complex_pref {rPref}

Complex Preferences

Description

Complex preferences are used to compose different preference orders. For example the Pareto composition (via operator *) is the usual operator to compose the preference for a Skyline query. The Skyline is also known as Pareto frontier. All complex preferences are mathematically strict partial orders (irreflexive and transitive).

Usage

## S4 method for signature 'preference,preference'
e1 * e2

## S4 method for signature 'preference,preference'
e1 & e2

## S4 method for signature 'preference,preference'
e1 | e2

## S4 method for signature 'preference,preference'
e1 + e2

reverse(p)

is.complex_pref(x)

Arguments

p, e1, e2

Preference objects (they can be either base preferences, see base_pref, or complex preferences)

x

An object to be tested if it is a complex preference.

Skylines

The most important preference composition operator is the Pareto operator (p1 * p2) to formulate a Skyline query. A tuple t1 is better than t2 w.r.t. p1 * p2 if it is strictly better w.r.t. one of the preferences p1, p2 and is better or equal w.r.t. the other preference.

The syntactical correspondence to other query languages supporting Skylines/preferences to rPref is given as follows:

• A query in the syntax from Börzsönyi et. al (2001) like

  "... SKYLINE OF a MAX, b MIN, c MAX"

  corresponds in rPref to the preference

  high(a) * low(b) * high(c).

• A query in the syntax from Kiessling (2002) like

  "... PREFERRING a LOWEST AND (b HIGHEST PRIOR TO c LOWEST)"

  corresponds in rPref to

  low(a) * (high(b) & low(c)).

• A query in the syntax of the "Skyline" feature of the commercial database "EXASOL EXASolution 5" like

  "... PREFERRING LOW a PLUS (b = 1 PRIOR TO LOW c))"

  corresponds in rPref to

  low(a) * (true(b == 1) & low(c)).

Note that preferences in rPref can be translated to some of this query dialects by show.query.

Definition of Additional Preference Operators

Additionally, rPref supports the following preference composition operators:

p1 & p2

Prioritization (lexicographical order): A tuple t1 is better than t2 w.r.t. p1 & p2 if it is strictly better w.r.t. p1 or is equal w.r.t. p1 and is better w.r.t. p2.

p1 | p2

Intersection preference: A tuple t1 is better than t2 w.r.t. p1 | p2 if it is strictly better w.r.t. both preferences. This is a stricter variant of the Pareto operator. The evaluation of psel(df, p1 | p2) is always a subset of psel(df, p1 * p2).

p1 + p2

Union preference: A tuple t1 is better than t2 w.r.t. p1 + p2 if it is strictly better w.r.t. to one of the preferences. Note that this can violate the strict partial order property, if the domains (the tuples on which p1 and p2 define better-than-relationships) of the preferences are not disjoint.

reverse(p1) or -p1

Reverse preference (converse relation): A tuple t1 is better than t2 w.r.t. -p1 if t2 is better than t1 w.r.t. p1. The unary minus operator, i.e. -p1, is a short hand notation for reverse(p1).

The function is.complex_pref returns TRUE if x is a complex preference object (i.e., was constructed by one of these binary operators or the unary operator reverse) and FALSE otherwise.

Associated Data Sets

If one of the preferences for a binary operator are associated with a data set (see base_pref), then this association is propagated. For example, the preference

p <- high(mpg, df = mtcars) * high(hp)

as well as

p <- high(mpg) * high(hp, df = mtcars)

both result in the same complex preference which is associated with mtcars. A partial evaluation is also invoked for all preferences which are added. For example, using this p,

p <- p * true(cyl == max(mtcars$cyl))

generates the following console output:

[Preference] high(mpg) * high(hp) * true(cyl == 8)
* associated data source: data.frame "mtcars" [32 x 11]

We see that the association with the data set is propagated and max(mtcars$cyl) is partially evaluated.

References

S. Borzsonyi, D. Kossmann, K. Stocker (2001): The Skyline Operator. In Data Engineering (ICDE '01), pages 421-430.

W. Kiessling (2002): Foundations of Preferences in Database Systems. In Very Large Data Bases (VLDB '02), pages 311-322.

S. Mandl, O. Kozachuk, M. Endres, W. Kiessling (2015): Preference Analytics in EXASolution. 16th Conference on Database Systems for Business, Technology, and Web.

See Also

See base_pref for the construction of base preferences. See general_pref for functions applicable to all kind of preferences. See psel for the evaluation of preferences.

Examples

# Defines a preference for cars with low consumption (high mpg-value)
# and simultaneously high horsepower.
p1 <- high(mpg) * high(hp)  

# Performs the preference search.
psel(mtcars, p1)

# Alternative way: create preference with associated data set.
p2 <- high(mpg, df = mtcars) * high(hp)  
peval(p2)