Progressive Approximate Aggregate Queries with a Multi-Resolution Tree Structure

Iosif Lazaridis, Sharad MehrotraUniversity of California, Irvine

SIGMOD 2001, Santa Barbara

Talk Outline

Aggregate QueriesMotivation for Approximate AnsweringMulti-Resolution Aggregate Tree (MRA-

Tree)Progressive Algorithm with Error BoundsExperimental EvaluationSummary and Future Work

Aggregate Queries

9

6

3

8

27

Q

S

minQ = 2

maxQ = 7

countQ = 3

sumQ = 2+7+6 = 15

avgQ = 15/3 = 5

Evaluating Aggregate Queries

Exact answering Scan all points of D checking each against Q Retrieve points in Q via a multi-dimensional index

on D

Both linear/index scan can be very expensive

Approximate answering Many applications (selectivity estimation, data

analysis, visualization) do not require exact answers

Motivating Examples

My boss needs to see theincome aggregates

in 10 minutes!

How many tanks 10 miles from me?

Boss

Techniques for Approximate Aggregate Queries

Online estimation (Interactive) Sampling

Offline estimation (Data Synopsis) Sampling, Histograms, Wavelets

Our Technique: Online estimator via a scan of a modified

multi-dimensional index (MRA-Tree) Allows incremental tradeoff of accuracy for

response time, with guaranteed error bounds

Multi-Resolution Aggregate Tree (MRA-Tree)

An MRA-Tree can be instantiated with any of the popular multi-dimensional index trees (R-Tree, quadtree, Hybrid tree, etc.)

A non-leaf node contains (for each of its subtrees) four aggregates {MIN,MAX,COUNT,SUM}

A leaf node contains the actual data pointsTree operations are identical with those of the

plain (non-MRA) tree with the consideration that aggregates must be maintained

MRA-Tree Example

min

max

count

sum

Non-Leaf Node

Leaf Nodes

2 4

3

54

2

3

9

4

4

2

9

5

1

4

4

2

6

1

6

6

1

2

2

1

6

Progressive Algorithm Outline

We want Best answer for given time Shortest time for given precision of the answer Refine an answer at will, trading time for precision

How we achieve it Do a prioritized traversal of nodes of the MRA-

tree Maintain an estimate of the answer E(aggQ) Maintain a 100% interval of confidence I = [L,

H], such that L aggQ H

Generic Algorithm (1)

QN

disjoint

contains

QNQN

is contained

QN

partially overlaps

Two sets of nodes: NP (partial contribution to the query) NC (complete contribution)

Generic Algorithm (2)

Initialize NP with the root At each iteration: Remove one node N from NP

and for each Nchild of its children

discard, if Nchild disjoint with Q

insert into NP if Q is contained or partially overlaps with Nchild

“insert” into NC if Q contains Nchild (we only need to maintain aggNC)

QN

Generic Algorithm (3)

Node in NP

Node in NC

To instantiate the algorithm for {MIN,MAX,COUNT,SUM,AVG}:

Error Bounds. Interval I=[L, H] : L aggQ

H

Traversal Policy. Which node from NP to

explore next? Minimize |I|

Estimation. Provide an estimate of the

answer: E(aggQ)

MIN (and MAX)

3

9

4

5

Interval

minNC = min { 4, 5 } = 4

minNP = min { 3, 9 } = 3

L = min {minNC, minNP} = 3

H = minNC = 4

hence, I = [3, 4]

Estimate

Lower bound:

E(minQ) = L = 3

Traversal

Choose N NP:

minN = minNP

COUNT (and SUM)

10

20%

25%

8

6

9

Interval

countNC = 9+6 = 15

countNP = 8+10 = 18

L = countNC = 15

H = countNC + countNP = 33

hence, I = [15, 33]

Estimate

E(countQ) = L + 0.258 + 0.210 = 19

Traversal

Choose N NP: countNcountM, M NP

AVG

A

B

min max count sumA 5 10 5 35 B – – 10 55

Interval

Current avgNC = 55/10 = 5.5

10

5

5

5

10

Estimate

E(avgQ) = E(sumQ)/ E(countQ)

Traversal

– max countN

– max (maxN-avgNC), (avgNC-minN)

Distribution of Values {5, 5, 5, 10, 10}

Maximum possible: (55+210) / (10+2) = 6.25

Minimum possible: (55+35) / (10+3) = 5.38

hence, I = [5.38, 6.25]

Experiments

Synthetic datasets 2-4DReal datasets: 2D spatial (USGS) and 4D

(UCI KDD Forest Cover)MRA-quadtree and MRA-Rtree indicesWe study

MRA-tree Vs. “plain” tree MRA-tree Vs. online sampling Accuracy of estimation Scalability with database size

MRA-Quadtree (Nodes Visited)

MRA-Quadtree Vs. "plain" Quadtree (2D, USGS, exact, COUNT)

02000

40006000

800010000

12000

0 10 20 30

Spatial Selectivity (% space)

No

de

s V

isit

ed

Quadtree

MRA-Quadtree

MRA-Quadtree (Error Reduction)

Absolute Relative Error Vs. Nodes Visited(2D, USGS, COUNT, 25% spatial sel.)

00.20.40.6

0.81

1.21.4

0 100 200 300 400 500 600

Nodes Visited

Ab

so

lute

Re

lati

ve

Err

or

Absolute Relative Error = )aggmax(1,

|)E(aggagg|Q

QQ

MRA-Rtree (2D, USGS) I/O Performance

0

5

10

15

20

25

0 5 10 15Selectivity (% space)

Pa

ge

I/O

s (

% D

B S

ize

)

MRA-Rtree (exact)

R-Tree

MRA-Rtree (10%max. rel. err.)

DB Size = size(page)

)size(pointpoints#

Estimation vs. Maximum Error (4D, Forest Cover, sel. 16% / axis)

0.0001

0.001

0.01

0.1

1

10

100

1000

0 500 1000 1500

Page I/Os

Abs

olut

e R

elat

ive

Err

or

Estimation Error

Maximum PotentialError

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

50 100 150 200 250

Page I/Os

Abs

olut

e R

elat

ive

Err

or

MRA-Rtree

Online Sampling

MRA-Rtree vs. Online SamplingEstimation Accuracy (4D, Forest Cover)

Database Size (3D Synthetic, exact, 10% spatial sel.)

5

7

9

11

13

15

17

19

21

0 200 400 600 800 1000 1200 1400Number of Points (x1000)

Pag

e I/

Os

(% D

B S

ize)

Summary

MRA-Tree is a modified multi-dimensional index for approximate answering of aggregate queries

For exact answer faster than “plain” index

Advantages over offline estimators Progressively improving answers Error bounds

Advantages over sampling Better estimate for same I/O

Algorithm scales gracefully with database size

Future Work (QUASAR Project, UC Irvine)

Scalability with high dimensionality, by using a dedicated high-D index structure

Scalability in high update rate environments Approximate query processing of general SQL

queries using dedicated data structures, similar to MRA-tree