Slides
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slides: slide/2021-02-18-QueryAlgorithms.html
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- date: "Feb. 23"
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topic: "Extended Relational Algebra"
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materials:
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slides: slide/2021-02-23-ExtRA.html
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- date: "Feb. 25"
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topic: "Physical Data Layout"
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materials:
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slides: slide/2021-02-25-PhysicalLayout.html
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- date: "Mar. 2"
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topic: "Indexes: Tree-Based, Hash"
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- date: "Mar. 4"
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src/teaching/cse-562/2021sp/slide/2021-02-23-ExtRA.erb
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src/teaching/cse-562/2021sp/slide/2021-02-23-ExtRA.erb
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---
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template: templates/cse4562_2019_slides.erb
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title: "Extended Relational Algebra"
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date: February 13, 2019
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textbook: "Ch. 5.2, 15.4"
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---
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<section>
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<section>
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<h3>Extended Relational Algebra</h3>
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<dl>
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<dt class="fragment highlight-grey" data-fragment-index="1">Set/Bag Operations</dt>
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<dd class="fragment highlight-grey" data-fragment-index="1">Select ($\sigma$), Project ($\pi$), Join ($\bowtie$), Union ($\cup$)</dd>
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<dt>Bag Operations</dt>
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<dd>Distinct ($\delta$), Outer Joins (⟗)</dd>
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<dt>List Operations</dt>
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<dd>Sort ($\tau$), Limit (L)</dd>
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<dt>Arithmetic Operations</dt>
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<dd>Extended Projection ($\pi$), Aggregation ($\Sigma$), Grouping ($\gamma$)</dd>
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</dl>
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<dl>
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</dl>
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</section>
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<section>
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<h3>Extended Projection</h3>
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<p style="margin: 50px;">Like normal projection, but can create new columns</p>
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$\pi_{M \leftarrow A+B*C,\; N \leftarrow 2}(R)$
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<p>produces 1 row for every row of R, with 2 columns: M and N</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>Outer Join</h3>
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<p class="fragment">... but first</p>
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</section>
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<section>
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<h3><code>NULL</code> Values</h3>
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<ul>
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<li>Field values can be unknown or inapplicable.<ul>
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<li>A tree with an unknown species.</li>
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<li>The street of a tree in a park.</li>
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<li>The '.' on many of your ID cards</li>
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</ul></li>
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<li>SQL provides a special <code>NULL</code> value for these cases</li>
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</ul>
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<p class="fragment">NULL makes things more complicated.</p>
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</section>
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<section>
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$$\textbf{Trees.SPC_COMMON} = \texttt{'Brooklyn'}$$
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<p style="margin: 50px;" class="fragment">
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What happens if <b>Trees.SPC_COMMON</b> is NULL?
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</p>
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<div class="fragment">
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$$\texttt{NULL} = \textbf{'Brooklyn'} \equiv \textbf{Unknown}$$
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</div>
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</section>
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<section>
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<table>
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<tr><td>Unknown</td><td>AND</td><td>Unknown</td><td>$\equiv$</td><td>Unknown</td></tr>
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<tr class="fragment" data-fragment-index="1"><td>Unknown</td><td>AND</td><td>True</td><td>$\equiv$</td><td>Unknown</td></tr>
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<tr class="fragment" data-fragment-index="2"><td>Unknown</td><td>AND</td><td>False</td><td>$\equiv$</td><td>False</td></tr>
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<tr><td colspan="5"></td></tr>
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<tr class="fragment" data-fragment-index="3"><td>Unknown</td><td>OR</td><td>Unknown</td><td>$\equiv$</td><td>Unknown</td></tr>
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<tr class="fragment" data-fragment-index="3"><td>Unknown</td><td>OR</td><td>True</td><td>$\equiv$</td><td>True</td></tr>
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<tr class="fragment" data-fragment-index="3"><td>Unknown</td><td>OR</td><td>False</td><td>$\equiv$</td><td>Unknown</td></tr>
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<tr><td colspan="5"></td></tr>
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<tr class="fragment" data-fragment-index="4"><td></td><td>NOT</td><td>Unknown</td><td>$\equiv$</td><td>Unknown</td></tr>
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</table>
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<p class="fragment"><code>WHERE</code> clauses eliminate all non-True rows</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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$$Streets \bowtie_{StreetName} Trees$$
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<p style="margin-top: 100px;" class="fragment">What happens if some streets have no trees?</p>
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</section>
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<section>
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<h3>Outer Join (⟗, ⟕, ⟖)</h3>
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<ol>
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<li>Include all results from the normal (inner) join.</li>
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<li>Also include rows that don't get joined.</li>
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</ol>
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</section>
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<section>
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<h3>Outer Join</h3>
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<dl>
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<dt>Inner Join</dt>
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<dd>Normal, plain, simple join</dd>
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<dt>Left Outer Join (⟕)</dt>
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<dd>Include un-joined rows from the left hand side</dd>
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<dt>Right Outer Join (⟖)</dt>
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<dd>Include un-joined rows from the right hand side</dd>
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<dt>[Full] Outer Join (⟗)</dt>
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<dd>Include un-joined rows from either side</dd>
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</dl>
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</section>
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<section>
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<div style="display: inline-block; margin-right: 50px;">
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<p style="text-decoration: underline;">Streets</p>
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<table>
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<tr><th>District</th><th>StreetName</th></tr>
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<tr><td>A</td><td>Apple St.</td></tr>
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<tr><td>B</td><td>Border St.</td></tr>
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<tr><td>C</td><td>Coventry St.</td></tr>
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<tr><td>D</td><td>Durham St.</td></tr>
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</table>
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</div>
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<div style="display: inline-block;">
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<p style="text-decoration: underline;">Trees</p>
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<table>
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<tr><th>TreeId</th><th>StreetName</th></tr>
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<tr><td>1</td><td>Apple St.</td></tr>
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<tr><td>2</td><td>Border St.</td></tr>
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<tr><td>3</td><td>Durham St.</td></tr>
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<tr><td>4</td><td>Elicott St.</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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<p style="text-decoration: underline;">Streets ⟗ Trees</p>
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<table>
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<tr><th>District</th><th>StreetName</th> <th>TreeId</th><th>StreetName</th></tr>
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<tr><td>A</td> <td>Apple St.</td> <td>1</td> <td>Apple St.</td></tr>
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<tr><td>B</td> <td>Border St.</td> <td>2</td> <td>Border St.</td></tr>
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<tr class="fragment highlight-red" data-fragment-index="1"><td>C</td> <td>Coventry St.</td><td>NULL</td> <td>NULL</td></tr>
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<tr><td>D</td> <td>Durham St.</td> <td>3</td> <td>Durham St.</td></tr>
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<tr class="fragment highlight-blue" data-fragment-index="2"><td>NULL</td> <td>NULL</td> <td>4</td> <td>Elicott St.</td></tr>
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</table>
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<p class="fragment" data-fragment-index="1" style="color: red;">Only LEFT outer join</p>
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<p class="fragment" data-fragment-index="2" style="color: blue;">Only RIGHT outer join</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>Sort / Limit</h3>
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<p>
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$$\tau_{A}(R)$$
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The tuples of $R$ in ascending order according to 'A'
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</p>
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<p>
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$$\textbf{L}_{n}(R)$$
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The first $n$ tuples of R
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<div style="font-size: 60%">(Typically combined with sort. If not, pick arbitrarily.)</div>
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</p>
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</section>
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<section>
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<h3>Sort</h3>
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<p style="margin-top: 100px;">
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Pick your favorite sort algorithm.
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</p>
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<p class="fragment">What happens if you don't have enough memory?</p>
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</section>
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<section>
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<p><b>Key Idea:</b> Merging 2 sorted lists requires $O(1)$ memory.</p>
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</section>
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<section>
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<h3>2-Way Sort</h3>
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<dl>
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<dt>Pass 1</dt>
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<dd>Create lots of (small) sorted lists.</dd>
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<dt>Pass 2+</dt>
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<dd>Merge sorted lists of size $N$ into sorted lists of size $2N$</dd>
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</dl>
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</section>
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<section>
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<h3>Pass 1: Create Sorted Runs</h3>
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<svg data-src="graphics/2021-02-23-Sort-Run.svg" style="margin: 100px;" />
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</section>
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<section>
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<h3>Pass 2: Merge Sorted Runs</h3>
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<svg data-src="graphics/2021-02-23-Merge-Run.svg" style="margin: 100px;" />
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</section>
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<section>
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<p>Repeat Pass 2 As Needed.</p>
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</section>
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<section>
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<p>What's the bottleneck?</p>
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<p class="fragment">IO Cost: $O(N \cdot \lceil\log_2(N)\rceil)$<br/>(with $N$ blocks)</p>
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</section>
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<section>
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<svg data-src="graphics/2021-02-23-Merge-Tree.svg" />
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</section>
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<section>
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<h3>Using More Memory</h3>
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<dl>
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<dt>Pass 1</dt>
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<dd>Sort Bigger Buffers</dd>
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<dd class="fragment">Re-Use Memory When Done</dd>
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<dt>Pass 2</dt>
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<dd>Merge $K$ Runs Simultaneously: $O(N \cdot \lceil\log_K(N)\rceil)$ IO</dd>
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</dl>
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-1.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-2.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-3.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-4.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-5.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-6.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-7.svg" />
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</section>
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<section>
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<h3>Replacement Sort</h3>
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<img src="graphics/2021-02-23-ReplSort-8.svg" />
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</section>
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<section>
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<p>On average, we'll get runs of size $2 \cdot |WS|$</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>Aggregation</h3>
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<dl>
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<div class="fragment">
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<dt>Normal Aggregates</dt>
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<dd><code class="sql">SELECT COUNT(*) FROM R</code></dd>
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<dd><code class="sql">SELECT SUM(A) FROM R</code></dd>
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</div>
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<div class="fragment">
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<dt>Group-By Aggregates</dt>
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<dd><code class="sql">SELECT A, SUM(B) FROM R GROUP BY A</code></dd>
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</div>
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<div class="fragment">
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<dt>Distinct</dt>
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<dd><code class="sql">SELECT DISTINCT A FROM R</code></dd>
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<dd class="fragment"><code class="sql">SELECT A FROM R GROUP BY A</code></dd>
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</div>
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</dl>
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</section>
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<section>
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<h3>Normal Aggregates</h3>
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<table style="font-size: 80%;" class="fragment">
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<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>TREE_DBH</th></tr>
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<tr><td>180683</td><td>'red maple'</td><td>'Queens'</td><td>3</td></tr>
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<tr><td>315986</td><td>'pin oak'</td><td>'Queens'</td><td>21</td></tr>
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<tr><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>3</td></tr>
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<tr><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>10</td></tr>
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<tr><td>189565</td><td>'American linden'</td><td>'Brooklyn'</td><td>21</td></tr>
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<tr><td style="font-weight: bold;" colspan="3">... and 683783 more</td></tr>
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</table>
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<div style="margin-top: 60px">
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<code class="sql fragment">SELECT COUNT(*) FROM TREES</code>
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</div>
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</section>
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<section>
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<table style="font-size: 70%;">
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<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>TREE_DBH</th></tr>
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<tr class="fragment" data-fragment-index="1"><td colspan="4">COUNT = 0</td></tr>
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<tr class="fragment" data-fragment-index="2" style="font-size: smaller;"><td>180683</td><td>'red maple'</td><td>'Queens'</td><td>3</td></tr>
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<tr class="fragment" data-fragment-index="3"><td colspan="4">COUNT = 1</td></tr>
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<tr class="fragment" data-fragment-index="4" style="font-size: smaller;"><td>315986</td><td>'pin oak'</td><td>'Queens'</td><td>21</td></tr>
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<tr class="fragment" data-fragment-index="5"><td colspan="4">COUNT = 2</td></tr>
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<tr class="fragment" data-fragment-index="6" style="font-size: smaller;"><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>3</td></tr>
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<tr class="fragment" data-fragment-index="6"><td colspan="4">COUNT = 3</td></tr>
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<tr class="fragment" data-fragment-index="7" style="font-size: smaller;"><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>10</td></tr>
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<tr class="fragment" data-fragment-index="7"><td colspan="4">COUNT = 4</td></tr>
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<tr class="fragment" data-fragment-index="8" style="font-size: smaller;"><td>189565</td><td>'American linden'</td><td>'Brooklyn'</td><td>21</td></tr>
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<tr class="fragment" data-fragment-index="8"><td colspan="4">COUNT = 5</td></tr>
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<tr class="fragment" data-fragment-index="9"><td style="font-weight: bold;" colspan="3">... and 683783 more</td></tr>
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<tr class="fragment" data-fragment-index="9"><td colspan="4">COUNT = 683788</td></tr>
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</table>
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</section>
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<section>
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<div style="margin-bottom: 60px">
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<code class="sql">SELECT SUM(TREE_DBH) FROM TREES</code>
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</div>
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<table style="font-size: 70%;">
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<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>TREE_DBH</th></tr>
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<tr class="fragment" data-fragment-index="1"><td colspan="4">SUM = 0</td></tr>
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<tr class="fragment" data-fragment-index="2" style="font-size: smaller;"><td>180683</td><td>'red maple'</td><td>'Queens'</td><td>3</td></tr>
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<tr class="fragment" data-fragment-index="3"><td colspan="4">SUM = 3</td></tr>
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<tr class="fragment" data-fragment-index="4" style="font-size: smaller;"><td>315986</td><td>'pin oak'</td><td>'Queens'</td><td>21</td></tr>
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<tr class="fragment" data-fragment-index="5"><td colspan="4">SUM = 24</td></tr>
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<tr class="fragment" data-fragment-index="6" style="font-size: smaller;"><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>3</td></tr>
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<tr class="fragment" data-fragment-index="6"><td colspan="4">SUM = 27</td></tr>
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||||
<tr class="fragment" data-fragment-index="7" style="font-size: smaller;"><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>10</td></tr>
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||||
<tr class="fragment" data-fragment-index="7"><td colspan="4">SUM = 37</td></tr>
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||||
<tr class="fragment" data-fragment-index="8" style="font-size: smaller;"><td>189565</td><td>'American linden'</td><td>'Brooklyn'</td><td>21</td></tr>
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||||
<tr class="fragment" data-fragment-index="8"><td colspan="4">SUM = 58</td></tr>
|
||||
<tr class="fragment" data-fragment-index="9"><td style="font-weight: bold;" colspan="3">... and 683783 more</td></tr>
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</table>
|
||||
</section>
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||||
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||||
<section>
|
||||
<h3>Basic Aggregate Pattern</h3>
|
||||
<p class="fragment" style="font-size: 70%">This is also sometimes called a "fold"</p>
|
||||
<dl>
|
||||
<dt>Init</dt>
|
||||
<dd>Define a starting value for the accumulator</dd>
|
||||
<dt>Fold(Accum, New)</dt>
|
||||
<dd>Merge a new value into the accumulator</dd>
|
||||
</dl>
|
||||
</section>
|
||||
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||||
<section>
|
||||
<h3>COUNT(*)</h3>
|
||||
<dl>
|
||||
<dt>Init</dt>
|
||||
<dd class="fragment">$0$</dd>
|
||||
<dt>Fold(Accum, New)</dt>
|
||||
<dd class="fragment">$Accum + 1$</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>SUM(A)</h3>
|
||||
<dl>
|
||||
<dt>Init</dt>
|
||||
<dd class="fragment">$0$</dd>
|
||||
<dt>Fold(Accum, New)</dt>
|
||||
<dd class="fragment">$Accum + New$</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>AVG(A)</h3>
|
||||
<dl>
|
||||
<dt>Init</dt>
|
||||
<dd class="fragment">$\{ sum = 0, count = 0 \}$</dd>
|
||||
<dt>Fold(Accum, New)</dt>
|
||||
<dd class="fragment">$\{ sum = Accum.sum + New, \\\;count = Accum.count + 1\}$</dd>
|
||||
<dt class="fragment">Finalize(Accum)</dt>
|
||||
<dd class="fragment">$\frac{Accum.sum}{Accum.count}$</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Basic Aggregate Pattern</h3>
|
||||
<dl>
|
||||
<dt>Init</dt>
|
||||
<dd>Define a starting value for the accumulator</dd>
|
||||
<dt>Fold(Accum, New)</dt>
|
||||
<dd>Merge a new value into the accumulator</dd>
|
||||
<dt>Finalize(Accum)</dt>
|
||||
<dd>Extract the aggregate from the accumulator.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Basic Aggregate Types</h3>
|
||||
<p class="fragment" style="font-size: 60%">Grey et. al. "Data Cube: A Relational Aggregation Operator Generalizing Group-By, Cross-Tab, and Sub-Totals</p>
|
||||
|
||||
<dl>
|
||||
<dt>Distributive</dt>
|
||||
<dd>Finite-sized accumulator and doesn't need a finalize (COUNT, SUM)</dd>
|
||||
<dt>Algebraic</dt>
|
||||
<dd>Finite-sized accumulator but needs a finalize (AVG)</dd>
|
||||
<dt>Holistic</dt>
|
||||
<dd>Unbounded accumulator (MEDIAN)</dd>
|
||||
</dl>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<section>
|
||||
<h3>Group-By Aggregates</h3>
|
||||
<div style="margin-top: 60px">
|
||||
<code class="sql">SELECT SPC_COMMON, COUNT(*) FROM TREES GROUP BY SPC_COMMON</code>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Naive Idea:</b> Keep a separate accumulator for each group</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<table style="font-size: 70%;">
|
||||
<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>TREE_DBH</th></tr>
|
||||
<tr class="fragment" data-fragment-index="1"><td colspan="4">{}</td></tr>
|
||||
<tr class="fragment" data-fragment-index="2" style="font-size: smaller;"><td>180683</td><td>'red maple'</td><td>'Queens'</td><td>3</td></tr>
|
||||
<tr class="fragment" data-fragment-index="3"><td colspan="4">{ 'red maple' = 1 }</td></tr>
|
||||
<tr class="fragment" data-fragment-index="4" style="font-size: smaller;"><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>10</td></tr>
|
||||
<tr class="fragment" data-fragment-index="5"><td colspan="4">{ 'red maple' = 1, 'honeylocust' = 1 }</td></tr>
|
||||
<tr class="fragment" data-fragment-index="6" style="font-size: smaller;"><td>315986</td><td>'pin oak'</td><td>'Queens'</td><td>21</td></tr>
|
||||
<tr class="fragment" data-fragment-index="7"><td colspan="4">{ 'red maple' = 1, 'honeylocust' = 1, 'pin oak' = 1 }</td></tr>
|
||||
<tr class="fragment" data-fragment-index="8" style="font-size: smaller;"><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>3</td></tr>
|
||||
<tr class="fragment" data-fragment-index="9"><td colspan="4">{ 'red maple' = 1, 'honeylocust' = 2, 'pin oak' = 1 }</td></tr>
|
||||
</table>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>What could go wrong?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Alternative Grouping Algorithms</h3>
|
||||
|
||||
<dl>
|
||||
<dt>2-pass Hash Aggregate</dt>
|
||||
<dd>Like 2-pass Hash Join: Distribute groups across buckets, then do an in-memory aggregate for each bucket.</dd>
|
||||
|
||||
<dt>Sort-Aggregate</dt>
|
||||
<dd>Like Sort-Merge Join: Sort data by groups, then group elements will be adjacent.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<svg data-src="graphics/2021-02-23-2PassHashAgg.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<table style="font-size: 70%;">
|
||||
<tr><th>TREE_ID</th><th>SPC_COMMON</th><th>BORONAME</th><th>TREE_DBH</th></tr>
|
||||
<tr class="fragment"><td colspan="4">{}</td></tr>
|
||||
<tr class="fragment" style="font-size: smaller;"><td>204337</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>10</td></tr>
|
||||
<tr class="fragment"><td colspan="4">{ 'honeylocust' = 1 }</td></tr>
|
||||
<tr class="fragment" style="font-size: smaller;"><td>204026</td><td>'honeylocust'</td><td>'Brooklyn'</td><td>3</td></tr>
|
||||
<tr class="fragment"><td colspan="4">{ 'honeylocust' = 2 }</td></tr>
|
||||
<tr class="fragment"><td colspan="4" style="font-weight: bold">... and more</td></tr>
|
||||
<tr class="fragment" style="font-size: smaller;"><td>315986</td><td>'pin oak'</td><td>'Queens'</td><td>21</td></tr>
|
||||
<tr class="fragment"><td colspan="4">{ <span class="fragment highlight-grey">'honeylocust' = 3206,</span> 'pin oak' = 1 }</td></tr>
|
||||
<tr class="fragment"><td colspan="4" style="font-weight: bold">... and more</td></tr>
|
||||
<tr class="fragment" style="font-size: smaller;"><td>180683</td><td>'red maple'</td><td>'Queens'</td><td>3</td></tr>
|
||||
<tr class="fragment"><td colspan="4">{ <span class="fragment highlight-grey">'pin oak' = 53814,</span> 'red maple' = 1 }</td></tr>
|
||||
</table>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<dl>
|
||||
<dt>Set/Bag Operations</dt>
|
||||
<dd>Select ($\sigma$), Project ($\pi$), Join ($\bowtie$), Union ($\cup$)</dd>
|
||||
|
||||
<dt>Bag Operations</dt>
|
||||
<dd>Distinct ($\delta$), Outer Joins (⟗)</dd>
|
||||
|
||||
<dt>List Operations</dt>
|
||||
<dd>Sort ($\tau$), Limit (L)</dd>
|
||||
|
||||
<dt>Arithmetic Operations</dt>
|
||||
<dd>Extended Projection ($\pi$), Aggregation ($\sigma$), Grouping ($\gamma$)</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Next Time...</h3>
|
||||
<p>Physical Layouts</p>
|
||||
</section>
|
||||
|
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Loading…
Reference in a new issue