Aquery: A DATABASE SYSTEM FOR ORDER

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Aquery: A DATABASE SYSTEM FOR ORDER. Dennis Shasha, joint work with Alberto Lerner {lerner,shasha}@cs.nyu.edu. Motivation The need for ordered data. Queries in finance, signal processing etc. depend on order. SQL 99 has extensions but they are clumsy.

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### Aquery: A DATABASE SYSTEM FOR ORDER

Dennis Shasha, joint work with Alberto Lerner {lerner,shasha}@cs.nyu.edu

MotivationThe need for ordered data
• Queries in finance, signal processing etc. depend on order.
• SQL 99 has extensions but they are clumsy.
Moving Averages is algorithmically linear but…

Sales(month, total)

SELECT t1.month+1 AS forecastMonth, (t1.total+ t2.total + t3.total)/3 AS 3MonthMovingAverageFROM Sales AS t1, Sales AS t2, Sales AS t3WHERE t1.month = t2.month - 1 AND t1.month = t3.month – 2

Can optimizer make a 3-way (in general, n-way) join linear time?

Ref: Data Mining and Statistical Analysis Using SQL

Trueblood and LovettApress, 2001

Moving Averages in SQL99 is awkward

Sales(month, total)

Alberto, please put in SQL 99

Choose query from Jennifer

Alberto, I suggest query 2 because that may be hard in sql 99.

MotivationProblems Extending SQL with Order
• Queries are hard to read
• Cost of execution is often non-linear (would not pass basic algorithms course)
• Few operators preserve order, so optimization hard.
Idea
• Whatever can be done on a table can be done on an ordered table (arrable). Not vice versa.
• Aquery – query language on arrables; operations on columns.
• Upward compatible from SQL 92.
• Optimization ideas are new, but natural.
SQL + OrderMoving Averages

Sales(month, total)

SELECT month, avgs(8, total)FROM Sales ASSUMING ORDER month

avgs: vector-to-vector function, order-dependant and size-preserving

order to be used on vector-to-vector functions

• Execution (Sales is an arrable):
• FROM clause – enforces the order in ASSUMING clause
• SELECT clause – for each month yields the moving average (window size 8) ending at that month.
SQL + OrderTop N

Employee(ID, salary)

SELECT first(N, salary) FROM Employee ASSUMING ORDER Salary

first: vector-to-vector function, order-dependant and non size-preserving

• Execution:
• FROM clause – orders arrable by Salary
• SELECT clause – applies first() to the ‘salary’ vector, yielding first N values of that vector given the order. Could get the top earning IDs by saying first(N, ID).
SQL + OrderVector-to-Vector Functions

size-preserving

non size-preserving

prev, next, \$, []

avgs(*), prds(*), sums(*), deltas(*), ratios(*), reverse,

drop, first, last

order-dependant

rank, tile

min, max, avg, count

non order-dependant

Given a table with securities’ price ticks, ticks(ID, date, price, timestamp), find the best possible profit if one bought and sold security ‘S’ in a given day

Note that max – min doesn’t work, because must buy before you sell (no selling short).

Challenge

In a given day, what would be the maximum difference between a buying and selling point of each security?

SELECT ID, max(price – mins(price))FROM Ticks ASSUMING ORDER timestampWHERE tradeDate = ‘99/99/99’GROUP BY ID

max

running min

min

• Execution:
• For each security, compute the running minimum vector for price and then subtract from the price vector itself; result is a vector of spreads.
• Note that max – min would overstate spread.
Best-profit query

SELECT max(price – mins(price))

FROM ticks

ASSUMING ORDER timestamp

WHERE day = ‘12/09/2002’

AND ID = ‘S’

price 15 19 13 7 4 5 4 7 12 2

mins 15 15 13 7 4 4 4 4 4 4

- 0 4 0 0 0 1 0 3 8 2

[AQuery]

SELECT max(price–mins(price))

FROM ticks ASSUMING timestamp

WHERE ID=“x”

[SQL:1999]

SELECT max(rdif)

price - min(price)

OVER

(PARTITION BY ID

ORDER BY timestamp

ROWS UNBOUNDED

PRECEDING) AS rdif

FROM Ticks ) AS t1

WHERE ID=“x”

Best-profit query comparison
Complex queries: Crossing averages part I

When does the 21-day average cross the 5-month average?

INSERT INTO temp FROMSELECT ID, tradeDate, avgs(21 days, closePrice) AS a21, avgs(5 months, closePrice) AS a5, prev(avgs(21 days, closePrice)) AS pa21, prev(avgs(5 months, closePrice)) AS pa5FROM TradedStocks NATURAL JOIN Market ASSUMING ORDER tradeDateGROUP BY ID

Complex queries: Crossing averages part uses non 1NF
• Execution:
• FROM clause – order-preserving join
• GROUP BY clause – groups are defined based on the value of the Id column
• SELECT clause – functions are applied; non-grouped columns become vector fields so that target cardinality is met. Violates first normal form

Vectorfield

groups in ID and non-grouped column

grouped ID and non-grouped column

two columns withthe same cardinality

Complex queries: Crossing averages part II

Get the result from the resulting non first normal form relation temp

SELECT ID, tradeDateFROM flatten(temp)WHERE a21 > a5 AND pa21 <= pa5

• Execution:
• FROM clause – flatten transforms temp into a first normal form relation (for row r, every vector field in r MUST have the same cardinality). Could have been placed at end of previous query.
• Standard query processing after that.
SQL + OrderRelated Work: Research
• SEQUIN – Seshadri et al.
• Sequences are first-class objects
• Difficult to mix tables and sequences.
• SRQL – Ramakrishnan et al.
• Elegant algebra and language
• No work on transformations.
• SQL-TS – Sadri et al.
• Language for finding patterns in sequence
• But: Not everything is a pattern!
SQL + OrderRelated Works: Products
• RISQL – Red Brick
• Some vector-to-vector, order-dependent, size-preserving functions
• Low-hanging fruit approach to language design.
• Analysis Functions – Oracle 9i
• Quite complete set of vector-to-vector functions
• But: Can only be used in the select clause; poor optimization (our preliminary study)
• KSQL – Kx Systems
• Arrable extension to SQL but syntactically incompatible.
• No cost-based optimization.
Order-related optimization techniques
• Starburst’s “glue” (Lohman, 88) and Exodus/Volcano “Enforcers” (Graefe and McKeena, 93)
• DB2 Order optimization (Simmen et al., 96)
• Top-k query optimization (Carey and Kossman, 97)
• Hash-based order-preserving join (Claussen et al., 01)
• Temporal query optimization addressing order and duplicates (Slivinskas et al., 01)
Interchange sorting + order preserving operators

SELECT ts.ID, ts.Exchange, avgs(10, hq.ClosePrice)FROM TradedStocks AS ts NATURAL JOIN HistoricQuotes AS hq ASSUMING ORDER hq.TradeDateGROUP BY Id

avgs

avgs

avgs

g-by

sort

g-by

op

avgs

op

sort

g-by

g-by

op

op

sort

op

(1) Sort then joinpreserving order

(2) Preserve existingorder

(3) Join then sortbefore grouping

(4) Join then sortafter grouping

UDFs evaluation order

Gene(geneId, seq)SELECT t1.geneId, t2.geneId, dist(t1.seq, t2.seq)FROM Gene AS t1, Gene AS tWHERE dist(t1.seq, t2.seq) < 5 AND posA(t1.seq, t2.seq)

posA asks whether sequences have Nucleo A in same position. Dist gives edit distance between two Sequences.

posA

dist

Switch dynamicallybetween (1) and (2) depending on the execution history

dist

posA

(1)

(2)

(3)

Transformations Building Blocks
• Order optimization
• Simmens et al. `96 – push-down sorts over joins, and combining and avoiding sorts
• Order preserving operators
• KSQL – joins on vector
• Claussen et al. `00 – OP hash-based join
• Push-down aggregating functions
• Chaudhuri and Shim `94, Yan and Larson `94 – evaluate aggregation before joins
• UDF evaluation
• Hellerstein and Stonebraker ’93 – evaluate UDF according to its ((output/input) – 1)/cost per tuple
• Porto et al. `00 – take correlation into account
SELECT last(price)FROM ticks t,base b ASSUMING ORDER name, timestampWHERE t.ID=b.ID AND name=“x”Original Query

last(price)

Name=“x”

sort

name,timesamp

ID

ticks

base

Then, push sort

sortA(r1 r2) A sortA(r1) lop r2

sortA(r) A r

Last price for a name query

last(price)

sort

name,timesamp

ID

ticks

Name=“x”

base

The projection is carrying an implicit selection: last(price) = price[n], where n is the last index of the price array

f(r.col[i])(r) order(r)f(r.col)(pos()=i(r))

Last price for a name query

price

last(price)

pos()=last

lop

ID

ticks

Name=“x”

base

Can we somehow push the last selection down the join?

Last price for a name query

price

pos()=last

lop

ID

ticks

Name=“x”

base

We can take the last position of each ID on ticks to reduce cardinality, but we need to group by ticks.ID first

But trading a join for a group by is usually a good deal?!

One more step: make this an “edge by”

Last price for a name query

price

safety

lop

ID

pos()=last

Name=“x”

Gby

base

ID

ticks

The “edgeby” operator
• The pattern edge condition(Gby(r)) can be efficiently implemented without grouping the whole arrable
• An edgeby is a kind of scan that take as parameters the grouping column(s), the direction to scan, a position, and if that position is the last (scan up to) or the first one the group (scan from this on).
• In the previous example, to find the last rows for each ID in ticks one would use, respectively, `ID,\' `backwards,\' ` 1,\' and \'up to.\'
Conclusion
• Arrable-based approach to ordered databases may be scary – dependency on order, vector-to-vector functions – but it’s expressive and fast.
• SQL extension that includes order is possible and reasonably simple.
• Optimization possibilities are vast.
Streaming
• Aquery has no special facilities for streaming data, but it is expressive enough.
• Idea for streaming data is to split the tables into tables that are indexed with old data and a buffer table with recent data.
• Optimizer works over both transparently.
Arrables’ Shape Properties
• The cardinality of an arrable’s array-columns must be the same
• An arrable’s tuple can contain single or array-valued column values.
• A 1NF arrable contains only single values
• A flatten-able ¬1NF arrable may contain array-valued columns, but they have the same cardinality
• A non-flatten-albe ¬1NF arrable is one that is not in the previous categories
Arrables Primitives
• Single-value indexingr[k] retrieves the tuple formed by the k-th position value of each of the arrables array-columns
• Multi-value indexingr[k1 k2 … kn] is similar to forming an arrable by insertion of r[k1], r[k2], …, r[kn]
Array-columns Expressions
• Expressions in AQuery are column-oriented, as opposed to row-oriented

price

xy

z

price

xy

z

prev(price)

-

xy

result

T

F

T

result

-

y-x

z-y

scalar =

-

=

>

Built-in functions
• prevv[i] = null, if i=0 or v[i-1], for 0<i|v|-1
• firstv,n[i] = v[i], for 0i<n
• minsv[i] = v[i], if i=0 or min(v[i],mins[i-1]), for 0<i|v|-1
• indexbv[i] = i such that bv[i] is true
Each modifier
• Built-in functions are applied over array-columns
• The each keyword makes the function application occur in each tuple level
Group and flatten operations
• An arrable can be grouped over a grouping expression that assigns a group value to each row. The net effect is to nest an 1NF arrable into a flatten-able ¬1F one
• The flatten operation undoes the operation
• But there are data-ordering issues!
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

Arrables mentioned in the FROM clause are combined using cartesian product. This is operation is order-oblivious

AQuery syntax and execution
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

The ASSUMING ORDER is enforced. From this point on any order-dependent expression can be used, regardless of the clause.

AQuery syntax and execution
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

The WHERE expression is executed. It must generate a boolean vector that maps each row of the current arrable to true or false. The false rows are eliminated and the resulting arrable is passed along.

AQuery syntax and execution
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

The GROUP BY expression is computed. It must map each tuple to a group. The resulting arrable has as many rows as there are groups. The columns that did not participate on the grouping expression now have array-valued values

AQuery syntax and execution
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

The HAVING clause expression must result in a boolean vector that maps each group, i.e, row, to a boolean value. The rows that map to false are excluded.

AQuery syntax and execution
It’s just SQL, really. But column-oriented.

SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING

Finally, each expression on the SELECT clause is executed, generating a new array-column. The resulting arrable is achieved by “zipping” these arrays side-by-side.

AQuery syntax and execution
SELECT ID, index(flags=‘FIN’)-index(flags=‘ACK’)

FROM netmgmt

ASSUMING ORDER timestamp

GROUP BY ID

Translating queries into Operations

each

Order-preservingportion of the plan

Gby

sort

netmgmt

Non- and Order-preserving Variations
• Operators may or may not be required to be preserve order, depending on their location.
• For instance:
• a group by based on hashing is an implementation of a non-order preserving group-by
• A nested-loops join is an implementation of a left-order preserving join
• There are performance implications in preserving or not order
Early sort and order-preserving group by

Early, non-order preserving group by and sort

Pick the best

each

each

GbysessionID

sorttimestamp

each

sorttimestamp

GbysessionID

netmgmt

netmgmt

AQuery Optimization
• Optimization is cost based
• The main strategies are:
• Define the extent of the order-preserving region of the plan, considering (correctness, obviously, and) the performance of variation of operators
• Apply efficient implementations of patterns of operators
• There are generic techniques yet each query category presents unique opportunities and challenges
• Equivalence rules between operators can be used in the process
Order Equivalence
• Let Order(r) return the attributes that the arrable r is “ordered by.”
• Two arrables r1 and r2 are order-equivalent with respect to the list of attributes A, denoted r1 Ar2 if A is a prefix of Order(r1) and of Order(r2), and some permutation of r1 is identical to r2
• We denote by r1 {}r2 the case where r1 and r2 are multiset-equivalent (i.e. they contain the same setof tuples and for each tuple in the set, they contain the same number of instances of that tuple).
Sort Elimination
• Often sort can be eliminated or can be done over less fields
• sortA(r) A r if A is prefix of Order(r)
• sortA•B(r) A•B sort’A(r) if B is prefix of Order(r) and sort’ is stable
• sortA(sortB(r)) A•B sortAif A B is a valid functional dependency
• sortA(sortB(r)) A•B sortA•(B-A) (r)
• oper-path(sort(r)) {} oper-path(r)if oper-path has no order-dependent operation
Conclusion
• Arrables offer a simple way to express order-based predicates and expressions
• Queries tend to be lucid, concise and, consequently, order idioms stand out
• Optimization naturally extend the know corpus of rules
• New possibility of operator implementing algorithm that take order into consideration
Order preserving joins

select lineitem.orderid, avgs(10, lineitem.qty), lineitem.lineid

from order, lineitem assuming order lineid

where order.date > 45

and order.date < 55 and lineitem.orderid = order.orderid

• Basic strategy 1: restrict based on date. Create hash on order. Run through lineitem, performing the join and pulling out the qty.
• Basic strategy 2: Arrange for lineitem.orderid to be an index into order. Then restrict order based on date giving a bit vector. The bit vector, indexed by lineitem.orderid, gives the relevant lineitem rows.
• The relevant order rows are then fetched using the surviving
• lineitem.orderid.
• Strategy 2 is often 3-10 times faster.