment-index="4">Data-Model Dependent</dt>
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<dd class="fragment" data-fragment-index="4">It's important to be clear whether we're talking about sets, <span class="fragment highlight-blue">bags</span>, or lists.</dd>
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</dl>
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</section>
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<section>
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<h3>In summary...</h3>
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<p style="font-size: 80%;">
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We say that $Q_1 \equiv Q_2$ if and only if<br/>
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we can guarantee that the <i>bag</i> of tuples produced by $Q_1(R, S, T, \ldots)$ <br/>
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is the same as the <i>bag</i> of tuples produced by $Q_2(R, S, T, \ldots)$ <br/>
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for any combination of valid inputs $R, S, T, \ldots$.
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</p>
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<p style="font-size: 70%;" class="fragment">
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... that satisfy any necessary properties.
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</p>
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</section>
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<section>
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<h3>Starting Rules</h3>
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<table style="font-size: 80%">
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<tr><th colspan="2" style="padding-top: 20px;">Selection</th></tr>
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<tr>
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<td>$\sigma_{c_1 \wedge c_2}(R) \equiv \sigma_{c_1}(\sigma_{c_2}(R))$</td>
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<td>(Decomposability)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Projection</th></tr>
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<tr>
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<td>$\pi_{A}(R) \equiv \pi_{A}(\pi_{A \cup B}(R))$</td>
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<td>(Idempotence)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Cross Product</th></tr>
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<tr>
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<td>$R \times (S \times T) \equiv (R \times S) \times T$</td>
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<td>(Associativity)</td>
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</tr>
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<tr>
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<td>$R \times S \equiv S \times R$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Union</th></tr>
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<tr>
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<td>$R \cup (S \cup T) \equiv (R \cup S) \cup T$</td>
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<td>(Associativity)</td>
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</tr>
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<tr>
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<td>$R \cup S \equiv S \cup R$</td>
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<td>(Commutativity)</td>
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</tr>
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</table>
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</section>
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<section>
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<h3>Try it!</h3>
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<p class="fragment highlight-grey" data-fragment-index="1">
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Show that
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$$R \times (S \times T) \equiv T \times (S \times R)$$
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</p>
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<div class="fragment highlight-grey" data-fragment-index="2">
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<p class="fragment" data-fragment-index="1">
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Show that
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$$\sigma_{c_1}(\sigma_{c_2}(R)) \equiv \sigma_{c_2}(\sigma_{c_1}(R))$$
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</p>
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</div>
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<div class="fragment highlight-grey" data-fragment-index="3">
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<p class="fragment" data-fragment-index="2">
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Show that
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$$R \bowtie_{c} S \equiv S \bowtie_{c} R$$
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</p>
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</div>
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<p class="fragment" data-fragment-index="3">
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Show that
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$$\sigma_{R.B = S.B \wedge R.A > 3}(R \times S) \equiv \sigma_{R.A > 3}(R \bowtie_{B} S)$$
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</p>
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</section>
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</section>
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<section>
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<section>
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<h3>Cross-Operator Rules</h3>
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</section>
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<section>
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<table style="font-size: 90%; margin-bottom: 50px;">
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<tr><th colspan="2" style="padding-top: 20px;">Selection + Projection</th></tr>
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<tr>
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<td>$\pi_{A}(\sigma_{c}(R)) \equiv \sigma_{c}(\pi_{A}(R))$</td>
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<td>(Commutativity)</td>
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</tr>
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</table>
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<p style="font-size: 80%;" class="fragment">... but only if $A$ and $c$ are <u>compatible</u></p>
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<p style="font-size: 80%;" class="fragment">$A$ must include all columns referenced by $c$ ($cols(c)$)</p>
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<div class="fragment" style="margin-top: 50px;">
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<h3>Try it!</h3>
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<p>
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Show that
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$$\pi_A(\sigma_c(R)) \equiv \pi_A(\sigma_c(\pi_{(A \cup cols(c))}(R)))$$
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</p>
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</div>
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</section>
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<section>
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<table style="font-size: 90%; margin-bottom: 50px;">
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<tr><th colspan="2" style="padding-top: 20px;">Selection + Cross Product</th></tr>
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<tr>
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<td>$\sigma_c(R \times S) \equiv (\sigma_{c}(R)) \times S$</td>
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<td>(Commutativity)</td>
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</tr>
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</table>
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<p style="font-size: 80%;" class="fragment">... but only if $c$ references only columns of $R$</p>
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<p style="font-size: 60%;" class="fragment">$cols(c) \subseteq cols(R)$</p>
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<div class="fragment" style="margin-top: 50px;">
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<h3>Try it!</h3>
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<p>
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Show that
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$$\sigma_{R.B = S.B \wedge R.A > 3}(R \times S) \equiv (\sigma_{R.A > 3}(R)) \bowtie_{B} S$$
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<div style="font-size: 70%;" class="fragment">When is this rewrite a good idea?</div>
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</p>
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</div>
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</section>
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<section>
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<table style="font-size: 90%; margin-bottom: 50px;">
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<tr><th colspan="2" style="padding-top: 20px;">Projection + Cross Product</th></tr>
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<tr>
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<td>$\pi_A(R \times S) \equiv (\pi_{A_R}(R)) \times (\pi_{A_S}(S))$</td>
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<td>(Commutativity)</td>
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</tr>
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</table>
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<p style="font-size: 80%;">... where $A_R$ and $A_S$ are the columns of $A$ from $R$ and $S$ respectively.</p>
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<p style="font-size: 60%;">$A_R = A \cap cols(R)$ $A_S = A \cap cols(S)$</p>
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<div class="fragment" style="margin-top: 50px;">
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<h3>Try it!</h3>
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<p>
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Show that
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$$\pi_{A}(R \bowtie_c S) \equiv (\pi_{A_R}(R)) \bowtie_c (\pi_{A_S}(S))$$
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<div style="font-size: 70%;" class="fragment">When does this condition hold?</div>
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</p>
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</div>
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</section>
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<section>
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<table style="font-size: 80%; margin-bottom: 50px;">
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<tr><th colspan="2" style="padding-top: 20px;">Intersection</th></tr>
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<tr>
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<td>$R \cap (S \cap T) \equiv (R \cap S) \cap T$</td>
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<td>(Associativity)</td>
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</tr>
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<tr>
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<td>$R \cap S \equiv S \cap R$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Selection + <u> </u></th></tr>
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<tr>
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<td>$\sigma_c(R \cup S) \equiv (\sigma_c(R)) \cup (\sigma_c(R))$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr>
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<td>$\sigma_c(R \cap S) \equiv (\sigma_c(R)) \cap (\sigma_c(R))$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Projection + <u> </u></th></tr>
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<tr>
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<td>$\pi_A(R \cup S) \equiv (\pi_A(R)) \cup (\pi_A(R))$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr>
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<td>$\pi_A(R \cap S) \equiv (\pi_A(R)) \cap (\pi_A(R))$</td>
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<td>(Commutativity)</td>
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</tr>
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<tr><th colspan="2" style="padding-top: 20px;">Cross Product + Union</th></tr>
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<tr>
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<td>$R \times (S \cup T) \equiv (R \times S) \cup (R \times T)$</td>
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<td>(Distributivity)</td>
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</tr>
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</table>
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</section>
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</section>
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<section>
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<section>
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<h3>Example</h3>
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<pre style="display: inline-block; width: 300px; vertical-align: middle;"><code class="sql">
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SELECT R.A, T.E
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FROM R, S, T
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WHERE R.B = S.B
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AND S.C < 5
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AND S.D = T.D
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</code></pre>
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-1.svg" style="vertical-align: middle;"/>
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-1.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-2.svg" style="vertical-align: middle;"/>
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-2.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-3.svg" style="vertical-align: middle;" />
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-3.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-4.svg" style="vertical-align: middle;" />
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-4.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-5.svg" style="vertical-align: middle;" />
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-5.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-6.svg" style="vertical-align: middle;" />
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</section>
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<section>
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<img src="graphics/2021-02-11-RA-Opt-6.svg" style="vertical-align: middle;" />
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<span style="vertical-align: middle; margin: 50px; font-size: 300%">➔</span>
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<img src="graphics/2021-02-11-RA-Opt-7.svg" style="vertical-align: middle;" />
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</section>
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</section>
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<section>
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<section>
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<h3>General Query Optimizers</h3>
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<p><b>Input:</b> Dumb translation of SQL to RA</p>
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<p>⬇︎</p>
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<p>Apply rewrites</p>
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<p>⬇︎</p>
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<p><b>Output:</b> Better, but equivalent query</p>
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</section>
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<section>
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<p>Which rewrite rules should we apply?</p>
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</section>
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<section>
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<dl style="font-size: 75%">
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<dt class="fragment highlight-grey" data-fragment-index="2">Selection Pushdown</dt>
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<dd class="fragment highlight-grey" data-fragment-index="2"><b>Always</b> commute Selections as close to the leaves as possible.</dd>
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<dt class="fragment highlight-grey" data-fragment-index="2">Join Construction</dt>
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<dd class="fragment highlight-grey" data-fragment-index="2">Joins are <b>always</b> better than cross-products. <span class="fragment" data-fragment-index="1">(if there's a good join algorithm)</span></dd>
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<dt class="fragment highlight-grey" data-fragment-index="2">(Optional) Projection Pushdown</dt>
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<dd class="fragment highlight-grey" data-fragment-index="2">Commuting Projections down to the leaves removes redundant columns, and <b>may</b> be beneficial for some systems.</dd>
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<dt>Join Algorithm Selection</dt>
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<dd>Joins can be implemented differently, depending on the join predicate.</dd>
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<dt>Join/Union Ordering</dt>
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<dd>The order in which joins are evaluated <b>may</b> affect query runtimes.</dd>
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<dt>Access Paths</dt>
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<dd>$(\sigma_c(R))$ and $(Q(\ldots) \bowtie_c R)$ are special cases that we can make fast!</dd>
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</dl>
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<p class="fragment" data-fragment-index="2" style="font-size: 60%">Some rewrites are situational... we need more information to decide when to apply them.</p>
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</section>
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<section>
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<h3>General Query Optimizers</h3>
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<ol style="font-size: 60%">
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<li class="fragment" data-fragment-index="1">Apply blind heuristics (e.g., push down selections)</li>
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<li class="fragment" data-fragment-index="2">Enumerate all possible <i>execution plans</i> by varying <span class="fragment" data-fragment-index="3" style="font-size: 80%;">(or for a reasonable subset)</span>
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<ul>
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<li>Join/Union Evaluation Order (commutativity, associativity, distributivity)</li>
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<li class="fragment" data-fragment-index="4">Algorithms for Joins, Aggregates, Sort, Distinct, and others</span></li>
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<li class="fragment" data-fragment-index="4">Data Access Paths</li>
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</ul>
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</li>
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<li class="fragment" data-fragment-index="5">Estimate the cost of each execution plan</li>
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<li class="fragment" data-fragment-index="6">Pick the execution plan with the lowest cost</li>
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</ol>
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</section>
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<section>
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<h2>In Spark...</h2>
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</section>
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<section>
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<pre><code class="scala">
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plan.transform {
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case Filter(condition, Project(columns, child)) =>
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if(/* condition is compatible with columns */){
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Project(columns, Filter(condition, child))
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} else { plan }
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}
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</code></pre>
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<p class="fragment"><tt>match</tt>/<tt>case</tt> lets you find patterns.</p>
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<p class="fragment"><tt>transform</tt> lets you apply rewrite rules.</p>
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</section>
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<section>
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<p>Should this transformation be applied<br/>top-down, or bottom-up?</p>
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</section>
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<section>
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<h3>Spark</h3>
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<pre><code class="scala">
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plan.transformDown {
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case Filter(condition, Project(columns, child)) =>
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if(/* condition is compatible with columns */){
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Project(columns, Filter(condition, child))
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} else { plan }
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}
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</code></pre>
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<p><tt>transformUp</tt>: Require bottom-up tree traversal.</p>
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||||
<p><tt>transformDown</tt>: Require top-down tree traversal.</p>
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</section>
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<section>
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<h3>Spark</h3>
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|
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<pre><code class="scala">
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plan.transform {
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case Filter(condition, Union(children)) =>
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Union(
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children.map { child =>
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Filter(condition, child)
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||||
}
|
||||
)
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||||
}
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||||
</code></pre>
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</section>
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<section>
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<h3>Spark's Optimizer</h3>
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<p class="fragment">(In vastly oversimplified form)</p>
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<ol>
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<li class="fragment">Pick a transformation rule and apply it.</li>
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<li class="fragment">Repeat until the tree stops changing.</li>
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||||
</ol>
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||||
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||||
<p></p>
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</section>
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</section>
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<section>
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<b>Next time on Bat-Database Systems:</b><br/> Checkpoint 1 overview.
|
||||
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After Width: | Height: | Size: 7.9 KiB |
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Reference in a new issue