Probabilistic Contextual Skylines

D. Sacharidis1, A. Arvanitis12, T. Sellis12

1Institute for the Management of Information Systems — “Athena” R.C., Greece2National Technical University of Athens, Greece

static vs. dynamic skyline•(another) hotels example•Price and Distance are Statically Preferred (SP) attributes with fixed preferences: lower is better•Ignore Amenity (assume all amenities are equally preferable)

•h4 and h5 are in the static skyline

•Amenity is a Relatively Preferred (RP) attribute, preferences are defined per query

•h3, h4 and h5 are in the dynamic skyline

h3

h4

h5

DistanceP

rice

h1

h2

(P)(G)

(S)

(I)

(I)

h3

h4

h5

Distance

Pri

ce

h1

h2

contextual skyline

•just like dynamic skyline, but preferences are associated with some context

•what if no preferences are specified for the current context?•two issues:

•can we extract them from previous situations?•what does it mean to be in the skyline?

extract preferences

•key idea is to combine preferences from similar contexts to the current

•first assess the similarity between the current context Cq and all past contexts Cj :

•contexts may have conflicting preferences, and we model uncertainty with probabilities

•value u is better than v for the context Ci

with some probability

•probabilities can be extracted based on context similarities

probabilistic contextual skylines

dominance relationships are uncertain (assuming independence among attributes)

•tuple t dominates t’ for the context Ci with probability

the skyline probability of a tuple is defined as

probabilistic contextual skyline query, p-CSQ, returns all tuples with

example

skyline probability

(1)

(2)

(2)

(3)

non-indexed algorithms (1/2)• for RP attributes (unlike standard and tuple-probabilistic skylines)

•no monotonic visit order exists•transitivity is not preserved

we have to apply BNL-like methods (not SFS, BBS, etc.)

Basic Iterative Algorithm (BIA)•for each tuple scan the database and compute skyline probability (abort when below threshold)

S

I P

3/4

1/4

0S

I P

3/4

3/4

0

non-indexed algorithms (2/2)

Candidate Selection Algorithm (CSA)•it identifies candidates•group tuples by their values on the RP attributes

•tuples that are dominated in an RP-group have 0 probability•tuples that are in the skyline w.r.t. only the SP attributes have probability 1

h3

h4

h5

Distance

Pri

ce

h1

h2

(P)(G)

(S)

(I)

(I)

h6

(P)

probability 1

probability 0

CSA applies BIA only for the candidates(needs to check them against all tuples, though)

index-based algorithms (1/3)the algorithms only consider the candidates

Basic Group Counting (BGC)idea: tuples in an RP group that dominate a candidate t contribute the same probability•use COUNT aR-tree per RP group

•but, don’t just issue a range query per tree… we don’t care about the exact count if tuple’s probability is below threshold•instead visit nodes from all trees in parallel and use a single priority queue•the node from the tree which has the highest expected probability of dominating tuple t has the largest priority

index-based algorithms (2/3)

Super Group Counting (SGC)

•there can be a lot of RP groups with only a few tuples •to mitigate this, assign groups to super groups•use a GROUP-COUNT aR-tree per super-group

•entry: < ei, MBBi, c[g1], c[g2], … >

•where c[gj] is the number of tuples beneath node ei that belong to the j-th group

•same algorithm as BGC… you only need to redefine the expected dominance probability to take into account multiple groups

index-based algorithms (3/3)

Batch Counting Algorithm (BCA)

•all previous algorithms compute the skyline probability of one tuple at a time•BCA examines candidates in batches (as many as fit in memory)•extra bookkeeping with each heap entry to avoid double counting

e1 is deheaped

e1+ dominates t1 but not t2

entire e1 contributes to t1, but for t2 we need to expand e1 and enheap its children e2, e3

also remember e1+ with e2, e3

t1

t2e2

e1 e3

e1+

Experiments

Non-indexed BIA, CSA Index-based SGC, BCA

Total Time vs. Dataset Cardinality

Non-indexed BIA, CSA Index-based SGC, BCA

Total Time vs. RP domain size

Non-indexed CSA Index-based SGC, BCA

Total Time vs. Dimensionality

Non-indexed CSA Index-based SGC, BCA

thank you!