Composite Subset Measures

Lei Chen, Paul Barford, Bee-Chung Chen, Vinod Yegneswaran

University of Wisconsin - Madison

Raghu RamakrishnanYahoo! Research and University of Wisconsin – Madison

09.12.2006

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Motivation

Consider this query: “For each year and each country, compute the ratio of the

average personal incomes between richest city and poorest city . Then find the number of countries where such ratio continuously decrease between 1990-2000“

It is Hard to write in SQL Hard to optimize/understand the SQL query

This kind of queries is increasingly common: Multi-step aggregation Must scale to very large datasets, often distributed

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Contributions

A new framework for expressing such compositional aggregate queries Key contribution is how we look at the

computation, in terms of aggregating over related regions in “cube space”

An efficient evaluation framework based on sorted scans that take into account of multiple aggregation steps Experimental results

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Background Computing “measures”

Measures summarize some characteristic of data subsets (e.g., SUM, std dev, beta-value of a portfolio)

Approaches: Group by, data cubes, Hancock, Sawzall

Cube space Partition feature space using attribute values; domain

hierarchies organize this space into nested collections of regions

Regions: (2006, Korea), (2006/09, Seoul) Region sets: (Year, Country), (Month, City)

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Composite Subset Measures

The measure of a cube region is computed by: Aggregating data in a region directly (e.g.,

sales volumes for each day), or Summarizing the measures for related

regions, e.g.: The maximum of daily volumes within a year The ratio of average personal incomes

between the richest and poorest cities in a country

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What is “Related” in Cube Space

Focus on relationships which are commonly used can be efficiently evaluated

Self Parent/Child

E.g., Year/Day

Child/Parent E.g., Day/Year

Sibling E.g., Today/Tomorrow

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Examples (Network Analysis)

Data involved: Stream of data records for IP packet information Time (t), Source (U), Destination (D) , Size (s)

Queries: For every minute, the number of outgoing packets

from each given source IP For every hour, the maximum number of minutely

outgoing packets from a given source IP

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Expression Algebra

Each measure entity is defined as a collection of region/value pairs Regions should belong to same region set

Fact Table Aggregation Selection Match join Combine join

( )cond T

, ( )G aggg T

,| cond aggS T

1 2( , ,..., )fc nS T T T

D

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Example: Aggregation

For every hour and every unique IP, compute the number of outgoing packets

( : , : ), (*)C t hour U IP countS g D

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Example: Selection

For every hour, compute the sum of outgoing packets from those source IP with at least five packets in that hour (High traffic count)

( : ), (*) 5( )S t hour count M CS g S

time

Source

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Example: Match

For each six hour time window, compute the average of the high traffic count

1 2 2 21 2

( . [ . , . 6]), ( . )|

S S S Savg S SS t S t S t avg S M

S S S

1SS

2SS

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Example: Combine

For each hour, compute the ratio between the six hour average and the high traffic count

. / .avg Sratio avg S M S M SS S S

SS

avgS

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Aggregation Workflows

A diagrammatic way to express multiple composite subset measure expressions Semantically equivalent to the algebra

Rectangles: Region sets Ellipses: Measures associated with the

Region sets Arcs: Computational dependencies

among measures

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Example

U:IPt:hour

Countcount(*)

Savg

Avg(Count)

Count.t=Sbase.t

t:hour

Region set

Measure name

Aggregation formulaSelection condition

Match condition

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Example (cont.)

U:IPt:hour

MAXSmax(s)

MINSmin(s)

RatioMAXS/MINS

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Multi-step Execution Plan Evaluation based on the topology order

of the aggregation workflow Materialize non-dependent measures Then evaluate dependent measures

following the arcs of the aggregation workflow

May need to perform join Problem

Intermediate measures: extra I/O

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Simple Scan Execution[*]

Build one hash table for each measure “Insert” data into hash tables of low-level

measures Propagate the measures upwards after the

scan is over Distributive or algebraic aggregation function Problem

Each hash table keeps all the entries Bottleneck: Memory capacity

[*] T. Johnson and D. Chatziantoniou, Extending complex ad-hoc OLAP, in CIKM, 1999, 170-179.

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Sort/Scan Execution Simple scan requires large memory

For each hash table, we need to keep all the entries during the scan

When the data is ordered Some hash entries can be flushed out before

the scan is finished The memory footprint can be reduced One pass scan becomes feasible CPU cost is reduced

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Evaluation

t:Day t:DayU:IP

t:MonthU:IP

Sort by day

month 1 month 2

Output stream for each hash table is still ordered!

COUNT0count(*)

COUNT2count(*)

COUNT3count(*)

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Evaluation

t:Day t:DayU:IP

t:MonthU:IP

Sort by month

month 1 month 2

COUNT0count(*)

COUNT2count(*)

COUNT3count(*)

All the output stream is ordered by month!

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Evaluation

t:MonthU:IP

Data are sortedby (t:month, U:IP)

month 1 month 2

COUNT3count(*)

1 1 1 2

By carefully choosing the sort order of the raw data, we can greatly reduce the memory footprint

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Order and Slack Order

How the records are ordered in the stream E.g., <t:day, U:IP>

Slack The gap between the output stream of the measure

and the scan progress of raw data E.g., <t:day:[-3,+3]>

We have developed a mechanism to Calculate the order/slack Take advantage of the order/slack information during

evaluation

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Evaluation Network

M2

M3 M4

M1

order key:<t:Day, U:IP>slack: t:[-1,+1]

order key:<t:Hour, U:IP>slack: t:[-1,+1]order key:<t:Hour, U:IP>

slack: <>

hashtables

Scan sorted data

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Optimization How to find a good sort order?

Enumerate all possible orders For each order estimate the memory usage Use sort orders with minimal usage

Evaluation with multiple passes What measure to compute during each

pass? What order to use in each pass?

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Experiments 64 million records Synthetic data set Scenario 1

The measures of a region are computed by combining the aggregated measures for different kinds of child region sets

Scenario 2 The measures of a region are computed by

aggregating the measures of multiple chained siblings

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Experimental Results (cont.)

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9000

2 3 4 5 6

#dependent child measures

Ex

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DB

SortScan

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Experimental Results

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2 3 4 5 6 7

Size of the Sibling Chain

Ex

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SortScan

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Conclusions Composite measures as building blocks for

complicated analysis process Algebra provides the semantic foundation Aggregation workflow offers intuitive interface Sort/Scan execution plan evaluates multiple

dependent measures in the same run and hence improve the evaluation performance