using trees to depict a forest

Using Trees to Depict a Forest

Bin Liu, H. V. JagadishEECS, University of Michigan, Ann Arbor

1

Motivation – Too Many Results

• In interactive database querying, we often get more results than we can comprehend immediately

• Try search a popular keyword

• When do you actually click over 2-3 pages of results?– 85% of users never go to the second page [1,2]

2

Why IR Solutions Do NOT Apply

• Sorting and ranking are standard IR techniques– Search engines show most relevant hits in the first

page• However, for a database query, all tuples in

the query result set are equally relevant– For example, Select * from Cars where price < 13,000– All matching results should be available to user– What to do when there are millions of results?

3

Make the First Page Count

• If no user preference information available, how to best arrange results?– Sort by attribute?– Random selection?– Others?

• Show the most “representative” results– Best help users learn what is in the result set– User can decide further actions based on

representatives4

Our Proposal – MusiqLens Experience

5

6

Suppose a user wants a 2005 Civic but there are too many of them…

7

MusiqLens on the Car DataID MODEL PRICE YEAR MILEAGE CONDITION

872 Civic $12,000 2005 50,000 Good 122 morelike this

901 Civic $16,000 2005 40,000 Excellent 345 morelike this



132 Civic $9,500 2005 86,000 Fair 185 morelike this


8

MusiqLens on the Car DataID MODEL PRICE YEAR MILEAGE CONDITION





132 Civic $9,500 2005 86,000 Fair 185 morelike this


9

Zooming in: 2005 Honda Civics ~ ID 132

ID MODEL PRICE YEAR MILEAGE CONDITION342 Civic $9,800 2005 72,000 Good 25 more

like this768 Civic $10,000 2005 60,000 Good 10 more

like this132 Civic $9,500 2005 86,000 Fair 63 more




like this

10

Now Suppose User Filters by “Price < 9,500”

ID MODEL PRICE YEAR MILEAGE CONDITION342 Civic $9,800 2005 72,000 Good 25 more






like this

11

ID MODEL PRICE YEAR MILEAGE CONDITION123 Civic $9,100 2005 81,000 Fair 40 more






like this

After Filtering by “Price < 9,500”

Challenges

• Metric challenge– What is the best set of representatives?

• Representative finding challenge– How to find them efficiently?

• Query challenge– How to efficiently adapt to user’s query

operations?

12

Finding a Suitable Metric

• Users should be the ultimate judge– Which metric generates the representatives

that I can learn the most from

• User study– Use a set of candidates– Users observe the representatives– Users estimate more data points in the data– Representatives lead to best estimation wins

13

Metric Candidates

• Sort by attributes• Uniform random sampling• Density-biased sampling [3]• Sort by typicality [4]• K-medoids

– Average– Maximum

14

Density-biased Sampling

• Proposed by C. R. Palmer and C. Faloutsos [3]• Sample more from sparse regions, less from

dense regions• To counter the weakness of uniform sampling

where small clusters are missed

15

Sort by Typicality

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Proposed by Ming Hua, Jian Pei, et al [4]

Figure source: slides from Ming Hua

Metric Candidates - K-medoids

• A medoid of a cluster is the object whose average or maximum dissimilarity to others is smallest– Average medoid and max medoid

• K-medoids are k objects, each from a cluster where the object is the medoid

• Why not K-means– K-means cluster centers do not exist in

database– We must present real objects to users

17

C

Plotting the Candidates

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.10.20.30.40.50.60.70.80.9

1Random

Data: Yahoo! Autos, 3922 data points. Normalized price and mileage to 0-1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.10.20.30.40.50.60.70.80.9

1Density Biased

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Typical

Plotting the Candidates - Typicality

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Plotting the Candidates – k-medoids

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max-Medoids

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Avg-Medoids

User Study Procedure

• Users are given– 7 sets of data, generated using the 7 candidate methods– Each set consists of 8 representative points

• Users predict 4 more data points– That are most likely in the data set– Should not pick those already given

• Measure the predication error

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22

Predication Quality Measurement

P1

P2

D1

D2

So

For data point So:MinDist: D1

MaxDist: D2

AvgDist: (D1+D2)/2

Performance – AvgDist and MaxDist

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0

0.2

0.4

0.6

0.8

1

1.2AvgDist MaxDist

For AvgDist: Avg-Medoid is the winner.

For MaxDist: Max-Medoid is the winter.

Performance – MinDist

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Random

Avg-M

ed

Sort-

Mile

Density

Sort-

Price

Max-Med

Typica

l0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Avg-Medoid seems to be the winner

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Verdict

• Although result is insignificant in MinDist, overall AvgMeoid is better than Density

• Based on AvgDist and MinDist: Avg-Medoid• Based on MaxDist: Max-Medoid• In this paper, we choose average k-medoids

– Our algorithm can extend to max-medoids with small changes

• Statistical Significance of Result

Challenges




operations?

26

Cover Tree Based Algorithm

• Cover Tree was proposed by Beygelzimer, Kakade, and Langford in 2006 [5]

• Briefly discuss Cover Tree properties• Cover Tree based algorithms for computing k-

medoids

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Cover Tree Properties (1)

28Figure modified from slides of Cover Tree authors

Ci

Ci+1

Points in the Data (One Dimension)

Nesting: for all i, 1 ii CCAssume all pair-wise distance <= 1.


29

Ci

Ci+1

Covering: node in Ci is within distance of to its children in Ci+1

i2/1

Distance from node to any descendant is less than This value is called the “span” of the node.

12/1 i


30Figure modified from slides of Cover Tree authors

Ci

Ci+1

Points in the Data

Separation: nodes in Ci are separated by at least i2/1

s1s2

s10

s8s6

s7

s3

s5

s3 s8s5

s6s1 s2 s7

s8s5

s9s4s5 s8

s9

s5

s4

s3

s2

s10

s7s3

Additional Stats for Cover Tree (2D Example)

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Density (DS): number of points in the subtree

DS = 10

DS = 3

Centroid (CT): geometric center of points in the subtree

p

k-medoid Algorithm Outline

• We descend the cover tree to a level with more than k nodes

• Choose an initial k points as first set of medoids (seeds)– Bad seeds can lead to local minimums with a high

distance cost• Assigning nodes and repeated update until

medoids converge

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Cover Tree Based Seeding

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• Descend the cover tree to a level with more than k nodes (denote as level m)

• Use the parent level (m-1) as starting point for seeds– Each node has a weight, calculated as product of

span and density (the contribution of the subtree to the distance cost)

– Expand nodes using a priority queue– Fetch the first k nodes from the queue as seeds

A Simple Example: k = 4

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s1s2

s10

s8s6

s7

s3

s5

s3 s8s5

s6s1 s2 s7

s8s5

s9s4s5 s8

s9

s5

s4

s3

s2

s10

s7s3

Span = 2

Span = 1

Span = 1/2

Span = 1/4

Priority Queue on node weight (density * span):

S3 (5), S8 (3), S5 (2)

S8 (3/2), S5 (1), S3 (1), S7 (1), S2 (1/2)

Final set of seeds

Update Process

1. Initially, assign all nodes to closest seed to form k clusters

2. For each cluster, calculate the geometric center

• Use centroid and density information to approximate subtree

3. Find the node that is closest to the geometric center, designate as a new medoid

4. Repeat from step 1 until medoids converge35

Challenges




operations?

36

Query Adaptation

• Handle user actions– Zooming– Selection (filtering)

• Zooming– Expand all nodes assigned to the medoid– Run k-medoid algorithm on the new set of nodes

37

Selection

• Effect of selection on a node– Completely invalid– Fully valid– Partially valid

• Estimate the validity percentage (VG) of each node

• Multiply the VG with weight of each node

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50

150

A

Mileage

S1

S2

S3 S4

S5

S6

S7

a

Pric

e

1200030

201

4557

90b

Experiments – Initial Medoid Quality

• Compare with R-tree based method [6]• Data sets

– Synthetic dataset: 2D points with zipf distribution– Real dataset: LA data set from R-tree Portal, 130k

points• Measurement

– Time to compute the medoids– Average distance from a data point to its medoid

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Results on Synthetic Data

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256K 512K 1024K 2048K 4096K-1.73472347597681E-18

0.002

0.004

0.006

0.008

0.01

R-tree

Cover Tree

Cardinality

Tim

e (s

econ

ds)

256K 512K 1024K 2048K 4096K0

100

200

300

400

500

600

700

800

R-tree

Cover Tree

CardinalityDi

stan

ce

For various sizes of data, Cover-tree based method outperforms R-tree based method

Time Distance

Result on Real Data

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2 8 32 128 5120

200

400

600

800

1000

1200

1400

1600

R-tree

Cover Tree

k

Dist

ance

2 8 32 128 5120

0.01

0.02

0.03

0.04

0.05

0.06

R-tree

Cover Tree

k

Tim

e (s

econ

ds)

For various k values, Cover-tree based method outperforms R-tree based method on real data

Query Adaptation

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0.8 0.6 0.4 0.20

100

200

300

400

500

600Re-Compute

Incremental

Selectivity

Dist

ance

0.8 0.6 0.4 0.20

50

100

150

200

250

300

350Re-Compute

Incremental

Selectivity

Dist

ance

Synthetic Data Real Data

Compare with re-building the cover tree and running the k-medoid algorithm from scratch.

Time cost of re-building is orders-of-magnitude higher than incremental computation.

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Related Work

• Classic/textbook k-medoid methods– Partition Around Medoids (PAM) and Clustering LARge

Applications (CLARA), L. Kaufman and P. Rousseeuw, 1990– CLARANS, R. T. Ng and J. Han, TKDE 2002

• Tree-based methods– Focusing on Representatives (FOR), M. Ester, H. Kriegel,

and X. Xu, KDD 1996– Tree-based Partitioning Querying (TPAQ), K. Mouratidis, D.

Papadias, and S. Papadimitriou, VLDBJ 2008

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Related Work (2)• Clustering methods

– For example, BIRCH, T. Zhang, R. Ramakrishnan, and M. Livny, SIGMOD 1996

• Result presentation methods– Automatic result categorization, K.Chakrabarti,

S.Chaudhuri, and S.wonHwang, SIGMOD 2004– DataScope, T. Wu, et al, VLDB 2007

• Other recent work– Finding representative set from massive data, ICDM 2005– Generalized group by, C. Li, et al, SIGMOD 2007– Query result diversification, E. Vee et al., ICDE 2008

Conclusion

• We proposed MusiqLens framework for solving the many-answer problem

• We conducted user study to select a metric for choosing representatives

• We proposed efficient method for computing and maintaining the representatives under user actions

• Part of the database usability project at Univ. of Michigan– Led by Prof. H.V. Jagadish– http://www.eecs.umich.edu/db/usable/

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Thank you.

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Bin Liu,[email protected]

Questions?

References[1] E. Agichtein, E. Brill, S. T. Dumais, and R. Ragno, Learning user interaction models

for predicting web search result preferences. SIGIR, 2006[2] B. J. Jansen and A. Spink. How are we searching the world wide web? a comparison

of nine search engine transaction logs. Inf. Process. Manage., 42(1), 2006[3] C. R. Palmer and C. Faloutsos. Density biased sampling: An improved method for

data mining and clustering. In SIGMOD Conference, 2000[4] M. Hua, J. Pei, A. W.-C. Fu, X. Lin, and H. Fung Leung. Efficiently answering top-k

typicality queries on large databases. In VLDB, pages 890{901, 2007.[5] A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In

ICML, 2006.[6] K. Mouratidis, D. Papadias, and S. Papadimitriou. Tree-based partition querying: a

methodology for computing medoids in large spatial datasets. VLDB J., 17(4):923-945, 2008.

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using trees to depict a forest

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