SUBSKY: Efficient Computation of Skylines in Subspaces

1
SUBSKYEfficient Computation of Skylines in
Subspaces

• Authors Yufei Tao, Xiaokui Xiao, and Jian Pei
• Conference ICDE 2006
• Presenter Kamiru
• Superviosr Dr. Nikos Mamoulis

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Skyline Queries

• Given a set of d-dimenional points, a point p
  dominates another p if
• piltpi, for all i in d,
• and pjltpj, for any j in d
• Skyline queries aim to find the points that are
  not dominated by any point

turnover rate
1
For the NBA database, Low turnover rate and low
foul rate are two important factors for a defense
player
Best point
foul rate
0
1
player
3
Applications of Skyline Queries

• Find a good hotel to me according to distance and
  price

Hotel D must not a good hotel for this user,
since its price is higher and distance is farther
than other hotels
price 2000
price 500
D
C
B
price 1000
A
price 1500
4
Alternative applications of Skyline Queries - i

• Some top-k queries are calculated by Skyline
  queries
• A top-k query retrieves the k tuples in P with
  highest scores according to g
• where g must be a monotonic function, ex
• g(p) p.x p.y

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Alternative applications of Skyline Queries - i

• Please help me to find who are the top-2 NBA
  players according to sum of their points and
  assists in 2007-2008 season

The values are represented by right-top corner of
each player photo
points
PRUNED
20
Top-2 results must be in top-2 skyband
assists
10
10
0
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Alternative applications of Skyline Queries - ii

• Another interesting measurement is dominating
  count (DC)
• DC is counted by the number of dominating points
  to a query

turnover rate
1
Ex find the top-2 dominating players in the NBA
database according to turnover rate and fould rate
1
1
2
4
0
Best point
foul rate
0
1
player
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Skyline Computations

• Two categories of skyline computations
• Computing from scratch (no index)
• Relied on index
• Computing from scratch (no index)
• Advantage
• No any pre-computation
• Not to update any index when data changed
• Drawback
• Must calculate from scratch
• It must scan the entire data at least once

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Skyline Computations

• Relied on index
• Once you built, get to use it many times
• Lower query cost is occurred by performing the
  search on an appropriate structure
• B - tree
• R - tree
• Since all of us are database people, (I hope) we
  prefer method 2 more

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Related works

• Computing from scratch (no index)
• Block nested loop
• Sort filter skyline
• Divide and conquer
• Bitmap
• Linear elimination sort for skyline

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Related works

• Relied on index
• B tree approach
• index
• R tree approach
• Nearest neighbor (NN)
• Branch and bound skyline (BBS)
• BBS has been proved that is I/O optimal. It
  accesses fewer disk pages than any algorithm
  based on R-trees

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Related works

• index

Point p adds to list i if p has the smallest
value in dimension i
y
List x p50.1 p60.25 p20.3 p70.6
p5
p7
p8
List y p40.1 p10.2 p30.3 p80.6
p6
p2
1) Ssky p5
p3
2) Ssky p5,p4
p1
p4
x
3) Ssky p5,p4,p1
Best point

• All remaining elements in List x are pruned by p1
  since both coordinates of p6 is bigger than p1
• Due to the same reason, all remaining elements in
  List y are pruned by p1 too

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Related works

• BBS

Dominant region
M1
M2
M2
N3
p5
p7
N1
N2
N3
N4
p6
p8
N4
p1
p2
p3
p4
p5
p6
p7
p8
M1
p2
N2
p3
N1

• HNNp1,p2,N2,M2
• p1 is the first NN object from best point
• Dominant region of p1 shows in grey color

p4
p1
Best point

• p2 is pruned by dominating region
• Expand N2

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SUBSKY

• According to NBA database, we have more than 10
  different attributes for one player
• Skyline queries may be interested in some
  attributes only

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SUBSKY

• Build one R-tree and run BBS
• BBS is an I/O optimal algorithm based on R-tree
  index, but their approaches are optimized for a
  fixed set of dimensions
• Build R-trees for all elements in the power set
  of dimensions
• Hugh storage space

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SUBSKY for uniform data

• Anchor point Ac the maximal corner of the data
  space having maximum coordinate on all dimensions

maximum value of the coordinate
Ac
f(p1)
f(p)max(1-pi), where i is from 1 to d
y
1
p1
f(p2)
fsky(psky)
fsky(psky)min(1-pskyi), where i is from 1 to d
p2
psky
Pruning region of psky
No any point p satisfying f(p)ltfsky(psky) can
belong to the skyline
x
Best point
1
A similar result exists for the skyline of any
subspace
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SUBSKY for uniform data

• Skyline queries only apply on relevant dimensions
  SUB
• fsky(psky)min(1-pskyi),
• where i is in SUB
• Then,
• f(p) lt fsky(psky) lt fsky(psky)
• No any point p satisfying the above equation can
  belong to the skyline

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SUBSKY for uniform data

• Assume that our skyline query is interested in
  dimension x and y only
• First, we sort the data by f(pi)
• p3, p4, p1, p2, p5
• Sskyp3, fsky(p3)0.5 min(1-0.5,1-0.3)
• U0.5 (largest f value in Ssky)
• Sskyp3,p4, fsky(p4)0.1
• U0.5
• Sskyp1,p4, fsky(p1)0.8
• p3 is removed by adding p1, since it is dominated
  by p1
• U0.8

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Analysis

• Assume that you have 15D uniform distributed
  objects with cardinality 100k, and we want to
  retrieve the skyline in a subspace SUB containing
  any two dimensions.

1
Greater than 90 to find an object in area(?, ?),
where ?0.001, since (1- ?2)100k Therefore, fsky(
p) 1- ? 0.999 The volume evaluates to
0.9991598.5, that is, we only need to access
1.5 of the dataset
?
1
?
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General SUBSKY

• In practical, data are usually clustered
• If the data are clustered, then we should expect
  that one anchor point cannot give us enough
  pruning power

Ac
A1
1
psky
x
Best point
1
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General SUBSKY

• Anchors for different clusters

Ac
A1
psky
s3
cluster s1
A2
s2
s4
x

• Two questions
• How to find the anchors?
• How to assign points to anchors?

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Finding the Anchors

• First, let us see what a perfect anchor of a
  point p
• If p is assigned to A, then p can be pruned by
  any skyline point dominating p

Any point on this line is a perfect anchor of
point p
A3
A2
Major perpendicular plane
A1
p
Anti-dominant region of p
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Finding the Anchors
Major perpendicular plane

• For each point, find the projections to the plane
• Ex p1, p2
• Partition the projected points into m clusters
  using algorithm k-means, and formulate an anchor
  for each cluster

p2
p1
p2
a good anchor
p1
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Finding the Anchors

• How to decide an anchor for a cluster?

Blue points are assigned to cluster S. How can we
decide the anchor for S?
A

• Obtain point B, whose coordinate on each
  dimension equals the lowest coordinate of the
  points in S in their original space on this axis

B

• Then, the algorithm computes the smallest square
  opposite to B which covers all points in S

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Assigning Points to Anchors

• A naïve way is to assign points to their closest
  anchor point in the major perpendicular plane
  (projected space)
• It is not directly quantifies the benefit of an
  assignment

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Assigning Points to Anchors

• In order to assign a point to a good anchor, this
  paper introduces a new measurement which name is
  effective region (ER)

p
Pruning region of p2
Pruning region of p1
All points in yellow region (ER) can make a
pruning region to Ac that cover p
p1
p2
If ER-volume of p is larger, then p has more
chance to be pruned
ER of p
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Assigning Points to Anchors
A
p
p
p1
p1
ER of p
p2
p2
ER of p
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Assigning Points to Anchors

• The pruning volume size of a point p to an anchor
  point Aj is
• ?max(0,Aji-L8(p,Aj)),
• where i is from 1 to d
• Therefore, assign a point p to Aj that produces
  the largest pruning volume size

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Query example

• We use the same example in previous slide
• Assume that we have two anchors, one is Ac and
  the other A is found by K-means (m1)
• Ac(1,1,1) and A(1,1,0.8)
• First, we calculate the ER volume of each data
  point with respect to Ac and A

Unit 10-3
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Query example

• Sorted list by f
• Ac p4 p1 p2 p5
• A p3

• Sskyp4, fsky(p4)0.5
• U0.5

• Sskyp4, p1, fsky(p1)0.8
• U0.8

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Experiments

• 3 real datasets NBA, Household, and Color
• 2 synthetic data (10D)
• Uniform
• Clustered
• 10 cluster centroids
• For each centroid, it takes N/10 points whose
  coordinate on each axis follows a Gaussian
  distribution with variance 0.05 and a mean equal
  to the corresponding coordinate of the centroid

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Experiments
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Experiments
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Experiments
3D subspaces, full-space dimensionality is 10
3D subspaces, 1 million cardinality
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Conclusion

• The core of SUBSKY is a transformation that
  convert multi-dimensional data into 1D values
• Show better performance than a I/O optimized
  algorithm in the subspace skyline problem
• Some continuous monitoring cases are good to
  investigate
• How to adopt the set of anchor points if data
  update rapidly
• The f values could be stored in other index
  structure to support fast update

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Assigning Points to Anchors

• Therefore, we have two ways to assign points to
  the anchors
• Assign points to their closest anchor point in
  the major perpendicular plane (projected space)
• Assign points to their closest anchor point by
  ER-volume in original space
• The second approach is better than the first in
  the major perpendicular plane, because the
  ER-volume directly quantifies the benefit of an
  assignment