Using Trees to
Depict a Forest
Bin Liu, H. V. Jagadish
EECS, University of
Michigan, Ann Arbor
1
Presented by
Sergey Shepshelvich
2
Motivation
 In
interactive database querying, we often get
more results than we can comprehend
immediately
 When
do you actually click over 2-3 pages of
results?


85% of users never go to the second page!
What to display on the first page?
3
Standard solutions
 Sorting


by attributes
Computationally expensive
Similar results can be distributed many pages
apart
 Ranking

Hard to estimate of the user's preference.
 In
database queries, all tuples are equally
relevant!

What to do when there are millions of results?
4
Make the First Page Count
 Human
beings are very capable of learning
from examples
 Show


the most “representative” results
Best help users learn what is in the result set
User can decide further actions based on
representatives
5
The Proposal:
MusiqLens Experience
(Model-driven Usable Systems for Information Querying)
6
Suppose a user wants a 2005 Civic
but there are too many of them…
7
MusiqLens on the Car Data
Id
Model
Price
Year Mileage Condition
872
Civic
$12,000
2005
50,000
Good
901
Civic
$16,000
2005
40,000
Excellent
725
Civic
$18,500
2005
30,000
Excellent
423
Civic
$17,000
2005
42,000
Good
132
Civic
$9,500
2005
86,000
Fair
322
Civic
$14,000
2005
73,000
Good
122 more like
this
345 more like
this
86 more like
this
201 more like
this
185 more like
this
55 more like
this
8
MusiqLens on the Car Data
Id
Model
Price
Year Mileage Condition
872
Civic
$12,000
2005
50,000
Good
901
Civic
$16,000
2005
40,000
Excellent
725
Civic
$18,500
2005
30,000
Excellent
423
Civic
$17,000
2005
42,000
Good
132
Civic
$9,500
2005
86,000
Fair
322
Civic
$14,000
2005
73,000
Good
122 more like
this
345 more like
this
86 more like
this
201 more like
this
185 more like
this
55 more like
this
9
After Zooming in:
2005 Honda Civics ~ ID 132
Id
Model
Price
Year Mileage Condition
342
Civic
$9,800
2005
72,000
Good
768
Civic
$10,000
2005
60,000
Good
132
Civic
$9,500
2005
86,000
Fair
122
Civic
$9,500
2005
76,000
Good
123
Civic
$9,100
2005
81,000
Fair
898
Civic
$9,000
2005
69,000
Fair
25 more like
this
10 more like
this
63 more like
this
5 more like
this
40 more like
this
42 more like
this
10
After Filtering by “Price < 9,500”
Id
Model
Price
Year Mileage Condition
123
Civic
$9,100
2005
81,000
Fair
898
Civic
$9,000
2005
69,000
Fair
133
Civic
$9,300
2005
87,000
Fair
126
Civic
$9,200
2005
89,000
Good
129
Civic
$8,900
2005
81,000
Fair
999
Civic
$9,000
2005
87,000
Fair
40 more like
this
42 more like
this
33 more like
this
3 more like
this
20 more like
this
12 more like
this
11
Challenges
 Representation
metric

What is the best set of representatives?
 Representative

finding
How to find them efficiently?
 Query

Modeling: finding a suitable
Refinement
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 to evaluate different representation
modeling
14
Metric Candidates
 Sort
by attributes
 Uniform random sampling

Small clusters are missed
 Density-biased

Sample more from sparse regions, less from dense
regions
 Sort

sampling
by typicality
Based on probabilistic modeling
 K-medoids
16
Metric Candidates - K-medoids

A medoid of a cluster is the object
whose dissimilarity to others is smallest



Average medoid and max medoid
K-medoids are k objects, each from a
different 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
Plotting the Candidates
Data: Yahoo! Autos, 3922 data points. Price and mileage are
normalized to 0..1
1
1
Random
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.5
Density
Biased
0.9
1
0
0.5
1
18
Plotting the Candidates - Typicality
1
0.9
Typical
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
19
Plotting the Candidates –
k-medoids
1
1
0.9
Max-Medoids
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
Avg-Medoids
0
0.2
0.4
0.6
0.8
1
20
User Study Procedure
 Users


7 sets of data, generated using the 7 candidate
methods
Each set consists of 8 representative points
 Users


are given:
predict 4 more data points
That are most likely in the data set
Should not pick those already given
 Measure
the predication error
21
Verdict
 K-meoids
is the winner
 In this paper, authors choose average kmedoids

Proposed algorithm can be extended to maxmedoids with small changes
22
Challenges
 Representation
metric

What is the best set of representatives?
 Representative

finding
How to find them efficiently?
 Query

Modeling: finding a suitable
Refinement
How to efficiently adapt to user’s query
operations?
23
Cover Tree Based Algorithm
 Cover
Tree was proposed by Beygelzimer,
Kakade, and Langford in 2006
 Briefly discuss Cover Tree properties
 See Cover Tree based algorithms for computing
k-medoids
24
Cover Tree Properties (1)
Nesting: for all 𝑖, 𝐶𝑖 ⊂ 𝐶𝑖+1
𝐶𝑖
𝐶𝑖+1
Points in the Data (One Dimension)
25
Cover Tree Properties (2)
Covering: node in 𝐶𝑖 is within distance of 1 2𝑖 to its
children in 𝐶𝑖+1
𝐶𝑖
𝐶𝑖+1
Distance from node to any descendant is less than
1 2𝑖−1 . This value is called the “span” of the node.
26
Cover Tree Properties (3)
Separation: nodes in 𝐶𝑖 are separated by at least 1 2𝑖
𝐶𝑖
𝐶𝑖+1
Note: 𝑖 allowed to be negative to satisfy above conditions.
27
Additional Stats for Cover Tree
(2D Example)
s5
s2
s1
s3
s7
s6
s4
p
s5
s8
s10 s9
s3
s3 s2 s7
DS = 10
s5
s8
s5
s8
DS = 3
s1 s3 s2 s6 s7 s5 s4 s8 s9 s10
Density (DS): number of points in the subtree
Centroid (CT): geometric center of points in the subtree
28
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
29
Cover Tree Based Seeding
 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
30
A Simple Example: k = 4
s5
s2
s1
s3
s7
s6
s3
s4
s5
s8
s10 s9
s3 s2 s7
Span = 2
s5
s8
Span = 1
s5
s8
Span = 1/2
s1 s3 s2 s6 s7 s5 s4 s8 s9 s10
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
Span = 1/4
31
Update Process
1.
2.
Initially, assign all nodes to closest seed to form
𝑘 clusters
For each cluster, calculate the geometric
center
 Use
centroid and density information to approximate
subtree
3.
4.
Find the node that is closest to the geometric
center, designate as a new medoid
Repeat from step 1 until medoids converge
32
Challenges
 Representation
metric

What is the best set of representatives?
 Representative

finding
How to find them efficiently?
 Query

Modeling: finding a suitable
Refinement
How to efficiently adapt to user’s query
operations?
33
Query Adaptation
 Handle


user actions
Zooming
Selection (filtering)
34
Zooming
 Zooming


Expand all nodes assigned to the medoid
Run k-medoid algorithm on the new set of nodes
35
 Effect
node



of selection on a
Completely invalid
Fully valid
Partially valid
 Estimate
the validity
percentage (VG) of each
node
 Multiply the VG with
weight of each node
Price
Selection
A
50
12000
S1
30
150 S
3
57 S5
S2
45 S4
a
b
201
90 S6
S7
Mileage
37
Experiments – Initial Medoid
Quality
 Compare
with R-tree based method by M. Ester,
H. Kriegel, and X. Xu
 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
38
Results on Synthetic Data
Distance
Time
800
0.01
700
0.008
0.006
0.004
R-tree
Cover Tree
0.002
Distance
Time (seconds)
600
500
400
R-tree
300
Cover Tree
200
100
0
0
256K
512K
1024K
Cardinality
2048K
4096K
256K
512K
1024K
2048K
Cardinality
For various sizes of data, Cover-tree based
method outperforms R-tree based method
4096K
39
Results on Real Data
0.06
R-tree
0.05
1400
R-tree
1200
Cover Tree
1000
800
600
400
Time (seconds)
Distance
1600
Cover Tree
0.04
0.03
0.02
0.01
200
0
0
2
8
32
k
128
512
2
8
32
128
k
For various k values, Cover-tree based method
outperforms R-tree based method on real data
512
40
Query Adaptation
Compare with re-building the cover tree and running the k-medoid
algorithm from scratch.
Synthetic Data
Real Data
600
350
Re-Compute
Incremental
400
300
200
Re-Compute
300
Incremental
250
Distance
Distance
500
200
150
100
100
50
0
0
0.8
0.6
0.4
Selectivity
0.2
0.8
0.6
0.4
Selectivity
Time cost of re-building is orders-of-magnitude higher
than incremental computation.
0.2
41
Conclusion
 Authors
proposed MusiqLens framework for
solving the many-answer problem
 Authors conducted user study to select a metric
for choosing representatives
 Authors 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/