mining favorable facets

Mining Favorable FacetsRaymond Chi-Wing Wong (the Chinese University of

Hong Kong)Jian Pei (Simon Fraser University)

Ada Wai-Chee Fu (the Chinese University of Hong Kong)

Ke Wang (Simon Fraser University)

KDD’ 07, August 12-15, 2007, San Jose,California, USA

Outline

1. Introduction2. Skyline3. Algorithm4. Empirical Study5. Conclusion

1. IntroductionSuppose we want to look for a vacation package

3 packages

Package ID Price Hotel-class

aa 44

bb 24002400 11

c 3000

Suppose we compare package a and b

We want to have cheaper price.

We want have a higher hotel-class.

We know that package a is “better”

than package bbecause 1. Price of package a is smaller2. Hotel-class of package a is

higher

Package a “dominates” package b5

10001000

1. IntroductionPackage ID Price Hotel-

class

aa 10001000 44

bb 24002400 11

c 3000 5

Thus, we do not need to consider package b.

We know that 1. Package a has a cheapest price2. Package c has a highest hotel-class

Packge a and c don’t dominate by other points

Thus, package a and package c are all of the “best” possible choices.

We call that package a and package c are skyline points.

Package ID

Price Hotel-class

Hotel-group

a 10001000 4 T (Tulips)

b 2400 1 T (Tulips)

c 3000 55 H (Horizon)

d 3600 4 H (Horizon)

e 2400 2 M (Mozilla)

f 3000 3 M (Mozilla)

6 packages

Suppose we want to look for a vacation package

Different customers may have different preferences on Hotel-group.

Suppose a customer have the

following preferences. H < T < MThe skyline points are packages a and c.

Suppose another customerhave the following

preferences. H < M < TThe skyline points are packages a, c and e.

In other words, different preferences give differentn skyline points.

1. Introduction

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)




ff 30003000 33 M (Mozilla)M (Mozilla)

Customer

Preference on Hotel-group

Skyline

Alice T < M {a, c}

Bob No special preference{a, c, e, ff}

Chris H < M {a, c, e}

David H < M < T {a, c, e}

Emily H < T < M {a, c}

Fred M < T {a, c, e, ff}

What preferences makepackage f a skyline point?

Suppose hotel-group Mozillawants to promote its own packages (e.g., package f) topotential customers.

Bob and Fred are the potential customers.

1. Introduction

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)





Problem: Given a package, we want to

find what preferences or conditions that

this package is a skyline point?

Favorable facetsFavorable facets

1. Introduction

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)






find what preferences or favorable facets

that this package is a skyline point?

{}SKY={a, c, e, f}

SKY={a,c}{T < M} {H < M}

SKY={a,c,e}{T < H}

SKY={a,c,e,f}{H < T}SKY={a,c,e,f}

{M < T}SKY={a,c,e,f} SKY={a,c,e,f}

{M < H}

{T < M, H < M} {T < M, T < M} {H < T, H < M} {T < H, M < H} …SKY={a,c} SKY={a,c} SKY={a,c,e} SKY={a,c,e,f}

{T < M, T < M, H < M} {T < M, T < M, M < H}SKY={a,c} SKY={a,c}

TSKY={}

We can solve the problem by a naive method: Lattice Search

1. Introduction

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)








{}SKY={a, c, e, f}

SKY={a,c}{T < M} {H < M}

SKY={a,c,e} SKY={a,c,e,f} SKY={a,c,e,f} SKY={a,c,e,f} SKY={a,c,e,f}

{T < M, H < M} {T < M, T < M} {H < T, H < M} …SKY={a,c} SKY={a,c} SKY={a,c,e}


TSKY={}


Consider package f

Preferences:{}, {T < H}, {H < T}, {M < T}{M < H}

,, {T < H, M < H}

SKY={a,c,e,f}

{T < H}

{H < T}

{M < T} {M < H}

{T < H, M < H}

We need to compute all skyline points for each possible preference

There are many preferences which qualify package f as a skyline point

This approach has two disadvantages. 1. Computation is costly. 2. It is difficult to interpret the results.

1. Introduction

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)








{}SKY={a, c, e, f}

SKY={a,c}{T < M} {H < M}

SKY={a,c,e} SKY={a,c,e,f} SKY={a,c,e,f} SKY={a,c,e,f} SKY={a,c,e,f}

{T < M, H < M} {T < M, T < M} {H < T, H < M} …SKY={a,c} SKY={a,c} SKY={a,c,e}


TSKY={}


Consider package f

SKY={a,c,e,f}

{T < H}

{H < T}

{M < T} {M < H}

{T < H, M < H}

border for f

We find that whenever the preference contains “T < M” or “H <

M”, package f is not a skyline

point.

We can say that “T < M” or “H < M” is a minimal disqualifying condition (MDC).

3. Algorithm

How to find MDCs of a point?


find what minimal conditions thatthis package is NOT a skyline point?

3. Algorithm

Package ID

Price Hotel-class

Hotel-group

a 1000 4 T (Tulips)

b 2400 1 T (Tulips)




f 3000 3 M (Mozilla)Point q is said to quasi-dominate point p ifall attributes of point q are NOT worse than those of point p.

e.g. Package a quasi-dominates package f because

1. Package a has a lower (or better) price than package f2. Package a has a higher (or better) hotel-class than package f

If package a quasi-dominates package f, we define Raf as follows. {T < M}

3. Algorithm

Two Algorithms MDC-O: Computing MDC On-the-fly

Does not store MDCs of points Compute MDC of a given points on-the-fly

MDC-M: A Materialization Method Store MDCs of all points

Indexing Method for Speed-up R*-tree



3.1 MDC-O: Computing MDC On-the-fly On-the-fly Algorithm

Given data point p

Variable MDC(p): minimal disqualifying condition

Algorithm MDC(p) For each data point q which quasi-dominates p

if MDC(p) does not contain Rqp

insert Rqp to MDC(p) Return MDC(p)



3.2 MDC-M: A Materialization Method

Materialization Algorithm Variable

MDC(p): minimal disqualifying condition Algorithm MDC(p)

For each data point p For each data point q which quasi-dominates p

if MDC(p) does not contain Rqp

then insert Rqp to MDC(p) Store MDC(p)



4. Empirical Study Datasets

Synthetic Dataset Real Dataset (from UCI)

Nursery Dataset Automobile Dataset

Default Values (Synthetic) No. of tuples = 500K No. of numeric dimensions = 3 No. of categorical dimensions = 1 No. of values in a nominal dimension = 20

4. Empirical Study

Without indexing: MDC-O: Slowest Search Time MDC-M: Faster Search Time Storage of MDC: 8MBWith indexing: MDC-O and MDC-M: Fast Search Time

4. Empirical Study Automobile

Three car models

Car MDC

Honda “Toyota < Honda”

Mitsubishi“Honda < Mitsubishi” or “Toyota < Mitsubishi”

Toyota -

A salesperson should NOT promote this car to the customer who prefers Toyota to Honda.

A salesperson should NOT promote this car to the customer who prefers Toyota to Honda.

A salesperson should promote this car to ANY customers.

5. Conclusion

Skyline Favorable Facets

Minimal Disqualifying Condition Algorithm

On-the-fly Materialization

Empirical Study

mining favorable facets

Documents