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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
< / div >
< div class = "slides" >
< section >
< h1 > Relational Algebra< / h1 >
< h3 > CSE 4/562 – < / h3 >
< h5 > February 5, 2018< / h5 >
< / section >
< section >
< section >
< h3 > The running theme...< / h3 >
< p >
Replace < b > [Thing A]< / b > with < u > better< / u > , but < u > equivalent< / u > < b > [Thing B]< / b > .
< ol >
< li class = "fragment" > How can we tell if < b > [Thing B]< / b > is equivalent to < b > [Thing A]< / b > ?< / li >
< li class = "fragment" > How can we tell if < b > [Thing B]< / b > is better than < b > [Thing A]< / b > ?< / li >
< / ol >
< / p >
< / section >
< section >
< p >
Replace < b > [Query A]< / b > with < u > better< / u > , but < u > equivalent< / u > < b > [Query B]< / b > .
< ol >
< li > How can we tell if < b > [Query B]< / b > is equivalent to < b > [Query A]< / b > ?< / li >
< li > How can we tell if < b > [Query B]< / b > is better than < b > [Query A]< / b > ?< / li >
< / ol >
< / p >
< p class = "fragment" > ... but first a few definitions.< / p >
< / section >
< section >
< h3 > Relational Data< / h3 >
< dl >
< dt > Relation (Table)< / dt >
< dd > A collection of Tuples of Values< / dd >
< dd > All tuples have the same set of attributes, or schema< / dd >
< / dl >
< / section >
< section >
< h3 > Relational Data< / h3 >
Types of relations...
< table > < tr >
< td class = "fragment" >
< h4 > Set< / h4 >
< table style = "font-size: small;" >
< tr > < th > SPC_COMMON< / th > < th > BORONAME< / th > < / tr >
< tr > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'American linden'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > < / td > < td > < / td > < / tr >
< tr > < td > < / td > < td > < / td > < / tr >
< / table >
< p style = "font-size: 60%;" > (Unique Only)< / p >
< / td >
< td >
< h4 > Bag< / h4 >
< table style = "font-size: small;" >
< tr > < th > SPC_COMMON< / th > < th > BORONAME< / th > < / tr >
< tr > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'American linden'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < / tr >
< / table >
< / td >
< td class = "fragment" >
< h4 > List< / h4 >
< table style = "font-size: small;" >
< tr > < th > SPC_COMMON< / th > < th > BORONAME< / th > < / tr >
< tr > < td > 'American linden'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < / tr >
< tr > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < / tr >
< / table >
< p style = "font-size: 60%;" > (Order Matters)< / p >
< / td >
< / tr > < / table >
< / table >
< / section >
< section >
< h3 > Declarative Languages< / h3 >
< table >
< tr > < th > Declarative< / th > < th > Imperative< / th > < / tr >
< tr >
< td > Say < b > what< / b > you want< / td >
< td > Say < b > how< / b > you want to get it< / td >
< / tr >
< tr >
< td style = "vertical-align: middle;" > "Get me the TPS Reports"< / td >
< td >
< pre > < code > Look at every T report
For each week:
Sum up the sprocket count
Find that week's S report
... < / code > < / pre >
< / td >
< / tr >
< tr >
< td > SQL, RA, R, …< / td >
< td > C, Scala, Java, Python, …< / td >
< / tr >
< / table >
< / section >
< section >
< 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 >
< / section >
< / section >
< section >
< section >
< h3 > Declarative Queries< / h3 >
< dl >
< dt > SQL< / dt >
< dd > Readability: Focus on what and where from< / dd >
< dt > Relational Algebra< / dt >
< dd > Data Flow: Focus on transformations< / dd >
< dt > Relational Calculus< / dt >
< dd > Minimality: Focus on attribute relationships< / dd >
< / dl >
< / section >
< section >
< h3 > SQL< / h3 >
< pre > < code class = "sql" >
SELECT R.A FROM R, S
WHERE R.B = S.B AND S.C = 10
< / code > < / pre >
< / section >
< section >
< h3 > Relational Algebra< / h3 >
< img src = "graphics/2018-02-05-RA-Tree.svg" / >
$$\pi_{R.A}(\sigma_{S.C=10}(\bowtie_{R.B = S.B}(\textbf{R}, \textbf{S})))$$
< / section >
< section >
< h3 > Relational Calculus< / h3 >
< p style = "margin-top: 100px" >
$$\exists B : R(A, B), S(B, 10)$$
< / p >
< / section >
< section >
< h3 > Declarative Queries< / h3 >
< dl >
< dt class = "fragment highlight-grey" data-fragment-index = "1" > SQL< / dt >
< dd class = "fragment highlight-grey" data-fragment-index = "1" > Readability: Focus on what and where from< / dd >
< dt > Relational Algebra< / dt >
< dd > Data Flow: Focus on transformations< / dd >
< dt class = "fragment highlight-grey" data-fragment-index = "1" > Relational Calculus< / dt >
< dd class = "fragment highlight-grey" data-fragment-index = "1" > Minimality: Focus on attribute relationships< / dd >
< / dl >
< / section >
< / section >
< section >
< section >
< h3 style = "margin-bottom: 50px;" > Preliminaries< / h3 >
< p > We start with a database < b > instance< / b > with a fixed < b > schema< / b > < / p >
< p class = "fragment" style = "margin-top: 40px;" >
Queries are applied to Relations
$$Q(\textbf{Trees}, \textbf{SpeciesInfo})$$
< / p >
< p class = "fragment" style = "margin-top: 40px;" >
Queries are also Relations!
$$Q_2(\textbf{SpeciesInfo}, Q_1(\textbf{Trees}))$$
< span class = "fragment" style = "font-size: 50%; display: block; margin-top: 0px; margin-bottom: 10px;" > (Relational Algebra is < u > Closed< / u > )< / span >
< / p >
< / section >
< section >
< h3 > Relational Algebra< / h3 >
< table >
< tr > < th > Operation< / th > < th > Sym< / th > < th > Meaning< / th > < / tr >
< tr > < td > Selection< / td > < td > $\sigma$< / td > < td > Select a subset of the input rows< / td > < / tr >
< tr > < td > Projection< / td > < td > $\pi$< / td > < td > Delete unwanted columns< / td > < / tr >
< tr > < td > Cross-product< / td > < td > $\times$< / td > < td > Combine two relations< / td > < / tr >
< tr > < td > Set-difference< / td > < td > $-$< / td > < td > Tuples in Rel 1, but not Rel 2< / td > < / tr >
< tr > < td > Union< / td > < td > $\cup$< / td > < td > Tuples either in Rel 1 or in Rel 2< / td > < / tr >
< / table >
< p class = "fragment" >
< b > Also:< / b > Intersection, < u > Join< / u > , Division, Renaming
< / p >
< p class = "fragment" style = "font-size: 50%" > (Not essential, but can be useful)< / p >
< / section >
< section >
< table >
< tr >
< th > Input< / th >
< th > < / th >
< th > Query Language< / th >
< th > < / th >
< th > Output< / th >
< / tr >
< tr class = "fragment" data-fragment-index = "1" >
< td > Sets of Tuples< / td >
< td > $\rightarrow$< / td >
< td > < span class = "fragment highlight-grey" data-fragment-index = "2" > Set< / span > Relational Algebra< / td >
< td > $\rightarrow$< / td >
< td > Set of Tuples< / td >
< / tr >
< tr class = "fragment" data-fragment-index = "3" >
< td > Bags of Tuples< / td >
< td > $\rightarrow$< / td >
< td > Bag Relational Algebra< / td >
< td > $\rightarrow$< / td >
< td > Bag of Tuples< / td >
< / tr >
< tr class = "fragment" data-fragment-index = "4" >
< td class = "fragment highlight-grey" data-fragment-index = "5" > Lists of Tuples< / td >
< td class = "fragment highlight-grey" data-fragment-index = "5" > $\rightarrow$< / td >
< td class = "fragment highlight-grey" data-fragment-index = "5" > Extended Relational Algebra< / td >
< td class = "fragment highlight-grey" data-fragment-index = "5" > $\rightarrow$< / td >
< td class = "fragment highlight-grey" data-fragment-index = "5" > List of Tuples< / td >
< / tr >
< / table >
< p class = "fragment" data-fragment-index = "5" > First we focus on sets and bags.< / p >
< / section >
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< section >
< h3 > Selection ($\sigma_{c}$)< / h3 >
< p > Delete rows that fail the condition $c$.< / p >
< div class = "fragment" >
$$\sigma_{(BORONAME = \texttt{'Brooklyn'})} \textbf{Trees}$$
< table style = "font-size: small; margin-top: 30px;" >
< tr > < th > TREE_ID< / th > < th > SPC_COMMON< / th > < th > BORONAME< / th > < th > ...< / th > < / tr >
< tr > < td > 204026< / td > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < td > ...< / td > < / tr >
< tr > < td > 204337< / td > < td > 'honeylocust'< / td > < td > 'Brooklyn'< / td > < td > ...< / td > < / tr >
< tr > < td > 189565< / td > < td > 'American linden'< / td > < td > 'Brooklyn'< / td > < td > ...< / td > < / tr >
< tr > < td > 192755< / td > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < td > ...< / td > < / tr >
< tr > < td > 189465< / td > < td > 'London planetree'< / td > < td > 'Brooklyn'< / td > < td > ...< / td > < / tr >
< tr > < td style = "font-weight: bold;" colspan = "4" > ... and 177287 more< / td > < / tr >
< / table >
< / div >
< / section >
< section >
< h3 > Projection ($\pi_{A}$)< / h3 >
< p > Delete attributes not in the projection list $A$.< / p >
< div class = "fragment" >
$$\pi_{BORONAME}(Trees)$$
< table style = "font-size: small; margin-top: 30px;" >
< tr > < th > BORONAME< / th > < / tr >
< tr > < td > Queens< / td > < / tr >
< tr > < td > Brooklyn< / td > < / tr >
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< tr > < td > Manhattan< / td > < / tr >
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< tr > < td > Bronx< / td > < / tr >
< tr > < td > Staten Island< / td > < / tr >
< / table >
< / div >
< p class = "fragment" > Only 5 results... not 683788?< / p >
< p class = "fragment" > Set and Bag Projection are different< / p >
< / section >
< section >
< h3 > Reminder: Queries are Relations< / h3 >
< p > What are these queries schemas?< / p >
< div class = "fragment" style = "margin-top: 50px;" > $$\pi_{TREEID, SPC\_COMMON, BORONAME} \textbf{Trees}$$< / div >
< div class = "fragment" style = "margin-top: 50px;" > $$\sigma_{(BORONAME = \texttt{'Brooklyn'})} \textbf{Trees}$$< / div >
< div class = "fragment" style = "margin-top: 50px;" > $$\sigma_{(BORONAME = \texttt{'Brooklyn'})}(\pi_{TREEID, SPC\_COMMON, BORONAME} \textbf{Trees})$$< / div >
< / section >
< section >
< h3 > Union ($\cup$)< / h3 >
< p style = "margin-bottom: 0px;" > Takes two relations that are < u > union-compatible< / u > ...< / p >
< p style = "font-size: 60%; margin-top: 0px;" > (Both relations have the same number of fields with the same types)< / p >
< p > ... and returns all tuples appearing in < u > either< / u > relation< / p >
< div class = "fragment" style = "font-size: 70%;" >
$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} \textbf{Trees}) \cup (\sigma_{(BORONAME=\texttt{'Manhattan'})} \textbf{Trees})$$
< / div >
< p class = "fragment" > We use $\uplus$ if we explicitly mean < i > bag< / i > union< / p >
< / section >
< section >
< h3 > Intersection ($\cap$)< / h3 >
< p > Return all tuples appearing in < u > both< / u > < br / > of two union-compatible relations< / p >
< div class = "fragment" style = "font-size: 70%;" >
$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} (\pi_{SPC\_COMMON} \textbf{Trees})) \\ ~~~~~~~~~\cap (\sigma_{(BORONAME=\texttt{'Manhattan'})} (\pi_{SPC\_COMMON} \textbf{Trees}))$$
< p > What is this query asking?< / p >
< / div >
< / section >
< section >
< h3 > Set Difference< / h3 >
< p > Return all tuples appearing in the first, but not the second< br / > of two union-compatible relations< / p >
< div class = "fragment" style = "font-size: 70%;" >
$$(\sigma_{(BORONAME=\texttt{'Brooklyn'})} (\pi_{SPC\_COMMON} \textbf{Trees})) \\ ~~~~~~~~~- (\sigma_{(BORONAME=\texttt{'Manhattan'})} (\pi_{SPC\_COMMON} \textbf{Trees}))$$
< 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 >
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< 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 >
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< tr > < td style = "font-weight: bold;" colspan = "4" > ...< / td > < / tr >
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< 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 >
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< 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 >
2018-02-04 23:57:46 -05:00
< / section >
< / div > < / div >
< script src = "../reveal.js-3.6.0/js/reveal.js" > < / script >
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// Full list of configuration options available at:
// https://github.com/hakimel/../reveal.js#configuration
Reveal.initialize({
2018-02-06 22:44:57 -05:00
controls: true,
2018-02-04 23:57:46 -05:00
progress: true,
history: true,
center: true,
slideNumber: true,
transition: 'fade', // none/fade/slide/convex/concave/zoom
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radar: { borderColor: [ "rgba(20,220,220,.8)" , "rgba(220,120,120,.8)", "rgba(20,120,220,.8)" ]},
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