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<title>CSE 4/562 - Spring 2018</title>
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<meta name="author" content="Oliver Kennedy">
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CSE 4/562 - Database Systems
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</div>
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<div class="slides">
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<section>
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<h1>Relational Algebra</h1>
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<h3>CSE 4/562 – Database Systems</h3>
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<h5>February 5, 2018</h5>
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</section>
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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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Replace <b>[Thing A]</b> with <u>better</u>, but <u>equivalent</u> <b>[Thing B]</b>.
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<ol>
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<li class="fragment">How can we tell if <b>[Thing B]</b> is equivalent to <b>[Thing A]</b>?</li>
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<li class="fragment">How can we tell if <b>[Thing B]</b> is better than <b>[Thing A]</b>?</li>
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</ol>
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</p>
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</section>
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<section>
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<p>
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Replace <b>[Query A]</b> with <u>better</u>, but <u>equivalent</u> <b>[Query B]</b>.
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<ol>
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<li>How can we tell if <b>[Query B]</b> is equivalent to <b>[Query A]</b>?</li>
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<li>How can we tell if <b>[Query B]</b> is better than <b>[Query A]</b>?</li>
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</ol>
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</p>
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<p class="fragment">... but first a few definitions.</p>
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</section>
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<section>
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<h3>Relational Data</h3>
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<dl>
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<dt>Relation (Table)</dt>
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<dd>A collection of Tuples of Values</dd>
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<dd>All tuples have the same set of attributes, or schema</dd>
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</dl>
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</section>
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<section>
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<h3>Relational Data</h3>
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Types of relations...
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<table><tr>
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<td class="fragment">
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<h4>Set</h4>
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<table style="font-size: small;">
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<tr><th>SPC_COMMON</th><th>BORONAME</th></tr>
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<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
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<tr><td>'American linden'</td><td>'Brooklyn'</td></tr>
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<tr><td>'London planetree'</td><td>'Brooklyn'</td></tr>
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<tr><td> </td><td> </td></tr>
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<tr><td> </td><td> </td></tr>
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</table>
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<p style="font-size: 60%;">(Unique Only)</p>
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</td>
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<td>
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<h4>Bag</h4>
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<table style="font-size: small;">
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<tr><th>SPC_COMMON</th><th>BORONAME</th></tr>
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<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
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<tr><td>'London planetree'</td><td>'Brooklyn'</td></tr>
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<tr><td>'American linden'</td><td>'Brooklyn'</td></tr>
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<tr><td>'London planetree'</td><td>'Brooklyn'</td></tr>
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<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
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</table>
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</td>
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<td class="fragment">
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<h4>List</h4>
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<table style="font-size: small;">
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<tr><th>SPC_COMMON</th><th>BORONAME</th></tr>
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<tr><td>'American linden'</td><td>'Brooklyn'</td></tr>
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<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
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<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
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<tr><td>'London planetree'</td><td>'Brooklyn'</td></tr>
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<tr><td>'London planetree'</td><td>'Brooklyn'</td></tr>
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</table>
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<p style="font-size: 60%;">(Order Matters)</p>
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</td>
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</tr></table>
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</table>
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</section>
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<section>
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<h3>Declarative Languages</h3>
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<table>
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<tr><th>Declarative</th><th>Imperative</th></tr>
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<tr>
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<td>Say <b>what</b> you want</td>
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<td>Say <b>how</b> you want to get it</td>
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</tr>
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<tr>
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<td style="vertical-align: middle;">"Get me the TPS Reports"</td>
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<td>
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<pre><code>Look at every T report
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For each week:
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Sum up the sprocket count
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Find that week's S report
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... </code></pre>
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</td>
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</tr>
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<tr>
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<td>SQL, RA, R, …</td>
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<td>C, Scala, Java, Python, …</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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<p style="width: 600px; margin-left: auto; margin-right: auto;">Declarative languages make it easier to explore <u>equivalent</u> computations to find the <u>best</u> one.</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>Declarative Queries</h3>
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<dl>
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<dt>SQL</dt>
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<dd>Readability: Focus on what and where from</dd>
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<dt>Relational Algebra</dt>
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<dd>Data Flow: Focus on transformations</dd>
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<dt>Relational Calculus</dt>
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<dd>Minimality: Focus on attribute relationships</dd>
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</dl>
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</section>
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<section>
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<h3>SQL</h3>
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<pre><code class="sql">
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SELECT R.A FROM R, S
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WHERE R.B = S.B AND S.C = 10
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</code></pre>
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</section>
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<section>
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<h3>Relational Algebra</h3>
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<img src="graphics/2018-02-05-RA-Tree.svg" />
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$$\pi_{R.A}(\sigma_{S.C=10}(\bowtie_{R.B = S.B}(\textbf{R}, \textbf{S})))$$
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</section>
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<section>
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<h3>Relational Calculus</h3>
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<p style="margin-top: 100px">
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$$\exists B : R(A, B), S(B, 10)$$
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</p>
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</section>
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<section>
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<h3>Declarative Queries</h3>
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<dl>
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<dt class="fragment highlight-grey" data-fragment-index="1">SQL</dt>
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<dd class="fragment highlight-grey" data-fragment-index="1">Readability: Focus on what and where from</dd>
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<dt>Relational Algebra</dt>
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<dd>Data Flow: Focus on transformations</dd>
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<dt class="fragment highlight-grey" data-fragment-index="1">Relational Calculus</dt>
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<dd class="fragment highlight-grey" data-fragment-index="1">Minimality: Focus on attribute relationships</dd>
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</dl>
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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 style="margin-bottom: 50px;">Preliminaries</h3>
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<p>We start with a database <b>instance</b> with a fixed <b>schema</b></p>
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<p class="fragment" style="margin-top: 40px;">
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Queries are applied to Relations
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$$Q(\textbf{Trees}, \textbf{SpeciesInfo})$$
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</p>
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<p class="fragment" style="margin-top: 40px;">
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Queries are also Relations!
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$$Q_2(\textbf{SpeciesInfo}, Q_1(\textbf{Trees}))$$
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<span class="fragment" style="font-size: 50%; display: block; margin-top: 0px; margin-bottom: 10px;">(Relational Algebra is <u>Closed</u>)</span>
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</p>
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</section>
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<section>
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<h3>Relational Algebra</h3>
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<table>
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<tr><th>Operation</th><th>Sym</th><th>Meaning</th></tr>
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<tr><td>Selection</td><td>$\sigma$</td><td>Select a subset of the input rows</td></tr>
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<tr><td>Projection</td><td>$\pi$</td><td>Delete unwanted columns</td></tr>
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<tr><td>Cross-product</td><td>$\times$</td><td>Combine two relations</td></tr>
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<tr><td>Set-difference</td><td>$-$</td><td>Tuples in Rel 1, but not Rel 2</td></tr>
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<tr><td>Union</td><td>$\cup$</td><td>Tuples either in Rel 1 or in Rel 2</td></tr>
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</table>
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<p class="fragment">
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<b>Also:</b> Intersection, <u>Join</u>, Division, Renaming
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</p>
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<p class="fragment" style="font-size: 50%">(Not essential, but can be useful)</p>
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</section>
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<section>
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<table>
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<tr>
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<th>Input</th>
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<th></th>
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<th>Query Language</th>
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<th></th>
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<th>Output</th>
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</tr>
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<tr class="fragment" data-fragment-index="1">
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<td>Sets of Tuples</td>
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<td>$\rightarrow$</td>
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<td><span class="fragment highlight-grey" data-fragment-index="2">Set</span> Relational Algebra</td>
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<td>$\rightarrow$</td>
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<td>Set of Tuples</td>
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</tr>
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<tr class="fragment" data-fragment-index="3">
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<td>Bags of Tuples</td>
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<td>$\rightarrow$</td>
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<td>Bag Relational Algebra</td>
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<td>$\rightarrow$</td>
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<td>Bag of Tuples</td>
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</tr>
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<tr class="fragment" data-fragment-index="4">
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<td class="fragment highlight-grey" data-fragment-index="5">Lists of Tuples</td>
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<td class="fragment highlight-grey" data-fragment-index="5">$\rightarrow$</td>
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<td class="fragment highlight-grey" data-fragment-index="5">Extended Relational Algebra</td>
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<td class="fragment highlight-grey" data-fragment-index="5">$\rightarrow$</td>
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<td class="fragment highlight-grey" data-fragment-index="5">List of Tuples</td>
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</tr>
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</table>
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<p class="fragment" data-fragment-index="5">First we focus on sets and bags.</p>
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</section>
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<section>
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<h3>Selection ($\sigma_{c}$)</h3>
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<p>Delete rows that fail the condition $c$.</p>
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<div class="fragment">
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$$\sigma_{(BORONAME = \texttt{'Brooklyn'})} \textbf{Trees}$$
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<table style="font-size: small; margin-top: 30px;">
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<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>...</th></tr>
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<tr><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>...</td></tr>
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<tr><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>...</td></tr>
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<tr><td>189565</td><td>'American linden'</td><td>'Brooklyn'</td><td>...</td></tr>
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<tr><td>192755</td><td>'London planetree'</td><td>'Brooklyn'</td><td>...</td></tr>
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<tr><td>189465</td><td>'London planetree'</td><td>'Brooklyn'</td><td>...</td></tr>
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<tr><td style="font-weight: bold;" colspan="4">... and 177287 more</td></tr>
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</table>
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</div>
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</section>
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<section>
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<h3>Projection ($\pi_{A}$)</h3>
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<p>Delete attributes not in the projection list $A$.</p>
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<div class="fragment">
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$$\pi_{BORONAME}(Trees)$$
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<table style="font-size: small; margin-top: 30px;">
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<tr><th>BORONAME</th></tr>
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<tr><td>Queens</td></tr>
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<tr><td>Brooklyn</td></tr>
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<tr><td>Manhattan</td></tr>
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<tr><td>Bronx</td></tr>
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<tr><td>Staten Island</td></tr>
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</table>
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</div>
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<p class="fragment">Only 5 results... not 683788?</p>
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<p class="fragment">Set and Bag Projection are different</p>
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</section>
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<section>
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<h3>Reminder: Queries are Relations</h3>
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<p>What are these queries schemas?</p>
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<div class="fragment" style="margin-top: 50px;">$$\pi_{TREEID, SPC\_COMMON, BORONAME} \textbf{Trees}$$</div>
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<div class="fragment" style="margin-top: 50px;">$$\sigma_{(BORONAME = \texttt{'Brooklyn'})} \textbf{Trees}$$</div>
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<div class="fragment" style="margin-top: 50px;">$$\sigma_{(BORONAME = \texttt{'Brooklyn'})}(\pi_{TREEID, SPC\_COMMON, BORONAME} \textbf{Trees})$$</div>
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</section>
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<section>
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<h3>Union ($\cup$)</h3>
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<p style="margin-bottom: 0px;">Takes two relations that are <u>union-compatible</u>...</p>
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<p style="font-size: 60%; margin-top: 0px;">(Both relations have the same number of fields with the same types)</p>
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<p>... and returns all tuples appearing in <u>either</u> relation</p>
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<div class="fragment" style="font-size: 70%;">
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$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} \textbf{Trees}) \cup (\sigma_{(BORONAME=\texttt{'Manhattan'})} \textbf{Trees})$$
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</div>
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<p class="fragment">We use $\uplus$ if we explicitly mean <i>bag</i> union</p>
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</section>
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<section>
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<h3>Intersection ($\cap$)</h3>
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<p>Return all tuples appearing in <u>both</u> <br/>of two union-compatible relations</p>
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<div class="fragment" style="font-size: 70%;">
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$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} (\pi_{SPC\_COMMON} \textbf{Trees})) \\ ~~~~~~~~~\cap (\sigma_{(BORONAME=\texttt{'Manhattan'})} (\pi_{SPC\_COMMON} \textbf{Trees}))$$
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<p>What is this query asking?</p>
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</div>
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</section>
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|
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<section>
|
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<h3>Set Difference</h3>
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<p>Return all tuples appearing in the first, but not the second<br/>of two union-compatible relations</p>
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<div class="fragment" style="font-size: 70%;">
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$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} (\pi_{SPC\_COMMON} \textbf{Trees})) \\ ~~~~~~~~~- (\sigma_{(BORONAME=\texttt{'Manhattan'})} (\pi_{SPC\_COMMON} \textbf{Trees}))$$
|
||
<p>What is this query asking?</p>
|
||
</div>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Union, Intersection, Set Difference</h3>
|
||
|
||
<p style="margin-top: 100px;">What is the schema of the result of any of these operators?</p>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Cross (Cartesian) Product ($\times$)</h3>
|
||
|
||
<p>Create all pairs of tuples.</p>
|
||
|
||
<div class="fragment" data-fragment-index="1">
|
||
<div style="font-size: 70%">
|
||
$$\pi_{SPC\_COMMON, BORONAME} (\textbf{Trees}) \times \pi_{SPC\_COMMON, AVG\_HEIGHT} (\textbf{TreeInfo})$$
|
||
</div>
|
||
|
||
<table style="font-size: small; margin-top: 30px; display: inline-block; vertical-align: middle; margin-right: 50px;">
|
||
<tr><th>SPC_COMMON</th><th>AVG_HEIGHT</th></tr>
|
||
<tr class="fragment highlight-current-blue" data-fragment-index="2"><td>cedar elm</td><td>60</td></tr>
|
||
<tr class="fragment highlight-current-blue" data-fragment-index="3"><td>lacebark elm</td><td>45</td></tr>
|
||
<tr><td colspan="2" style="font-weight: bold;">... and more</td></tr>
|
||
</table>
|
||
|
||
<table style="font-size: small; margin-top: 30px; display: inline-block; vertical-align: middle;">
|
||
<tr><th>SPC_COMMON</th><th>BORONAME</th><th>SPC_COMMON</th><th>AVG_HEIGHT</th></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="3"><td>honeylocust</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="2">cedar elm</td><td>60</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="3"><td>honeylocust</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="2">cedar elm</td><td>60</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="3"><td>American linden</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="2">cedar elm</td><td>60</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="3"><td>London planetree</td><td>Manhattan</td><td class="fragment highlight-current-blue" data-fragment-index="2">cedar elm</td><td>60</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="3"><td>London planetree</td><td>Manhattan</td><td class="fragment highlight-current-blue" data-fragment-index="2">cedar elm</td><td>60</td></tr>
|
||
<tr><td style="font-weight: bold;" colspan="4">...</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="2"><td>honeylocust</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="3">lacebark elm</td><td>45</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="2"><td>honeylocust</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="3">lacebark elm</td><td>45</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="2"><td>American linden</td><td>Brooklyn</td><td class="fragment highlight-current-blue" data-fragment-index="3">lacebark elm</td><td>45</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="2"><td>London planetree</td><td>Manhattan</td><td class="fragment highlight-current-blue" data-fragment-index="3">lacebark elm</td><td>45</td></tr>
|
||
<tr class="fragment highlight-current-grey" data-fragment-index="2"><td>London planetree</td><td>Manhattan</td><td class="fragment highlight-current-blue" data-fragment-index="3">lacebark elm</td><td>45</td></tr>
|
||
<tr><td style="font-weight: bold;" colspan="4">... and more</td></tr>
|
||
</table>
|
||
</div>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Cross (Cartesian) Product ($\times$)</h3>
|
||
<div style="font-size: 70%; margin-top: 50px;">
|
||
$$\pi_{SPC\_COMMON,\ BORONAME} (\textbf{Trees}) \times \pi_{SPC\_COMMON,\ AVG\_HEIGHT} (\textbf{TreeInfo})$$
|
||
</div>
|
||
<p style="margin-top: 50px;">What is the schema of the resulting relation?</p>
|
||
<p class="fragment">The relation has a naming conflict<br/>(two attributes with the same name)</p>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Renaming ($\rho$)</h3>
|
||
<div style="font-size: 50%; margin-top: 50px;">
|
||
$$\rho_{TNAME,\ BORO,\ INAME,\ HEIGHT}\left( \pi_{SPC\_COMMON,\ BORONAME} (\textbf{Trees}) \times \pi_{SPC\_COMMON,\ AVG\_HEIGHT} (\textbf{TreeInfo})\right)$$
|
||
</div>
|
||
<p style="margin-top: 50px;">What is the schema of the resulting relation?</p>
|
||
<p style="margin-top: 50px;" class="fragment">When writing cross-products on the board,<br/>I will use implicit renaming</p>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Join ($\bowtie_c$)</h3>
|
||
<p>Pair tuples according to a condition c.</p>
|
||
<div style="font-size: 50%; margin-top: 50px;">
|
||
$$\pi_{SPC\_COMMON,\ BORONAME} (\textbf{Trees}) \bowtie_{T.SPC\_COMMON = TI.SPC\_COMMON} \pi_{SPC\_COMMON,\ AVG\_HEIGHT} (\textbf{TreeInfo})$$
|
||
</div>
|
||
|
||
<div class="fragment">
|
||
<div style="font-size: 50%; margin-top: 50px;">
|
||
Identical to...
|
||
$$\sigma_{T.SPC\_COMMON = TI.SPC\_COMMON}\left(\pi_{SPC\_COMMON,\ BORONAME} (\textbf{Trees}) \times \pi_{SPC\_COMMON,\ AVG\_HEIGHT} (\textbf{TreeInfo})\right)$$
|
||
</div>
|
||
</div>
|
||
|
||
<div class="fragment" style="margin-top: 100px">
|
||
$$R \bowtie_c S \equiv \sigma_c(R \times S)$$
|
||
</div>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Join Shorthands</h3>
|
||
<p><b>Equi-joins</b> are joins with only equality tests in the condition.</p>
|
||
<dl>
|
||
<dt class="fragment" data-fragment-index="1">Join on attribute(s)</dt>
|
||
<dd class="fragment" data-fragment-index="1">$R \bowtie_{A} S \equiv R \bowtie_{R.A = S.A} S$</dd>
|
||
<dd class="fragment" data-fragment-index="1">Same values on the listed attributes</dd>
|
||
<dt class="fragment" data-fragment-index="2">Natural Join</dt>
|
||
<dd class="fragment" data-fragment-index="2">$R \bowtie S \equiv R \bowtie_{attrs(R) \cap attrs(S)} S$</dd>
|
||
<dd class="fragment" data-fragment-index="2">Same values on all shared attributes</dd>
|
||
</dl>
|
||
</section>
|
||
|
||
<section>
|
||
<h4>Which operators can create duplicates?</h4>
|
||
<p style="font-size: 60%">(Which operators behave differently in Set- and Bag-RA?)</p>
|
||
|
||
<table>
|
||
<tr ><th>Operator </th><th>Symbol </th><th >Duplicates?</th></tr>
|
||
<tr ><td>Selection </td><td>$\sigma$ </td><td class="fragment" style="color: darkred;">No</td></tr>
|
||
<tr class="fragment"><td>Projection </td><td>$\pi$ </td><td class="fragment" style="color: darkgreen;">Yes</td></tr>
|
||
<tr class="fragment"><td>Cross-product </td><td>$\times$ </td><td class="fragment" style="color: darkred;">No</td></tr>
|
||
<tr class="fragment"><td>Set-difference</td><td>$-$ </td><td class="fragment" style="color: darkred;">No</td></tr>
|
||
<tr class="fragment"><td>Union </td><td>$\cup$ </td><td class="fragment" style="color: darkgreen;">Yes</td></tr>
|
||
<tr class="fragment"><td>Join </td><td>$\bowtie$</td><td class="fragment" style="color: darkred;">No</td></tr>
|
||
</table>
|
||
</section>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Group Work</h3>
|
||
<p>Find the <b>BORONAME</b>s of all boroughs that <b>do</b> have trees with an average height of below 45 inches</p>
|
||
|
||
<table style="font-size: small; margin-top: 30px; display: inline-block; vertical-align: middle; margin-right: 50px;">
|
||
<tr><th>SPC_COMMON</th><th>AVG_HEIGHT</th></tr>
|
||
<tr><td>cedar elm</td><td>60</td></tr>
|
||
<tr><td>lacebark elm</td><td>45</td></tr>
|
||
<tr><td colspan="2" style="font-weight: bold;">... and more</td></tr>
|
||
</table>
|
||
|
||
<table style="font-size: small; margin-top: 30px; display: inline-block; vertical-align: middle;">
|
||
<tr><th>SPC_COMMON</th><th>BORONAME</th></tr>
|
||
<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
|
||
<tr><td>'honeylocust'</td><td>'Brooklyn'</td></tr>
|
||
<tr><td>'American linden'</td><td>'Brooklyn'</td></tr>
|
||
<tr><td>'London planetree'</td><td>'Manhattan'</td></tr>
|
||
<tr><td>'London planetree'</td><td>'Manhattan'</td></tr>
|
||
<tr><td style="font-weight: bold;" colspan="2">... and more</td></tr>
|
||
</table>
|
||
|
||
<div class="fragment">
|
||
$$\pi_{BORONAME}(\sigma_{AVG\_HEIGHT < 45}(\textbf{Trees}\bowtie\textbf{TreeInfo}))$$
|
||
</div>
|
||
<div class="fragment">
|
||
$$\pi_{BORONAME}(\textbf{Trees}\bowtie\sigma_{AVG\_HEIGHT < 45}(\textbf{TreeInfo}))$$
|
||
</div>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Division ($/$)</h3>
|
||
|
||
<p>Not typically supported as a primitive operator,<br/>but useful for expressing queries like:</p>
|
||
<p style="font-size: 70%; font-weight: bold">Find species that appear in all boroughs</p>
|
||
<div style="font-size: 70%">
|
||
$$\pi_{BORONAME,\ SPC\_COMMON}(\textbf{Trees}) \;\;/\;\;\pi_{SPC\_COMMON}(\textbf{Trees})$$
|
||
(using set relational algebra)
|
||
</div>
|
||
<p>
|
||
$$R / S \equiv \{\; \left<\vec t\right> \;|\; \forall \left<\vec s\right> \in S, \left< \vec t \vec s \right> \in R \;\}$$
|
||
</p>
|
||
</section>
|
||
|
||
<section>
|
||
<table style="font-size: small; margin-top: 30px; display: inline-block; vertical-align: middle;">
|
||
<tr><th>BORO</th> <th>SPC_COMMON</th></tr>
|
||
<tr><td>Brooklyn</td> <td>honeylocust</td></tr>
|
||
<tr><td>Brooklyn</td> <td>American linden</td></tr>
|
||
<tr><td>Brooklyn</td> <td>London planetree</td></tr>
|
||
<tr><td>Manhattan</td> <td>honeylocust</td></tr>
|
||
<tr><td>Manhattan</td> <td>American linden</td></tr>
|
||
<tr><td>Manhattan</td> <td>pin oak</td></tr>
|
||
<tr><td>Queens</td> <td>honeylocust</td></tr>
|
||
<tr><td>Queens</td> <td>American linden</td></tr>
|
||
<tr><td>Bronx</td> <td>honeylocust</td></tr>
|
||
</table>
|
||
|
||
<table style="font-size: 40%; margin-left: 30px; display: inline-block; vertical-align: middle;">
|
||
<tr class="fragment"><td style="text-align: left; padding-bottom: 30px;">/ { honeylocust }</td> <td style="text-align: left;">= Brooklyn, Manhattan, Queens, Bronx</td></tr>
|
||
<tr class="fragment"><td style="text-align: left; padding-bottom: 30px;">/ { honeylocust, American linden }</td> <td style="text-align: left;">= Brooklyn, Manhattan, Queens</td></tr>
|
||
<tr class="fragment"><td style="text-align: left; padding-bottom: 30px;">/ { honeylocust, American linden, pin oak }</td><td style="text-align: left;">= Manhattan</td></tr>
|
||
</table>
|
||
</section>
|
||
|
||
<section>
|
||
<h3>Group Work</h3>
|
||
<p>If time permits: Implement division using other operators.</p>
|
||
</section>
|
||
</section>
|
||
|
||
<section>
|
||
<section>
|
||
<h3>Relational Algebra</h3>
|
||
<p>
|
||
A simple way to think about and work with<br/>
|
||
computations over collections.
|
||
</p>
|
||
<p>… simple → easy to evaluate</p>
|
||
<p>… simple → easy to optimize</p>
|
||
<p style="margin-top: 100px">
|
||
Next time, Optimizing RA
|
||
</p>
|
||
</section>
|
||
|
||
</div></div>
|
||
|
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<script src="../reveal.js-3.6.0/js/reveal.js"></script>
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<script>
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// https://github.com/hakimel/../reveal.js#configuration
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transition: 'fade', // none/fade/slide/convex/concave/zoom
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|