aquery: a database system for order dennis shasha, joint work with alberto lerner...

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 SQLTrueblood 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

Execution (Sales is an arrable):1. FROM clause – enforces the order in

ASSUMING clause2. SELECT clause – for each month yields

the moving average (window size 8) ending at that month.

avgs: vector-to-vector function, order-

dependant and size-preserving

order to be used on vector-to-vector

functions

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:1. FROM clause – orders arrable by Salary2. 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

prev, next, $, []

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

…

drop, first, lastorder-dependant

non order-dependant

size-preserving

non size-preserving

rank, tile min, max, avg, count

Challenge

• 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).

Complex queries: Best spreadIn a given day, what would be the maximum difference between a buying and selling point of each security?

Ticks(ID, price, tradeDate, timestamp, …)

SELECT ID, max(price – mins(price))FROM Ticks ASSUMING ORDER timestampWHERE tradeDate = ‘99/99/99’GROUP BY IDExecution:1. For each security, compute the running

minimum vector for price and then subtract from the price vector itself; result is a vector of spreads.

2. Note that max – min would overstate spread.

max

min

bestspread

running min

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

Best-profit query comparison

[AQuery]

SELECT max(price–mins(price))

FROM ticks ASSUMING timestamp

WHERE ID=“x”

AND tradeDate='99/99/99'

[SQL:1999]

SELECT max(rdif)

FROM (SELECT ID,tradeDate,

price - min(price)

OVER

(PARTITION BY ID

ORDER BY timestamp

ROWS UNBOUNDED

PRECEDING) AS rdif

FROM Ticks ) AS t1

WHERE ID=“x”

AND tradeDate='99/99/99'

Complex queries: Crossing averages part I

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

Market(ID, closePrice, tradeDate, …)TradedStocks(ID, Exchange,…)

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:1. FROM clause – order-preserving join2. GROUP BY clause – groups are defined

based on the value of the Id column3. SELECT clause – functions are applied;

non-grouped columns become vector fields so that target cardinality is met. Violates first normal form

groups in ID and non-grouped column

grouped ID and non-grouped column

Vector

field

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:1. 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.

2. 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)

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

Interchange sorting + order preserving operators

(1) Sort then joinpreserving order

(2) Preserve existingorder

(3) Join then sortbefore grouping

op

sort

g-by

avgs

op avgs

g-by

op

op

avgs

g-byop

sort

(4) Join then sortafter grouping

avgs

g-by

sort

TransformationsEarly sorting + order preserving

operators

0

2040

60

80

100120

140

110

020

030

040

050

058

1

Number of traded Securities (total of 581 securities and 127062 quotes)

Tim

e in

mil

isec

on

ds

Sort before op join

Existing order

Sort after a reg join

Sort after reg join andg-by

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

dist

posA

(2)(1) (3)

Switch dynamically

between (1) and (2)

depending on the

execution history

TransformationsUDFs Evaluation Order

110

1001000

10000100000

1000000

10 100 1000

Number of sequences

Tim

e in

mili

se

co

nd

s

dist then pos

pos then dist

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

Original Query

SELECT last(price)FROM ticks t,base b ASSUMING ORDER name, timestampWHERE t.ID=b.ID AND name=“x”

sort

baseticks

ID

name,timesamp

Name=“x”

last(price)

Last price for a name query

• The sort on name can be eliminated because there will be only one name

• Then, push sort– sortA(r1 r2) A sortA(r1) lop r2

– sortA(r) A r

sort

base

ticks

ID

name,timesamp

Name=“x”

last(price)


• 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))

base

ticks

ID

Name=“x”

last(price)

lop

pos()=last

price


• But why join the entire relation if we are only using the last tuple?

• Can we somehow push the last selection down the join?

base

ticks

ID

pos()=last

Name=“x”

price

lop


• 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”

base

ticks

ID

safety

Name=“x”

price

lop

pos()=last

GbyID

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

pricexyz

> scalar =

resultTFT

pricexyz

-

prev(price)-xy

=

result-

y-xz-y

Built-in functions

• prevv[i] = null, if i=0 or v[i-1], for 0<i|v|-1

• firstv,n[i] = v[i], for 0i<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!

AQuery syntax and execution

• It’s just SQL, really. But column-oriented.SELECTFROMASSUMING ORDERWHEREGROUP BYHAVING


• It’s just SQL, really. But column-oriented.SELECT

FROMASSUMING ORDERWHEREGROUP BYHAVING

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



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



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.



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



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.



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.

Translating queries into Operations

SELECT ID, index(flags=‘FIN’)-index(flags=‘ACK’)

FROM netmgmt

ASSUMING ORDER timestamp

GROUP BY ID

Gby

sort

netmgmt

Order-preservingportion of

the plan

each

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

Pick the best

• Early sort and order-preserving group by

• Early, non-order preserving group by and sort

GbysessionID

sorttimestamp

netmgmt

GbysessionID

sorttimestamp

netmgmt

each

eacheach

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 A r2 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 sortA

if 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 joinsselect lineitem.orderid, avgs(10, lineitem.qty), lineitem.lineidfrom order, lineitem assuming order lineidwhere order.date > 45and 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.

aquery: a database system for order dennis shasha, joint work with alberto lerner...

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