CoSky: A Practical Method for Ranking Skylines in Databases

Hana Alouaoui, Lotﬁ Lakhal, Rosine Cicchetti and Alain Casali

Laboratoire d’Informatique et Syst

eme, CRNS UMR 7020, Aix Marseille Universit

e, France

Keywords:

Databases, IR, MCDA, Ranking, Skylines.

Abstract:

Discovering Skylines in Databases have been actively studied to effectively identify optimal tuples/objects

with respect to a set of designated preference attributes. Few approaches have been proposed for ranking

the skylines to resolve the problem of the high cardinality of the result set. The most recent approach to

rank skylines is the dp-idp (dominance power- inverse dominance power) which extensively uses the Pareto-

dominance relation to determine the score of each skyline. The dp-idp method is in the very same spirit as tf-idf

weighting scheme from Information Retrieval. In this paper, we ﬁrstly make an Enrichment of dp-idp with

Dominance Hierarchy to facilitate the determination of Skyline scores, we propose then the CoSky method

(Cosine Skylines) for fast ranking skylines in Databases without computing the Pareto-dominance relation.

Cosky is a TOPSIS-like method (Technique for Order of Preference by Similarity to Ideal Solution) resulting

from the cross-fertilization between the ﬁelds of Information Retrieval, Multiple Criteria Decision Analysis,

and Databases. The innovative features of CoSky are principally: the automatic weighting of the normalized

attributes based on Gini index, the score of each skyline using the Saltons cosine of the angle between each

skyline object and the ideal object, and its direct implementation into any RDBMS without further structures.

Finally, we propose the algorithm DeepSky, a Multilevel skyline algorithm based on CoSky method to ﬁnd

Top-k ranked Skylines.

1 INTRODUCTION

Skyline computation, previously known as Pareto sets

and maximal vectors (Bentley et al., 1978), has re-

ceived a great attention in the statistical and mathe-

matical ﬁelds since many past decades. The skyline

computation is crucial to many multi-criteria deci-

sion making applications. Therefore, skyline queries

have attracted considerable attention in the context of

databases, especially with the introduction of skyline

operator by (Borzsonyi et al., 2001). These queries

are simple and expressive. They do not need user-

deﬁned scoring functions. They only require the user

preferences concerning the minimization or the maxi-

mization of attribute values. Suppose a customer who

wishes to buy a car and he is seeking for a car with

high power, low mileage and low price. Neverthe-

less, these criteria of selecting Cars are complemen-

tary since cars of higher power and lower mileage are

more expensive. In order to ﬁnd such cars, we must

query the corresponding Cars database relation (Ta-

ble 1). Let price, mileage (klm) and power be the at-

tributes of Cars, the users prefer to minimize the price

and the Mileage (klm) and maximize the power by

selecting items that are better than others regarding

Table 1: Database Relation Cars.

idcar price klm power

25 10 8

20 30 6

25 15 7

5 40 7

25 45 5

45 15 6

35 40 5

45 45 4

these three attributes.

Here is an example of a skyline query (with sky-

line operator) on the relation Cars:

SELECT ∗ FROM C a rs SKYLINE OF p r i c e MIN, klm MIN , pow er MAX;

The associated SQL query without skyline opera-

tor is:

SELECT ∗ FROM C a rs C ar1

WHERE NOT EXISTS (

SELECT ∗ FROM C a rs C ar2

WHERE (

Ca r2 . p r i c e =< Car1 . p r i c e

AND C ar 2 . klm =< Car1 . klm

508

Alouaoui, H., Lakhal, L., Cicchetti, R. and Casali, A.

CoSky: A Practical Method for Ranking Skylines in Databases.

DOI: 10.5220/0008363005080515

In Proceedings of the 11th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2019), pages 508-515

ISBN: 978-989-758-382-7

AND C ar 2 . po wer >= C ar 1 . po wer )

AND ( Ca r2 . p r i c e < Car1 . p r i c e

OR Car2 . klm < Car1 . klm

OR Car2 . power > C ar1 . power ) ) ;

As a result, a set of good cars is returned (C

, C

). Good cars in our case is the set of cars which

are as good or better in all dimensions (price, klm and

power) and better in at least one dimension. This set

of good points (cars) forms the Skylines. The Pareto-

dominance speciﬁes which data points belong to the

skyline and it can be formalized as follows:

Let r be a relational table, with A

, . . . , A

at-

tributes. A preference over A

is an expression of

one of two forms: Pref (A

) = min or Pref (A

)

= max. Let p and q be two tuples of r. We

say that p dominates q (denoted p≺

q) if and

only if p.A1≤q.A1, . . . , p.A

≤q.A

and ∃ j ∈ [1..m] :

p.A

<q.A

With:

(, ≺) =



(≤, <) iff pre f (A

) = min

(≥, >) iff pre f (A

) = max

(1)

In other words, an object (tuple) p dominates an-

other object q if it is as good in all attributes, and is

strictly better in at least one attribute. The Skyline is

a set S of tuples which are not dominated by any other

tuple. S = {t ∈ r|t is not dominated}

One of the major issues of the skyline operator is

the high cardinality of the result set which does not of-

fer any interesting insights. All objects are equally in-

teresting and there is no signiﬁcant discrimination be-

tween them. In order to face this obstacle, an efﬁcient

ranking of skyline objects has become a compelling

need. This solution is efﬁcient especially in the case

of high dimensional or anti-correlated data and partic-

ipates in reducing the huge size of the result set. Our

contribution lies within this scope. In this paper, we

introduce a novel approach that aims on one hand at

enriching an IR- style ranking mechanism based on

dp-idp scoring scheme. Our enrichment is founded

on the integration of a Dominance Hierarchy (DH) in

order to improve the calculation of skyline scores and

their afterward ranking. And on the other hand, we

propose our CoSky (Cosine Skylines) approach that

handles with skyline objects ranking without favor-

ing any dominance relationship. CoSky is a TOPSIS-

like method (the Technique for Order of Preference

by Similarity to Ideal Solution (Lai et al., 1994)) is a

cross-fertilization between the ﬁelds of Information

Retrieval, Multiple Criteria Decision Analysis, and

Databases. CoSky innovations are summarized by

the following steps: the attributes normalization using

Gini index, the automatic weighting of the normalized

attributes: they do not need user-deﬁned weighting at-

tributes as in TOPSIS method (Tscheikner-Gratl et al.,

2017), the calculation of each skyline score using the

Salton’s cosine of the angle between each skyline ob-

ject and the ideal object, and its direct implementation

into SQL without further structures.

The rest of the paper is organized as follows: sec-

tion 2 gives an overview of the methods proposed

in the literature to extract and rank the skyline ob-

jects. In section 3, we explain the dp-idp method

and we point out its weaknesses and the difﬁcul-

ties grasped while calculating the scores and deﬁn-

ing the layers of minima. In section 4, we describe

our proposal of enrichment and extension of dp-idp

Ranking mechanism based on Dominance Hierarchy

pre-computation. In section 5, we present our non-

dominance based approach The Cosky method, we

describe the main steps and we discuss the obtained

results given by a running example. The algorithm

DeepSky, a Multilevel skyline algorithm to ﬁnd Top-

k ranked Skylines that have k highest scores is de-

scribed in section 6. Finally, we sum up the main

conclusions of this paper as well as point out direc-

tions for future works.

2 RELATED WORK

In addition to the skyline queries, a panoply of al-

gorithms have been proposed in order to meet sky-

line constraints which vary with the studied computa-

tional domain. In (Borzsonyi et al., 2001), the Block

Nested Loop (BNL) algorithm is proposed in database

context; its principle is based on a window (memory

block with limited space) of size w. This window

stores the ﬁrst points that are undominated in each

pass. Passes are made over the data until obtaining all

the skyline points and each dominated point is elimi-

nated to not be read in the future. In case the window

is full, a temporary disk ﬁle is used to hold the can-

didate objects. The BNL algorithm is provided of a

timestamping mechanism allowing it to ﬁnd out when

a point is in the skyline and when all points it domi-

nates were eliminated. In (Spyratos et al., 2012), au-

thors propose an approach to compute the skyline of

a relational table taking into account preferences ex-

pressed over one or more attributes. This approach

does not consider the table structure or the tuples in-

dexing. It is based on query lattice concept presented

and explained in the paper. An algorithm is developed

to construct the skyline as the union of answers to a

subset of queries from that lattice without directly ac-

cessing the table R. To rank the query, they consider

the maximum path from the root query to another

query q. The higher the rank of a query the less the

tuples in its answer are preferred. Two index-based

CoSky: A Practical Method for Ranking Skylines in Databases

509

algorithms namely; Bitmap and Index are introduced

in (Tan et al., 2001). Bitmap uses the bitmap encod-

ing to determine the dominating points. The Index

approach is based on the partitioning of different ob-

jects into sorted lists. The sorting parameter is the

minimum coordinate. The lists are then indexed by

a B-tree. These algorithms return skyline points in a

ﬁxed order which cannot be adapted to the users pref-

erences. In (Papadias et al., 2005), the Branch and

Bound Skyline (BBS) algorithm is proposed, this al-

gorithm is based on nearest-neighbor search and only

nodes containing skyline points are accessed. BBS

is simple to implement due to its progressiveness and

I/O optimality. The SaLSa algorithm (Bartolini et al.,

2007) is a natural extension of SFS (Chomicki et al.,

2003) algorithm, whose originality is the ability of

computing the result without applying the computa-

tion of the Pareto-dominance relation to all the ob-

jects. This is achieved by pre-sorting the data us-

ing a monotone (as SFS) limiting function, and then

checking that unread data are all dominated by a so-

called stop point (object). A Randomized Skyline al-

gorithm (RAND) is presented in (Das Sarma et al.,

2009). RAND is a multi-pass streaming algorithm

which takes into account randomized I/Os. A compar-

ison between RAND and other known algorithms is

given and shows that it performs in the case of minor

and simple variations in the input (eg. Perturbations

of the data orders) while the other algorithms do not

return signiﬁcant results with such variations. As we

mentioned above, the huge cardinality of a skyline set

is a main obstacle that a decision maker faces. In or-

der to avoid it, an efﬁcient selection of skyline objects

has to be performed. Numerous works have been de-

voted to study the ranking of skylines. In (Chan et al.,

2006), a metric called skyline frequency is proposed

in order to rank the skyline by retaining the interest-

ing points with high skyline frequency. This method

scales well with dimensionality unlike other ranking

methods. Experiments show that the proposed algo-

rithm runs faster than other algorithms and returns

correct results even when considering a huge number

of dimensions

An alternative to skyline queries is presented in (Yiu

and Mamoulis, 2007); the top-k dominating queries.

These queries are proposed as a ranking method not

based on a scoring function. The top-k dominating

queries are evaluated in a multi-dimensional data con-

text.

In (Bartolini et al., 2007), a ranking approach applied

on an image Database is introduced. The technique is

based on the shaping of what the user is looking for

by specifying user deﬁned regions that dominate all

other regions. Authors show that the obtained results

are as good as those based on scoring functions and

that the proposed approach provides a perfect running

time.

(Vlachou and Vazirgiannis, 2010) propose a ranking

approach of skyline objects for a SKYCUBE (Lakhal

et al., 2017) in order to focus on the most informa-

tive objects. A Skyline graph is built up and captures

the dominance relationships between skyline objects

belonging to different subspaces. In (Chen et al.,

2015), a new operator is introduced in order to ﬁnd

the most desirable skyline object (MDSO). A ranking

criterion is formalized, it considers the number of the

non-skyline objects dominated by a skyline object s.

The higher this criterion is, the more interesting the

skyline object is. To process MDSO queries, three al-

gorithms are proposed namely; Cell Based algorithm

(CB), Sweep Based algorithm (SB), and Reuse Based

algorithm (RB). They return the most desirable k sky-

line objects.

3 SKYLINE RANKING BASED ON

DP-IDP

The dp-idp (dominance power- inverse dominance

power), is inspired by the tf-idf weighting scheme

from Information Retrieval which assigns to a term

t a weight in a document d. The idea is not to deter-

mine the number of occurrences of each query term

t in d, but instead the tf-idf weight of each term in

d. The aim is to ﬁnd important keywords in a docu-

ment corpus. In the skyline context, dominated points

impact skyline point differently. Consequently, these

points have not the same importance. Their contri-

bution depends on some local (per skyline point) and

some global characteristics (the entire skyline), simi-

larly to tf-idf.

The explored idea in (Valkanas et al., 2014) is that

a point’s importance has to be inversely proportional

to the number of skyline points that dominate it. The

dp-idp scheme takes into account the relative posi-

tions of the dominated points in order to differentiate

between them. It is centered on points that are not

dominated by many others: e.g. if sp≺

, sp≺

and p

do not dominate each other, they con-

tribute equally to sp. Otherwise, if p

≺

, then

score(p

) > score(p

), consequently the contribution

of p

is more considerable. The idp of a point p ∈

D\S) is the number of skyline points which dominate

p. A point p is important if it does not appear fre-

quently in a skylines point dominated set :

id p(p) = log

|S|

|{sp ∈ S : sp ≺

p}|

(2)

KDIR 2019 - 11th International Conference on Knowledge Discovery and Information Retrieval

510

To measure the d p of a dominated point p, its rel-

ative position to the skyline point sp is very important

and has to be taken into account. Hence, the same

dominated point may contribute differently to differ-

ent skyline points. For this reason, we have to ﬁnd the

layer of minimalm(p, sp) where the dominated point

p is located with respect to sp. The dominance power

of p is then given by the inverse of the layer where it

lies : The score (sp) which measures the importance

of a skyline point sp is given by the following for-

mula:

Score(sp) =

∑

p:sp≺

lm(p, sp)

log

|S|

|{sp

∈ S : sp

≺

p}|

(3)

The Baseline algorithm (Valkanas et al., 2014) is

proposed to rank the skyline on the basis of dp-idp

scheme. The main steps of this algorithm are:

1. Extracting the minimal layers of each skyline

point sp;

2. Considering each point (p) in each layer of min-

ima lm (p, sp), the number of skyline points dom-

inating it has to be found. The score sp is then

updated;

3. Sorting the skyline on the basis of the returned

scores.

This algorithm is time consuming and has lots of

shortcomings as mentioned in (Valkanas et al., 2014).

But the most important weakness according to us, is

the difﬁculty of the calculation of the layer of minima

which may lead to wrong results. For this reason, we

propose an enrichment of this approach in order to

make the Baseline algorithm faster .

At this state, our objective is to ameliorate the

dominance based approach (dp-idp) and to improve

the skyline ranking (Section 4). Further, in this paper,

we will introduce our non-dominance based approach

and we will discuss its performance in ranking sky-

line objects without referring to any dominance rela-

tion calculation (Section 5 and Section 6).

4 ENRICHMENT OF DP-IDP

WITH DOMINANCE

HIERARCHY

The dominance relation can be seen as an hierarchi-

cal sorting, i.e. A skyline point has necessarily a hi-

erarchical position superior to its dominated points.

This assumption motivated us to map the dominance

relationship, studied above, to a graph that we call

Dominance Hierarchy. The integration of DH into the

Figure 1: An example of a DH graph.

dp-idp skyline r anking method permits a better com-

putation of layers of minima (ﬁrst step in the base-

line algorithm) and consequently leads to better rank-

ing results. Our proposed graph is a kind of Directed

Acyclic Graph (DAG) used to give a presentation of

a partially ordered set (poset) by drawing its coverage

graph (i.e, graph which represents the same reacha-

bility relation of the main graph but with the fewest

possible edges). The DAG has a topological ordering

which may give an excellent presentation of the Hi-

erarchy evolution from a skyline point sp to its domi-

nated points.

Given a set of objects (D) and an order of domi-

nance ≺

, DH is the coverage graph of the ordered

set (D, ≺

Table 2: Scores Calculation.

sp Dominated Lm (p, sp) Score (sp)

points p

Lm(C

) = 2 0.297

Lm(C

) = 3

Lm(C

) = 3

Lm(C

) = 3

Lm(C

) = 4

Lm(C

) = 2 0

Lm(C

) = 2

Lm(C

) = 3

Lm(C

) = 2 0

Lm(C

) = 2

Lm(C

) = 3

The DH is a direct acyclic graph (c f . Figure 1)

contains as vertex the skyline point (sp). The order of

Dominance is illustrated by the edges between sp and

CoSky: A Practical Method for Ranking Skylines in Databases

511

Figure 2: An example of a DH graph.

its dominated points (p

, p

and p

We deﬁne the layer of minima(p,sp) as the num-

ber of vertices of the minimal path from sp to p in

DH. To compute the layer of minima(p3, sp), there

are two paths from sp to p

as shown in the graph

(c f . Figure 1):

1. First path: sp→p

→p

‘

2. Second path: sp→p

→p

The ﬁrst path contains 4 vertices and the second

one contains 3 vertices, then the minimal path from sp

to p

is the second one and lm(p

, sp) = 3. Lets con-

sider another example, the DH graph in Figure 2 il-

lustrates the dominance relations between the skyline

points and the dominated points of the Cars database

relation. I is the ideal point dominating all skylines.

To compute the score, we use formula 3. Thus, the

obtained rank is : C

or C

On the basis of the obtained results (c f . Table 2),

we conclude that the dp-idp method is unable to dis-

tinguish two skylines dominating objects which are

dominated by all skylines as it is the case here of the

skylines C

and C

5 THE COSKY METHOD

In order to solve the ranking problem, we propose

the CoSky method for ranking skylines in Databases.

CoSky (COsine Skylines) is a multi-steps approach

and it is not based on dominance relation calcula-

tion. CoSky is the ﬁrst TOPSIS-like method applied

to ranking the skylines. TOPSIS (Lai et al., 1994),

(Behzadian et al., 2012) is based on a vectorial nor-

malisation, a user-weight calculation of each attribute,

and the score of each object uses a geometric calcula-

tion of the distances between each alternative (object)

and the ideal/anti-ideal solutions. In CoSky method,

the normalisation of the attributes is based on the sum,

an automatic weighting of the normalized attributes

based on Gini index, and the score uses the Salton’s

cosine of the angle between each skyline object and

the ideal object. More calculation details are given

later in this section. In the rest of the paper, we con-

sider that i∈[1..n] and j∈[1..m] (where n is the number

of tuples and m is the number of attributes).

5.1 Step 1: Attributes Normalization

based on the Sum

The Skylines are normalized in the range between 0

and 1 to eliminate anomalies with different measure-

ment units and scales. This process transforms differ-

ent scales and units among various attributes (or cri-

terias) into common measurable units to make these

attributes comparables.

Let us consider the tuple p

= <v

, v

, .., v

> ∈

rSkyNorm (Normalized skylines), then we have :

i j

]

∑

i=1

]

, ∀t

∈ rSky(Skylines) (4)

5.2 Step 2: Automatic Weighting of

Normalized Attributes based on

Gini Index

Ranking the skylines aims principally to give an ex-

pressive discrimination between the selected objects.

In order to reach such ﬁnality, it is crucial to ﬁx a dis-

criminative measure. In the literature, several mea-

sures were proposed. The entropy concept was used

in various Multi-Attribute Decision Making prob-

lems. In (Huang, 2008) and (Hosseinzadeh Lotﬁ and

Fallahnejad, 2010), an entropy based method was pro-

posed. This method ﬁts well with our computation

context where we aim at differentiating attribute val-

ues in order to attempt a better decision making re-

lated to the skyline ranking. However, it has many

limits especially related to the entropy calculation.

Entropy requires logarithmic function computation

what presents a computational time issue. Then, it is

generally intended for attributes that occur in classes.

We rather propose another measure ’the Gini in-

dex’ which is faster than entropy and does not use logs

to insure the automatic weighting of attributes. More-

over, it is intended for continuous attributes. The Gini

index is employed to derive the weights of the eval-

uation criteria (attributes) in our proposed method.

KDIR 2019 - 11th International Conference on Knowledge Discovery and Information Retrieval

512

Hence, it determines the degree of divergence of at-

tribute values. The Gini index of A

is calculated by

the following equation:

Gini(A

) = 1 −

∑

i=1

]

(5)

The corresponding weight is given by the follow-

ing formula:

W (A

) =

Gini(A

)

∑

j=1

Gini(A

)

(6)

Let us consider the tuple p

= {a

, a

, . . . , a

} ∈

rSkyPond (rsky after attributes weighting), then we

have :

i j

= W (A

) ×t

], ∀t

∈ rSkyNorm (7)

5.3 Step 3: Determination of the Ideal

Skyline

The ideal, denoted I

, is an object that corresponds

optimally to the user preferences. For example, if

we consider the cars table, then searching an ideal

car combines conditions on the price which has to be

the smallest possible, the number of kilometers; the

smallest possible, and the power; the greatest possi-

ble.

Thus, if we consider I

=< v

, v

, .., v

>, then

we have:



max(A

) iff pre f (A

) = max

min(A

) iff pre f (A

) = min

(8)

5.4 Step 4: Calculation of Skyline

Scores on the Basis of the Salton’s

Cosine

This step aims to determine the score of a skyline ob-

ject on the basis of the Salton’s Cosine (or Similar-

ity Cosine). We calculate the cosine of the angle be-

tween the ideal and the skyline. The more the angle

is little (high cosine), the more the skyline object is

important. The Salton cosine ranges between 0 and

1 and is given in the formula below. Let us con-

sider the tuple p

=< x

, x

, .., x

>∈ rSkyPond and

=< y

, y

, .., y

> the ideal object, then we have :

[Score] = Cosine(p

, I

) =

||p

||.||I

(9)

[Score] =

∑

j=1

i j

∑

j=1

i j

∑

j=1

, ∀t

∈ rSkyScore

(10)

As a consequence of equation 10, t

[score] = 1

if and only if the skyline is the best object, and

[score] = 0 if and only if the skyline is the worst

object.

Remark: We can use the similarity principle of

TOPSIS to calculate the score of each skyline ob-

ject as following: Let us consider t

[scoreIdeal] =

Cosine(p

, I

) and ti[scoreAideal] = Cosine(p

, I

−

∈ rSkyPond, I

the ideal and I

−

the anti-ideal ob-

ject./ Thus, if we consider I

−

=< v

, v

, .., v

>, then

we have:



max(A

) iff pre f (A

) = min

min(A

) iff pre f (A

) = max

(11)

5.5 Step 5 : Ranking the Result by the

Score

This is the ﬁnal step in the CoSky process, it aims to

sort the skyline objects on the basis of the calculated

scores. The CoSky steps applied to the car database

relation can be calculated using a SQL queries (c f .

Appendix). The obtained results are given in Table 3.

The obtained rank is C

, C

Table 3: Skylines ranking with CoSky method.

idcar price klm power score

25 10 8 0.814

5 40 7 0.803

20 30 6 0.769

Unlike dp-idp, the CoSky method distinguishes

clearly the skyline scores. The skyline objects C

and

have explicit scores (score 6= 0 ) while using dp-

idp their score is 0. Thus we can rank them.

6 FINDING TOP-K RANKED

SKYLINES

The notion of Multilevel skylines for ﬁnding Top-

k Skylines (not ranked) is introduced in (Preisinger

and Endres, 2015). A Top-k Skyline query Qk on a

Database relation r computes the Top-k tuples with

respect to the skyline preferences. Let us consider

(r) the classical skyline set and Card(r) > k, then

we have:

1. If Card(S

(r)) > k, then Qk returns only k tuples

from S

(r);

2. If Card(S

(r)) = k, then Qk returns the skylines

(i.e. the set S

(r));

CoSky: A Practical Method for Ranking Skylines in Databases

513

3. If Card(S

(r))k, then the elements of S

(r) are

not enough numerous for an correct answer. A

Multilevel skylines approach has to be applied.

That means, not only all elements of the S

(r),

the ﬁrst level, are returned from (r\S

(r)), but also

some of the S

(r), the set of skylines result from

(r\(S

(r) ∪ S

(r)) and if the number of result tu-

ples is still less than k, then we have to build S

(r),

and so on . . .

The following algorithm DeepSky uses this multi-

level principle with our ranking method to ﬁnd Top-k

ranked Skylines. It returns the Multilevel k skylines

that have the k highest scores computed by the CoSky

procedure.

Input: The database relation r, Preferences

on attributes, and k

Output: the Top-k tuples/objects with best

scores : Topk

FS := 0;

rlayer := r;

while FS ≥k or rlayer =

0 do

rsky := CoSky(rLayer);

FS := FS + card (rsky);

if FS ¡ k then

Topk := Topk ∪ rsky;

rlayer := rlayer \ rsky;

end

else if FS ≥k then

Topk := the ﬁrst k skylines of rsky;

return Topk;

end

return Topk;

Algorithm 1: Algorithm DeepSky for ﬁnding the best Top-k

skylines.

Example: If we consider k = 4, the algorithm Deep-

Sky returns C

, C

, the ranked skylines from level

0 and C

the only skyline from level 1.

7 CONCLUSIONS

In this paper, we proposed novel techniques for rank-

ing skyline objects. Three contributions are de-

scribed: The ﬁrst is an enrichment of dp-idp method

by dominance hierarchy to fast scoring skylines. The

second is the CoSky method based on the renowned

TOPSIS schema from Multiple Criteria Decision

Analysis and Salton’s cosine similarity from Informa-

tion retrieval. An example of an SQL implementa-

tion of the proposed method was also described. Fi-

nally, we presented an algorithm for ﬁnding the Top-

k ranked skylines that have k highest scores using

the principle of Multilevel skylines and the CoSky

method. As a short term future work, we will im-

plement our approach in online sales applications in a

Big Data context.

REFERENCES

Bartolini, I., Ciaccia, P., Oria, V., and Ozsu, T. (2007). Flex-

ible integration of multimedia sub-queries with qual-

itative preferences. journal of Multimed Tools Appl,

33:275––300.

Behzadian, M., Otaghsara, S. K., Yazdani, M., and Ignatius,

J. (2012). A Review on state of the art survey of TOP-

SIS applications. Expert Systems with Applications,

39:13051—-13069.

Bentley, J. L., Kung, H. T., mellon Umversuy Putsburgh,

C., Schkolnick, M., and Thompson, C. D. (1978). On

the average number of maxima in a set of vectors and

applications. Journal of the ACM.

Borzsonyi, S., Kossmann, D., and Stocker, K. (2001). The

skyline Operator. In Proceedings of the ICDE Confer-

ence, page 421–430.

Chan, C. Y., Jagadish, H. V., Tan, K., Tung, A., and Zhang,

Z. (2006). On high dimensional skylines. In Pro-

ceedings of the International Conference on Extend-

ing Database Technology (EDBT), pages 478–495.

Chen, L., Gang, C., and Li, Q. (2015). Efﬁcient algorithms

for ﬁnding the most desirable skyline objects. Knowl-

edge Based Systems, 89:250–264.

Chomicki, J., Godfrey, P., Gryz, J., and Liang, D. (2003).

Skyline with presorting. pages 717– 719.

Das Sarma, A., Lall, A., Nanongkai, D., and Xu, J. (2009).

Randomized multi-pass streaming skyline algorithms.

PVLDB, 2:85–96.

Hosseinzadeh Lotﬁ, F. and Fallahnejad, R. (2010). Im-

precise shannon’s entropy and multi attribute decision

making. Entropy, 12.

Huang, J. (2008). Combining entropy weight and TOPSIS

method for information system selection. In Proceed-

ings of International Conference on Automation and

Logistics, pages 1184–1281.

Lai, Y., Liu, T., and Hwang, C. (1994). TOPSIS for MODM.

European Journal of Operational Research, 76:486–

500.

Lakhal, L., Nedjar, S., and Cicchetti, R. (2017). Multi-

dimensional skyline analysis based on agree concept

lattices. Intelligent Data Analysis, 21:1245—-1265.

Papadias, D., Tao, Y., Fu, G., and Seeger, B. (2005).

Progressive skyline computation in database systems.

ACM Trans. Database Syst., 30:41–82.

Preisinger, T. and Endres, M. (2015). Looking for the Best,

but not too Many of Them: Multi-Level and Top-k

Skylines. International Journal on Advances in Soft-

ware, 8:467–480.

KDIR 2019 - 11th International Conference on Knowledge Discovery and Information Retrieval

514

Spyratos, N., Sugibuchi, T., Simonenko, E., and Meghini,

C. (2012). Computing the skyline of a relational table

based on a query lattice. CEUR Workshop Proceed-

ings, 876:145–160.

Tan, K.-L., Eng, P.-K., and Chin Ooi, B. (2001). Efﬁcient

progressive skyline computation. pages 301–310.

Tscheikner-Gratl, F., Egger, P., Rauch, W., and Kleidor-

fer, M. (2017). Comparison of multi-criteria decision

support methods for integrated rehabilitation prioriti-

zation. Water, 9(2).

Valkanas, G., Papadopoulos, A., and Gunopulos, D. (2014).

Skyline ranking

a la ir. CEUR Workshop Proceedings,

1133:182–187.

Vlachou, A. and Vazirgiannis, M. (2010). Ranking the

sky: Discovering the importance of SKYLINE points

through subspace dominance relationships. Data and

Knowledge Engineering, 69:943–964.

Yiu, M. and Mamoulis, N. (2007). Efﬁcient Processing of

Top-k Dominating Queries on Multidimensional Data.

In Proceedings of the VLDB conference, pages 483–

494.

APPENDIX

WITH r Sky AS ( SELECT ∗ FROM C a rs C1

WHERE NOT EXISTS

(SELECT ∗ FROM Ca r s C2

WHERE ( C2 . p r i c e <= C1 . p r i c e

AND C2 . klm <= C1 . klm

AND C2 . pow er >= C1 . powe r )

AND ( C2 . p r i c e < C1 . p r i c e

OR C2 . klm < C1 . klm

OR C2 . p ower > C1 . powe r ) )

) ,

rSkyNorm AS (SELECT i d c a r ,

p r i c e / S p r i c e AS pr iceNo rm ,

klm / Sklm AS klmNorm ,

pow er / Spower AS powerNorm

FROM rSky ,

(SELECT SUM ( p r i c e ) AS S p ri c e ,

SUM ( klm ) AS Sklm ,

SUM ( pow er ) AS Sp ower FROM r Sk y )

) ,

r Sk y G in i AS ( SELECT

1− (SUM ( priceN or m ∗ p ri ce No rm ) ) AS p r i c e g i n i ,

1−(SUM ( klmNorm ∗ klmNorm ) ) AS k l mg i ni ,

1− (SUM ( powerNorm ∗ powerNorm ) ) AS p o w e r gin i

FROM rSkyNorm

) ,

rSkyW AS (SELECT

p r i c e g i n i / ( p r i c e g i n i + k l m g i ni + p o w e rg i n i ) AS p ri ce w ,

k l m gi n i / ( p r i c e g i n i + k l mgi n i + p o w e r g in i ) AS klmw ,

p o wer g i n i / ( p r i c e g i n i + k l m gin i + p o wer g i n i )AS powerw

FROM r Sk y G in i

) ,

rS kyP ond AS (SELECT

i d c a r ,

p ri c e w ∗ p ri ce No rm AS p r i ce p o nd ,

klmw ∗ klmNorm AS klmpond ,

powerw ∗ powerNorm AS pow erp ond

FROM rSkyNorm , rSkyW

) ,

i d e a l AS (SELECT

MIN ( p r i c e p o n d ) AS I DL p ri ce ,

MIN ( klmpon d ) AS IDLklm ,

MAX ( powe rpon d ) AS IDLpower

FROM r s ky Po n d

) ,

r Sk y Sco r e AS (SELECT

i d c a r ,

( I D Lp r i ce ∗ p r i c e p o nd + IDLklm ∗ klmpond +

IDLpower ∗ pow erp ond ) /

( s q r t ( p r i c e p o n d ∗ p r i c e p o n d + klmpond ∗ klm pond +

powerp ond ∗ power pond ) ) ∗

( s q r t ( I D Lp r i ce ∗ I DLp r ice + IDLklm ∗ IDLklm +

IDLpower ∗ IDLpower ) ) )

AS s c o r e

FROM i d e a l , r Sky Pon d

)

SELECT ca . i d c a r , p r i c e , klm , power , s c o r e FROM

rS ky ca , r S ky S co r e r s

WHERE c a . i d c a r = r s . i d c a r

ORDER BY s c o r e d es c ;

CoSky: A Practical Method for Ranking Skylines in Databases

515