2018-02-06 22:44:57 -05:00
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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 Equivalences< / h1 >
< h3 > CSE 4/562 – < / h3 >
< h5 > February 7, 2018< / h5 >
< / section >
< section >
< section >
< h3 > Recap: Relational Algebra< / h3 >
< table style = "font-size: 70%" >
< 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 >
< tr > < td > Intersection< / td > < td > $\cap$< / td > < td > Tuples in both Rel 1 and Rel 2< / td > < / tr >
< tr > < td > Join< / td > < td > $\bowtie$< / td > < td > Pairs of tuples matching a specified condition< / td > < / tr >
< tr style = "color: grey;" > < td > Division< / td > < td > $/$< / td > < td > "Inverse" of cross-product< / td > < / tr >
< / table >
< / 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%" class = "fragment" >
$$\pi_{BORONAME,\ SPC\_COMMON}(\textbf{Trees}) \;\;/\;\;\pi_{SPC\_COMMON}(\textbf{Trees})$$
(using set relational algebra)
< / div >
< p class = "fragment" >
$$R / S \equiv \{\; \left< \vec t\right> \;|\; \forall \left< \vec s\right> \in S, \left< \vec t \vec s \right> \in R \;\}$$
< / p >
< / section >
< section >
< h3 > Division ($/$)< / h3 >
< table style = "font-size: 60%; margin-top: 30px; display: inline-block; vertical-align: middle;" >
< tr > < th > BORO< / th > < th > SPC_COMMON< / th > < / tr >
< tr class = "fragment highlight-blue" data-fragment-index = "1" > < td > Brooklyn< / td > < td > honeylocust< / td > < / tr >
< tr class = "fragment highlight-red" data-fragment-index = "3" > < td > Brooklyn< / td > < td > American linden< / td > < / tr >
< tr > < td > Brooklyn< / td > < td > London planetree< / td > < / tr >
< tr class = "fragment highlight-blue" data-fragment-index = "1" > < td > Manhattan< / td > < td > honeylocust< / td > < / tr >
< tr class = "fragment highlight-red" data-fragment-index = "3" > < td > Manhattan< / td > < td > American linden< / td > < / tr >
< tr class = "fragment highlight-green" data-fragment-index = "5" > < td > Manhattan< / td > < td > pin oak< / td > < / tr >
< tr class = "fragment highlight-blue" data-fragment-index = "1" > < td > Queens< / td > < td > honeylocust< / td > < / tr >
< tr class = "fragment highlight-red" data-fragment-index = "3" > < td > Queens< / td > < td > American linden< / td > < / tr >
< tr class = "fragment highlight-blue" data-fragment-index = "1" > < 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" data-fragment-index = "1" > < td >
< span style = "font-size: 200%" > /< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > SPC_COMMON< / th > < / tr > < tr > < td class = "fragment highlight-current-blue" data-fragment-index = "1" > honeylocust< / td > < / tr > < / table >
< span class = "fragment" data-fragment-index = "2" >
< span style = "font-size: 200%; margin-left: 10px;" > =< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > BORO< / th > < / tr > < tr > < td > Brooklyn< / td > < / tr > < tr > < td > Manhattan< / td > < / tr > < tr > < td > Queens< / td > < / tr > < tr > < td > Bronx< / td > < / tr > < / table >
< / span >
< / td > < / tr >
< tr class = "fragment" data-fragment-index = "3" > < td >
< span style = "font-size: 200%" > /< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > SPC_COMMON< / th > < / tr > < tr > < td class = "fragment highlight-current-blue" data-fragment-index = "3" > honeylocust< / td > < / tr > < tr > < td class = "fragment highlight-current-red" data-fragment-index = "3" > American linden< / td > < / tr > < / table >
< span class = "fragment" data-fragment-index = "4" >
< span style = "font-size: 200%; margin-left: 10px;" > =< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > BORO< / th > < / tr > < tr > < td > Brooklyn< / td > < / tr > < tr > < td > Manhattan< / td > < / tr > < tr > < td > Queens< / td > < / tr > < / table >
< / span >
< / td > < / tr >
< tr class = "fragment" data-fragment-index = "5" > < td >
< span style = "font-size: 200%" > /< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > SPC_COMMON< / th > < / tr > < tr > < td class = "fragment highlight-current-blue" data-fragment-index = "5" > honeylocust< / td > < / tr > < tr > < td class = "fragment highlight-current-red" data-fragment-index = "5" > American linden< / td > < / tr > < tr > < td class = "fragment highlight-current-green" data-fragment-index = "5" > pin oak< / td > < / tr > < / table >
< span class = "fragment" data-fragment-index = "6" >
< span style = "font-size: 200%; margin-left: 10px;" > =< / span >
< table style = "display: inline-block; vertical-align: middle; margin-left: 10px; border: 1px solid black;" > < tr > < th > BORO< / th > < / tr > < tr > < td > Manhattan< / td > < / tr > < / table >
< / span >
< / td > < / tr >
< / table >
< / section >
< / section >
< section >
< section >
< h3 > The running theme< / h3 >
< p >
< span class = "fragment highlight-grey" data-fragment-index = "1" > If X and Y are < / span > < u > equivalent< / u > < span class = "fragment highlight-grey" data-fragment-index = "1" > and Y is < u > better< / u > ,< br / >
then replace all Xs with Ys< / span >
< / p >
< p class = "fragment" data-fragment-index = "1" style = "font-size: 70%;" > < b > Today's focus< / b > : Provable Equivalence for RA Expressions< / p >
< / section >
< section >
< h3 > Equivalence< / h3 >
$$Q_1 = \pi_{A}\left( \sigma_{c}( R ) \right)$$
$$Q_2 = \sigma_{c}\left( \pi_{A}( R ) \right)$$
< div class = "fragment" >
$$Q_1 \stackrel{?}{\equiv} Q_2$$
< / div >
< / section >
< section >
< h3 > Ground Rules< / h3 >
< dl >
< dt class = "fragment" data-fragment-index = "1" > Only Relational Values Matter< / dt >
< dd class = "fragment" data-fragment-index = "1" > Obviously $Q_1 \neq Q_2$. What we care about is whether $Q_1(R) = Q_2(R)$...< / dd >
< dt class = "fragment" data-fragment-index = "2" > Data Independent< / dt >
< dd class = "fragment" data-fragment-index = "2" > ... for < i > all< / i > valid input data $R$.< / dd >
< dd class = "fragment" data-fragment-index = "3" style = "font-size: 70%" > However, it's fair to talk about equivalence when we know the data has some properties. (more on this later)< / dd >
< dt class = "fragment" data-fragment-index = "4" > Data-Model Dependent< / dt >
< dd class = "fragment" data-fragment-index = "4" > It's important to be clear whether we're talking about sets, < span class = "fragment highlight-blue" > bags< / span > , or lists.< / dd >
< / dl >
< / section >
< section >
< h3 > In summary...< / h3 >
< p style = "font-size: 80%;" >
We say that $Q_1 \equiv Q_2$ if and only if< br / >
we can guarantee that the < i > bag< / i > of tuples produced by $Q_1(R, S, T, \ldots)$ < br / >
is the same as the < i > bag< / i > of tuples produced by $Q_2(R, S, T, \ldots)$ < br / >
for any combination of valid inputs $R, S, T, \ldots$.
< / p >
< p style = "font-size: 70%;" class = "fragment" >
... that satisfy any necessary properties.
< / p >
< / section >
< section >
< h3 > Starting Rules< / h3 >
< table style = "font-size: 80%" >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Selection< / th > < / tr >
< tr >
< td > $\sigma_{c_1 \wedge c_2}(R) \equiv \sigma_{c_1}(\sigma_{c_2}(R))$< / td >
< td > (Decomposability)< / td >
< / tr >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Projection< / th > < / tr >
< tr >
< td > $\pi_{A}(R) \equiv \pi_{A}(\pi_{A \cup B}(R))$< / td >
< td > (Idempotence)< / td >
< / tr >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Cross Product< / th > < / tr >
< tr >
< td > $R \times (S \times T) \equiv (R \times S) \times T$< / td >
< td > (Associativity)< / td >
< / tr >
< tr >
< td > $R \times S \equiv S \times R$< / td >
< td > (Commutativity)< / td >
< / tr >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Union< / th > < / tr >
< tr >
< td > $R \cup (S \cup T) \equiv (R \cup S) \cup T$< / td >
< td > (Associativity)< / td >
< / tr >
< tr >
< td > $R \cup S \equiv S \cup R$< / td >
< td > (Commutativity)< / td >
< / tr >
< / table >
< / section >
< section >
< h3 > Try it!< / h3 >
< p class = "fragment highlight-grey" data-fragment-index = "1" >
Show that
$$R \times (S \times T) \equiv T \times (S \times R)$$
< / p >
< div class = "fragment highlight-grey" data-fragment-index = "2" >
< p class = "fragment" data-fragment-index = "1" >
Show that
$$\sigma_{c_1}(\sigma_{c_2}(R)) \equiv \sigma_{c_2}(\sigma_{c_1}(R))$$
< / p >
< / div >
< div class = "fragment highlight-grey" data-fragment-index = "3" >
< p class = "fragment" data-fragment-index = "2" >
Show that
$$R \bowtie_{c} S \equiv S \bowtie_{c} R$$
< / p >
< / div >
< p class = "fragment" data-fragment-index = "3" >
Show that
$$\sigma_{R.B = S.B \wedge R.A > 3}(R \times S) \equiv \sigma_{R.A > 3}(R \bowtie_{B} S)$$
< / p >
< / section >
< / section >
< section >
< section >
< h3 > Rules for Multiple Operators< / h3 >
< / section >
< section >
< table style = "font-size: 90%; margin-bottom: 50px;" >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Selection + Projection< / th > < / tr >
< tr >
< td > $\pi_{A}(\sigma_{c}(R)) \equiv \sigma_{c}(\pi_{A}(R))$< / td >
< td > (Commutativity)< / td >
< / tr >
< / table >
< p style = "font-size: 80%;" class = "fragment" > ... but only if $A$ and $c$ are < u > compatible< / u > < / p >
< p style = "font-size: 80%;" class = "fragment" > $A$ must include all columns referenced by $c$ ($cols(c)$)< / p >
< div class = "fragment" style = "margin-top: 50px;" >
< h3 > Try it!< / h3 >
< p >
Show that
$$\pi_A(\sigma_c(R)) \equiv \pi_A(\sigma_c(\pi_{(A \cup cols(c))}(R)))$$
< / p >
< / div >
< / section >
< section >
< table style = "font-size: 90%; margin-bottom: 50px;" >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Selection + Cross Product< / th > < / tr >
< tr >
< td > $\sigma_c(R \times S) \equiv (\sigma_{c}(R)) \times S$< / td >
< td > (Commutativity)< / td >
< / tr >
< / table >
< p style = "font-size: 80%;" class = "fragment" > ... but only if $c$ references only columns of $R$< / p >
< p style = "font-size: 60%;" class = "fragment" > $cols(c) \subseteq cols(R)$< / p >
< div class = "fragment" style = "margin-top: 50px;" >
< h3 > Try it!< / h3 >
< p >
Show that
$$\sigma_{R.B = S.B \wedge R.A > 3}(R \times S) \equiv (\sigma_{R.A > 3}(R)) \bowtie_{B} S$$
< div style = "font-size: 70%;" class = "fragment" > When is this rewrite a good idea?< / div >
< / p >
< / div >
< / section >
< section >
< table style = "font-size: 90%; margin-bottom: 50px;" >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Projection + Cross Product< / th > < / tr >
< tr >
< td > $\pi_A(R \times S) \equiv (\pi_{A_R}(R)) \times (\pi_{A_S}(S))$< / td >
< td > (Commutativity)< / td >
< / tr >
< / table >
< p style = "font-size: 80%;" > ... where $A_R$ and $A_S$ are the columns of $A$ from $R$ and $S$ respectively.< / p >
< p style = "font-size: 60%;" > $A_R = A \cap cols(R)$ $A_S = A \cap cols(S)$< / p >
< div class = "fragment" style = "margin-top: 50px;" >
< h3 > Try it!< / h3 >
< p >
Show that
$$\pi_{A}(R \bowtie_c S) \equiv (\pi_{A_R}(R)) \bowtie_c (\pi_{A_S}(S))$$
< div style = "font-size: 70%;" class = "fragment" > When does this condition hold?< / div >
< / p >
< / div >
< / section >
< section >
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< table style = "font-size: 80%; margin-bottom: 50px;" >
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< tr > < th colspan = "2" style = "padding-top: 20px;" > Intersection< / th > < / tr >
< tr >
< td > $R \cap (S \cap T) \equiv (R \cap S) \cap T$< / td >
< td > (Associativity)< / td >
< / tr >
< tr >
< td > $R \cap S \equiv S \cap R$< / td >
< td > (Commutativity)< / td >
< / tr >
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< tr > < th colspan = "2" style = "padding-top: 20px;" > Selection + < u > < / u > < / th > < / tr >
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< tr >
< td > $\sigma_c(R \cup S) \equiv (\sigma_c(R)) \cup (\sigma_c(R))$< / td >
< td > (Commutativity)< / td >
< / tr >
< tr >
< td > $\sigma_c(R \cap S) \equiv (\sigma_c(R)) \cap (\sigma_c(R))$< / td >
< td > (Commutativity)< / td >
< / tr >
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< tr > < th colspan = "2" style = "padding-top: 20px;" > Projection + < u > < / u > < / th > < / tr >
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< tr >
< td > $\pi_A(R \cup S) \equiv (\pi_A(R)) \cup (\pi_A(R))$< / td >
< td > (Commutativity)< / td >
< / tr >
< tr >
< td > $\pi_A(R \cap S) \equiv (\pi_A(R)) \cap (\pi_A(R))$< / td >
< td > (Commutativity)< / td >
< / tr >
< tr > < th colspan = "2" style = "padding-top: 20px;" > Cross Product + Union< / th > < / tr >
< tr >
< td > $R \times (S \cup T) \equiv (R \times S) \cup (R \times T)$< / td >
< td > (Distributivity)< / td >
< / tr >
< / table >
< / section >
< / section >
< section >
< section >
< h3 > Example< / h3 >
< pre style = "display: inline-block; width: 300px; vertical-align: middle;" > < code class = "sql" >
SELECT R.A, T.E
FROM R, S, T
WHERE R.B = S.B
AND S.C < 5
AND S.D = T.D
< / code > < / pre >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-1.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-1.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-2.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-2.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-3.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-3.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-4.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-4.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-5.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-5.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-6.svg" style = "vertical-align: middle;" / >
< / section >
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< section >
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< img src = "graphics/2018-02-07-RA-Opt-6.svg" style = "vertical-align: middle;" / >
< span style = "vertical-align: middle; margin: 50px; font-size: 300%" > ➔< / span >
< img src = "graphics/2018-02-07-RA-Opt-7.svg" style = "vertical-align: middle;" / >
< / section >
< / section >
< section >
< section >
< h3 > General Query Optimizers< / h3 >
< p > < b > Input:< / b > Dumb translation of SQL to RA< / p >
< p > ⬇︎< / p >
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< p > Apply rewrites< / p >
< p > ⬇︎< / p >
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< p > < b > Output:< / b > Better, but equivalent query< / p >
< / section >
< section >
< p > Which rewrite rules should we apply?< / p >
< / section >
< section >
< dl style = "font-size: 75%" >
< dt class = "fragment highlight-grey" data-fragment-index = "1" > Selection Pushdown< / dt >
< dd class = "fragment highlight-grey" data-fragment-index = "1" > < b > Always< / b > commute Selections as close to the leaves as possible.< / dd >
< dt class = "fragment highlight-grey" data-fragment-index = "1" > Join Construction< / dt >
< dd class = "fragment highlight-grey" data-fragment-index = "1" > Joins are < b > always< / b > better than cross-products.< / dd >
< dt class = "fragment highlight-grey" data-fragment-index = "1" > (Optional) Projection Pushdown< / dt >
< dd class = "fragment highlight-grey" data-fragment-index = "1" > Commuting Projections down to the leaves removes redundant columns, and < b > may< / b > be beneficial for some systems.< / dd >
< dt > Join Algorithm Selection< / dt >
< dd > Joins can be implemented differently, depending on the join predicate.< / dd >
< dt > Join/Union Ordering< / dt >
< dd > The order in which joins are evaluated < b > may< / b > affect query runtimes.< / dd >
< dt > Access Paths< / dt >
< dd > $(\sigma_c(R))$ and $(Q(\ldots) \bowtie_c R)$ are special cases that we can make fast!< / dd >
< / dl >
< p class = "fragment" data-fragment-index = "1" style = "font-size: 60%" > Some rewrites are situational... we need more information to decide when to apply them.< / p >
< / section >
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< section >
< h3 > General Query Optimizers< / h3 >
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< ol style = "font-size: 60%" >
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< li class = "fragment" data-fragment-index = "1" > Apply blind heuristics (e.g., push down selections)< / li >
< li class = "fragment" data-fragment-index = "2" > Enumerate all possible < i > execution plans< / i > by varying < span class = "fragment" data-fragment-index = "6" style = "font-size: 80%;" > (or for a reasonable subset)< / span >
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< ul >
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< li > Join/Union Evaluation Order (commutativity, associativity, distributivity)< / li >
< li class = "fragment" data-fragment-index = "3" > < span class = "fragment highlight-blue" data-fragment-index = "8" > Algorithms< / span > for Joins, < span class = "fragment highlight-blue" data-fragment-index = "7" > Aggregates, Sort, Distinct< / span > , and others< / span > < / li >
< li class = "fragment" data-fragment-index = "3" > Data < span class = "fragment highlight-blue" data-fragment-index = "9" > Access Paths< / span > < / li >
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< / ul >
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< / li >
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< li class = "fragment" data-fragment-index = "4" > < span class = "fragment highlight-blue" data-fragment-index = "10" > Estimate< / span > the cost of each execution plan< / li >
< li class = "fragment" data-fragment-index = "5" > Pick the execution plan with the lowest cost< / li >
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< / ol >
< / section >
2018-02-06 22:44:57 -05:00
< section >
< b > Next Class:< / b > Extended Relational Algebra and Basic Join Algorithms
< / section >
< / section >
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