Estimation of the Maximum Domination Value in Multi-Dimensional Data Sets

Transcription

1 Proceeings of the 4th East-European Conference on Avances in Databases an Information Systems ADBIS) 200 Estimation of the Maximum Domination Value in Multi-Dimensional Data Sets Eleftherios Tiakas, Apostolos. Papaopoulos, an Yiannis Manolopoulos Department of Informatics, Aristotle University, 5424 Thessaloniki, Greece Abstract. The last years there is an increasing interest for query processing techniques that take into consieration the ominance relationship between objects to select the most promising ones, base on user preferences. Skyline an top-k ominating queries are examples of such techniques. A skyline query computes the objects that are not ominate, whereas a top-k ominating query returns the k objects with the highest omination score. To enable query optimization, it is important to estimate the expecte number of skyline objects as well as the maximum omination value of an object. In this paper, we provie an estimation for the maximum omination value for ata sets with statistical inepenence between their attributes. We provie three ifferent methoologies for estimating an calculating the maximum omination value, an we test their performance an accuracy. Among the propose estimation methos, our metho Estimation with Roots outperforms all others an returns the most accurate results. Introuction Top-k an skyline queries are two alternatives to pose preferences in query processing. In a top-k query a ranking function is require to associate a score to each object. The answer to the query is the set of k objects with the best score. A skyline query oes not require a ranking function, an the result is base on preferences minimization or maximization) pose in each attribute. The result is compose of all objects that are not ominate. For the rest of the work we eal with multiimensional points, where each imension correspons to an attribute. Formally, a multiimensional point p i = x i, x i2,..., x i ) D ominates another point p j = x j, x j2,..., x j ) D p i p j ) when: a {,..., } : x ia x ja b {,..., } : x ib < x jb where is the number of imensions. A top-k ominating query may be seen as a combination of a top-k an a skyline query. More specifically, a top-k ominating query returns the k objects with the highest omination scores. The omination value of an object p, enote as omp), equals the number of objects that p ominates [2, 3].

2 A B C D E F G H I J Distance to beach m) Price euros) F B C Price euros) G E J I A H D Distance to beach m) Fig.. The hotel ata set. The maximum omination value is the number of objects ominate by the top- best) object. More formally, let us assign to each item t of the ata set D a score, mt), which equals the number of items that t ominates: mt) = {q D : q t} Then, if p is the object with the maximum omination value we have: p = arg max{mt), t D} t An example is illustrate in Figure. A tourist wants to select the best hotel accoring to the attributes istance to the beach an price per night. The omination values of all hotels A, B, C, D, E, F, G, H, I, J are 0,, 0, 0, 2, 0, 4, 6, 2, 3 respectively, thus the hotel with the max omination value is H. This hotel is the best possible selection, whereas the next two best choices are hotels G an J. In this work, we focus on estimating the maximum omination value in a multiimensional ata uner the uniformity an inepenence assumptions. Estimating the maximum omination value contributes in: i) optimizing top-k ominating an skyline algorithms, ii) estimating the cost of top-k ominating an skyline queries, iii) eveloping pruning strategies for these queries an algorithms. Moreover, we show that the maximum omination value is closely relate to the carinality of the skyline set. The rest of the article is organize as follows. Section 2 briefly escribes relate work in the area. Section 3 stuies in etail ifferent estimation methos, whereas Section 4 contains performance evaluation results. Finally, Section 5 conclues the work. 2

3 2 Relate Work As we will show in the sequel, the maximum omination value is relate to the skyline carinality which has been stuie recently. There are two ifferent approaches for the skyline carinality estimation problem: i) the parametric methos, an ii) the non-parametric methos. Parametric methos use only main parameters of the ata set, like its carinality an its imensionality. Bentley et al. [] establishe that the skyline carinality is Oln ) ). Buchta [3] prove ) another asymptotic boun of the skyline carinality, which is: Θ ln ) )!. Bentley et al. [] an Gofrey [6, 7], uner the assumptions of attribute value inepenence an that all attributes in a imension are unique an totally orere, establishe that the skyline carinality can be estimate with harmonics: ŝ, = H,. Gofrey [6, 7] establishe that for sufficient large, ln ) ŝ, = H, )!. Lu et al. [0] establishe specific parametric formulae to estimate the skyline carinality over uniformly an arbitrary istribute ata, keeping the inepenence assumption between imensions. on-parametric methos use a sampling process in the ata set to capture its characteristics an estimate the skyline carinality. Chauhuri et al. [4] relax the assumptions of statistical inepenence an attribute value uniqueness, an they use uniform ranom sampling in orer to aress correlations in the ata. They assume that the skyline carinality follows the rule: s = A log B for some ln ) constants A, B which is an even more generalize formula of )! ), an using log sampling they calculate the A, B values. Therefore, this metho can be seen as a hybri metho both parametric an non-parametric). Zhang et al. [4] use a kernel-base non-parametric approach that it oes not rely on any assumptions about ata set properties. Using sampling over the ata set they erive the appropriate kernels to efficiently estimate the skyline carinality in any kin of ata istribution. Both irections sometimes prouce significant estimation errors. Moreover, in non-parametric methos there is a traeoff between the estimation accuracy an the sampling preprocessing cost over the ata. In this paper, we focus on estimating the maximum omination value using only parametric methos. To the best of our knowlege, this is the first work stuying the estimation of the maximum omination value an its relationship with the skyline carinality. 3 Estimation Methos In this section we present specific methos to estimate the maximum omination value of a ata set. We first explain how this maximum omination value is strongly connecte with the skyline carinality of the ata set. ext, we present two estimation methos inspire from [6, 7, 0], an finally we propose a novel metho that is much simpler, more efficient an more accurate than its opponents. For each presente estimation metho, the main task is to prouce a formula that inclues only the main ata set parameters, which are: the number of items 3

4 of the ata set carinality ), an the number of the existing attributes imensionality ). In this respect, several properties an results are erive for the maximum omination value an the item having this value. For the remaining part of this stuy we aopt the following assumptions: All attribute values in a single imension are istinct omain assumption). The imensions are statistically inepenent, i.e., there are no pair-wise or group correlations nor anti-correlations inepenence assumption). Let p i, i {,..., } be the items of the ata set, an x i, x i2,..., x i ) their corresponing attributes in the selecte imensions. Uner our assumptions, no two items share a value over any imension, thus the items can be totally orere on any imension. Therefore, it is not necessary to consier the actual attribute values of the items, but we can conceptually replace these values by their rank position along any imension. Thus, let r i, r i2,..., r i ) be the corresponing final istinct rank positions of item p i in the selecte imensions where r ij {,..., }). Without loss of generality, we assume that over the attribute values in a imension minimum is best. Then, the item with rank position will have the smallest value on that imension, whereas the item with rank position will have the largest one. 3. Maximum Domination Value an Skyline Carinality Here we stuy how the maximum omination value is relate to the skyline carinality of the ata set. Figure 2 reveals this relationship. Let p be the item of the ata set with the maximum omination value. A first important property is that p is efinitely a skyline point. This was first prove in [2] for any monotone ranking function over the ata set, an also shown in [2, 3] for the top- item y p q O x Fig. 2. Maximum omination value an skyline items. 4

5 in top-k ominating queries. Moreover, p ominates most of the items lying in the marke area. This area is calle the omination area of p. o other point ominates more items than p oes. Let om be the exact omination value of p, which is the number of all items that lie in its omination area i.e., the maximum omination value of the ata set). On the other han, the skyline items are the items that lie in the otte line. Let s be the number of the skyline items i.e., the skyline carinality). As p oes not ominate any item containe in the skyline, its omination value satisfies the relation: om s Therefore, a simple overestimation of the maximum omination value is om = s, an can be compute when the skyline carinality s is alreay known or it has been efficiently estimate om = ŝ). The error rate of this estimation epens only on the items that lie neither in the skyline nor in the omination area of p, like item q for example. These items are calle outliers. Moreover, as the ata set carinality increases, the number of outliers becomes significantly smaller than, an the estimation becomes more accurate. On the contrary, as the ata set imensionality increases, the number of outliers also increases, an the estimation becomes less accurate. 3.2 Estimation with Harmonics Here, we present an estimation approach using harmonic numbers an their properties, inspire from [6, 7]. The analysis reveals the intrinsic similarities between the maximum omination value an the skyline carinality, an shows that om = ŝ. Let om, be the ranom variable which measures the number of items ominate by the top- item the maximum omination value). We enote as om, the expecte value of om,. Theorem. In any ata set uner the omain an inepenence assumptions, the expecte value om, satisfies the following recurrence: om, = om, + om, for >, > 0, where om, = an om, = 0. Proof. If =, then we have only one imension an the item with rank position is the top- item that ominates all other items. Thus it hols that om, =. If =, then we have only one item an none item to ominate. Thus, om, = 0. In case that > an >, there is an item with rank position on imension. This item has the maximum omination value as it ominates all other items on that imension. The probability that this item will remain a top- item is the probability that no other item has a greater omination value in any other imension 2,..., ), given the inepenence assumption. However, om, is the maximum omination value out of these imensions. Thus, 5

6 as any item has equal probability to be place in rank position on imension, we have om, to be the probability that this item has the maximum omination value. Since, the first ranke item on imension cannot be ominate by any other item, the maximum omination value is etermine by the remaining items which is estimate by om,. Therefore, we have: om, = om, + om, The recurrence for om, is strongly relate to the harmonic numbers: The harmonic of a positive integer n is efine as: H n = n i= i. The k-th orer harmonic [] of a positive integer n for integers k > 0 is efine as: H k,n = n H k,i i= i, where H 0,n =, n > 0 an H k,0 = 0, k > 0. ote also that: H,n = H n, n > 0. In orer to retrieve the funamental relation of om, with the harmonic numbers, we compute om 2, an using mathematical inuction we erive the final formula. For the om 2, value we have: om 2, = om, + om 2, = + om 2, = = + om, + om 2, 2 = = = = i = = i= ) i = H ow, let us assume that the following equation hols for a specific k, i.e., om k, = H k, ). We will prove that the previous equation hols also for the next natural number k +. We have: om k+, = om k, + om k+, = i=2 = om k, + om k, + 2 om k, = = i= = i= i om k,i = H ) k,i i i= = 6 i i H k,i) = i= H k,i i = H k,

7 Therefore, for any >, > 0 it hols that: om, = H, ) Equation generates some important properties for the maximum omination value: om, is strongly relate to the skyline carinality of the ata set. As shown in [6, 7], if ŝ, is the expecte value of the skyline carinality, then it hols that: ŝ, = H, Therefore, we have: om, = ŝ, 2) In particular, om, an ŝ, share the same recurrence equation of Theorem but with ifferent initial conitions. as prove in [], it hols that lim H, =. Therefore, we have: lim om, = lim H, lim om, = 0 3) Equation 3 is a valiation of the fact that as the imensionality increases, the maximum omination value an consequently all the following omination values) ecreases until reaching zero. In particular, the imensionality beyon which all omination values become equal to zero, is a small number. We call that imension the eliminating imension, an enote it as 0. In the sequel, we focus in the computation of om,. Since it hols that om, = H,, the main task is the efficient computation of the harmonic term H,. There are three ifferent methos to follow for this task: Recursive Calculation: The calculation of H, can be achieve by running a recursive algorithm that follows the irect efinition formula: H k,n = n i= H k,i i where H 0,n = an k > 0. We can also use a look up table at run-time, however, these recurrence computations are expensive. The algorithmic time complexity is exponential: O ). As shown later in the experimental results section, the calculation time is not acceptable even for small imensionality values. Boun Approximation: This metho was propose in [6, 7] an is base on asymptotic bouns of H k,. Bentley et al. [] establishe that: ŝ, is Oln ) ). Bentley et al. [] an Gofrey [6, 7] establishe that: ŝ, = H,, thus: H, is Oln ) ) 7

8 Buchta [3] an Gofrey [6, 7] improve this asymptotic boun as follows: ) ln ) H, Θ )! Therefore, we can instantly estimate om, using the following formula: ) ln ) om, Θ )! or equivalently for an appropriate real number λ): ) ln ) om, λ )! 4) This is not a concrete estimation an generates a significant error rate. Moreover, by varying the imensionality range it will be shown that this estimation is not even a monotone function an changes its monotonicity after halving the eliminating imension i.e., for any > 0 2 ). Therefore, it provies wrong theoretical results. Generating Functions Approximation: This metho was also propose in [6, 7] an is base on Knuth s generalization via generating functions [8, 9], which establishe that: H k, = k c,c 2,...,c k i= H ci i, i c i ci!, c, c 2,..., c k 0 c + 2c kc k = k 5) where H i, is the i-th hyper-harmonic of an is efine as: H i, = j= j i H, = H, = H ) ote that c, c 2,..., c k are positive or zero) integer numbers, whereas the number of terms of the sum in Equation 5 stems from all possible combinations of c, c 2,..., c k that satisfy the equation c + 2c kc k = k. This number is k) an expresses the number of all possible ways to partition k as a sum of positive integers. Therefore, H k, can be expresse as a polynomial of k) terms which contain the first k hyper-harmonics H i,, i =,..., k). For example: H 2, = 2 H2, + 2 H 2, H 3, = 6 H3, + 2 H, H 2, + 3 H 3, This approximation of H k, is remarkably accurate. In particular, with this metho we reach almost exactly the theoretical values of H k, when compute 8

9 with the recursive approach. This will be also evaluate in the experimental results section. For any given imension, the time cost to compute the require hyper-harmonics is O). Then, having the previous formulae, we can immeiate calculate H,. The only requirement is to generate the appropriate formula for the imension with the ) terms. Gofrey [6, 7] mentione that this number of terms )) grows quickly, an, thus, it is not viable to compute the require formula this way, an suggests not using this approximation for large values of. However, motivate by the accuracy of this approximation of H k,, we evelope a ynamic-programming algorithm that efficiently prouces these equations. Due to lack of space we o not elaborate further. Therefore, we can almost instantly estimate om, with hyper-harmonics using the previous approximation formula: om, c,c 2,...,c i= H ci i, i c i ci! 6) by taking special care to all possible floating point overflow values, an by using the erive equations which recore through the automation. 3.3 Estimation with Multiple Summations In this section we present an estimation approach using a specific formula with multiple summations inspire from the stuy of [0]. For compatibility reasons we will keep all previous notations an variables. Y. Lu et al. [0] introuce an estimation approach of the skyline carinality that relaxes the omain assumption of our basic moel. The statistical inepenence assumption still remains, but now the ata can have uplicate attribute values. Their stuy is base in probabilistic methos, an it uses the value carinality of each imension. Their first main result is the following: ŝ, = c c 2 t = t 2=... c t = ) c i= i t j c j= j where,, an c j is the value carinality of the j-th imension. They also generalize this result in case of having the probability functions f j x) of the ata over each imension, but always keeping the inepenence assumption: ŝ, = c c 2 t = t 2=... c t = f t )f 2 t 2 )...f t ) t j j= x= f j x) However, in both cases, the computational complexity is Oc c 2... c ), which is not acceptable even if in few imensions the value carinality is high close to ). They also trie to relax this complexity cost by introucing high an low 7) 8) 9

10 carinality criteria, but this cost remains high, an this is why their experimental results are restricte to small imensionality an carinality variations =, 2, 3 an 000). We will see in our experimental results that even if we have high carinality in 3 imensions an up to 000 items the estimation time is not acceptable. Although the metho of [0] works efficiently only in small carinalities an imensionalities, it woul be very interesting to apply this metho in our moel an stuy its accuracy. Therefore, uner the omain assumption of our moel, all value carinalities c j will be equal to an Equation 7 gives: or equivalently: ŝ, =... t = t 2= ŝ, = t =... t = t 2 = ) t ) t 2...t t = t ) t 2...t Thus, using the property of the estimate maximum omination value of Equation 2, the final estimation formula is: om,... t = t 2= t = t ) t 2...t 9) which has an exponential computational complexity O )). In our experimental results we will see that Equation 9 returns values remarkably close to the harmonic H, values. In aition, by increasing, the returne values converge to H,, thus it must be relate somehow with the k-th orer harmonics. This strong relation remains unproven. Finally, as the two methos return almost the same estimations, their accuracy is similar. 3.4 Estimation with Roots In this section we present a novel estimation approach using a simple formula, which provies more accurate estimation results. Let p be the item with the maximum omination value, an r p, r p2,..., r p ) be its corresponing final rank positions in the total orering along any imension. Let also a be the maximum rank position of p through all imensions i.e., a = max{r p, r p2,..., r p }). Then, a splits the total orering of the items in two parts as in Figure 3: i) the a)-area, an ii) the a)-area. ow, any item q that all its rank positions lie in the a)-area, will be an outlier or a skyline item. ote that the opposite oes not hol, thus not any skyline or outlier item lie in the a)-area. The probability P a that an item lies in the a)-area is: P a = P p i lies in the a) area) = 0

11 a area -a) area i p q r i 2 i q p r... p q r Fig. 3. Total orering of items by rank positions = P r i a r i2 a... r i a) ow, ue to the inepenence assumption we have: P a = P r i a) P r i2 a)... P r im a) = a a... a = a a ) = Moreover, any item r that all its rank positions lie in the a)-area, will efinitely be ominate by p. Thus r lies in the omination area of p. ote that the opposite oes not hol, thus not any item of the omination area of p, lies also in the a)-area). The probability P a that an item lies in the a)-area is: P a = P p i lies in the a) area) = = P r i > a r i2 > a... r i > a) Due to the inepenence assumption we have: P a = P r i > a)... P r i > a) = a... a a) = = a ) Therefore, the number of items lying in the a)-area C a ), an in the a)- area C a ) will be: a C a = P a = ) an C a = P a = a ) respectively. However, as it hols that all items lying in the a)-area are ominate by p, we have omp) C a, or equivalently: omp) P a omp) a ) 0) which escribes a tight lower boun for the omination value of p. Aitionally, as p efinitely lies in the a)-area, at least one item must be insie that area, thus it must hol that C a, or equivalently: a ) P a )

12 To efficiently estimate the maximum omination value, we have to maximize the lower boun of Inequality 0 uner the a variable, respecting the conition of Inequality for the a variable. Therefore, let us efine the function fa) = ) a that expresses the lower boun values, where a [0, ]. It has f0) =, f) = 0 an the following relation erives: f a) = a ) We have f a) < 0, a 0, ), thus f is a monotone escening function in [0,], an returns values also in [0,]. Moreover, the conition of Inequality gives: a ) a a Thus, f must be restricte in [, ]. Due to the escening monotonicity of f, it takes its maximum value when a max =. Therefore, we have: fa max ) = a ) ) max ) = = an the final estimation of the maximum omination value is: ) om, 2) Generalization of the Root Metho To get even more accurate estimations, we can generalize the root metho by allowing the a variable to take values slightly smaller than. This left shift of the a variable breaks the conition of Inequality, but allows the possibility of taking into account items that are not lying in the a), a)-areas an are ominate by p increasing its omination value. We further stuie this estimation improvement making exhaustive experimental tests with ifferent a values, an we conclue that the estimation is very accurate when: a shifte = Then f takes the value: ) fa shifte ) = a ) shifte = = This hien square root factor enhances the estimation accuracy an provies the most accurate results for the maximum omination value. Thus, the final propose estimation formula is: om, ) 3) ) 2

13 4 Performance Evaluation To test the estimation accuracy, we perform several experiments using inepenent ata sets of = 00, K, 0K, 00K, M items an varying the imensionality from to values beyon the eliminating imension 0. We recor the exact average of 0 same type ata sets) an the estimate maximum omination values. For brevity, we present only a small set of representative results, which epict the most significant aspects. All experiments have been conucte on a Pentium 4 with 3GHz Qua Core Extreme CPU, 4GB of RAM, using Winows XP. All methos have been implemente in C++. Table summarizes the methos compare. otation Interpretation RealAvg Real Average Values o Estimation) HarmRecc Estimation with Harmonics Recursive Calculation) HarmBoun Estimation with Harmonics Boun Approximation) HarmGenF Estimation with Harmonics Generating Functions Approximation) CombSums Estimation with Multiple Summations Roots Estimation with Roots Simple) RootsGen Estimation with Roots Generalize with the square root) Table. Summary of methos evaluate. RealAvg HarmRecc HarmBoun HarmGenF CombSums Roots RootsGen Table 2. Maximum omination value estimation for = K Figure 4 epicts the maximum omination value estimation results for all estimation methos, varying the carinality an the imensionality of the ata 3

14 sets. Table 2 presents the etaile estimation values of the corresponing graph for =K, for further inspection. We have not recore the values where the MAX omination value estimation =00) MAX omination value estimation =000) RealAvg HarmRecc HarmBoun HarmGenF RootsGen CombSums Roots RealAvg HarmRecc HarmBoun HarmGenF RootsGen CombSums Roots MAX omination value estimation =0000) MAX omination value estimation =00000) RealAvg HarmRecc HarmBoun HarmGenF RootsGen CombSums Roots RealAvg HarmRecc HarmBoun HarmGenF RootsGen CombSums Roots MAX omination value estimation =000000) RealAvg HarmRecc HarmBoun HarmGenF RootsGen CombSums Roots Fig. 4. Maximum omination value estimation for =00, K, 0K, 00K, M. Estimation Error = ) 00% 90% 80% 70% 60% 50% 40% 30% HarmBoun HarmGenF Roots RootsGen 20% 0% 0% Fig. 5. Estimation error for =M. 4

15 computational time is more than 0 minutes. Figure 5 presents the estimation error of the 4 methos that return values into the full range for =M, for further inspection. Base on the previous results we observe the following: The HarmRecc metho, ue to the exponential computation complexity, returns values in short time only for small imensionality an carinality values. Moreover, after 3 imensions the estimation error is significant. The HarmBoun metho returns estimation results instantly, but it is the most inaccurate metho for estimation. The HarmGenF metho computes its results very efficiently. It prouces almost exactly the theoretical values of H k, when compute recursively. Therefore, it returns the same estimation results with the HarmRecc metho. However, we observe that as we increase the carinality of the ata set, the estimation error increases an becomes significant in almost all the imensionality range. The CombSums metho fails to return values even in small imensionality an carinality selections, ue to its exponential computational complexity we can see only 4 values when =00 an 2 values when =000). In aition, the returne values are remarkably close to those of HarmGenF an HarmRecc. By increasing the carinality, the values converge to the harmonic values of the previous methos, thus it must be relate somehow with the k-th orer harmonics. However, again the estimation error increases an becomes significant in the whole imensionality range. The Roots metho returns estimation results more efficiently than all the previous methos. By increasing the carinality, the estimation becomes more accurate, ue to the fact that the number of the outliers becomes significantly smaller than. On the contrary, as the imensionality increases, the number of outliers increases as well, an the estimation becomes less accurate. However, when we further move into the imensionality range an when we approach the eliminating imension, the estimation becomes again accurate. The RootsGen metho is the most efficient way to get estimation results an outperforms all previous methos. It manages to approximate the maximum omination value with the smallest estimation error in the whole imensionality an carinality range. 5 Conclusions This paper stuies parametric methos for estimating the maximum omination value in multi-imensional ata sets, uner the assumption of statistical inepenence between imensions an the assumption that there are no uplicate attribute values in a imension. The experimental results confirm that our propose estimation metho outperforms all other methos an achieves the highest estimation accuracy. Future work may inclue: i) further stuy of the eliminating imension 0, proviing an estimation formula for its calculation, ii) the stuy the estimation of the skyline carinality uner the Roots metho an its variants 5

16 an iii)the stuy of the maximum omination value estimation an the eliminating imension in arbitrary ata sets, by relaxing the assumptions of uniformity, inepenence an istinct values that use in this work. References. J.L. Bentley, H.T. Kung, M. Schkolnick an C.D. Thompson: On the Average umber of Maxima Set of Vectors an Applications, Journal of the ACM, Vol.25, o.4, pp , S. Borzsonyi, D. Kossmann an K. Stocker, The Skyline Operator, Proceeings 7th International Conference on Data Engineering ICDE), pp , Heielberg, Germany, C. Buchta, On the average number of maxima in a set of vectors., Information Processing Letters 33, pp.6365, S. Chauhuri,. Dalvi an R. Kaushik, Robust Carinality an Cost Estimation for the Skyline Operator, Proceeings 22n International Conference on Data Engineering ICDE), Atlanta, GA, J. Huang: Tuning the Carinality of Skyline, Proceeings 0th Asia-Pacific Web Conference APWeb) Workshops, Shenyang, China, P. Gofrey, Carinality Estimation of Skyline Queries: Harmonics in Data, Technical Report CS , York University, P. Gofrey, Skyline Carinality for Relational Processing, Proceeings 3r International Symposium of Founations of Information an Knowlege Systems FoIKS), pp.78-97, Wilhelminenburg Castle, Austria, R.L. Graham, D.E. Knuth an O. Patashnik, Concrete Mathematics, Aison- Wesley, D.E. Knuth, Funamental Algorithms: The Art of Computer Programming., Aison-Wesley, Y. Lu, J. Zhao, L. Chen, B. Cui an D. Yang, Effective Skyline Carinality Estimation on Data Streams, Proceeings 20th International Conference on Database an Expert Systems Applications DEXA), pp , Turin, Italy, S. Roman, The harmonic logarithms an the binomial formula, Journal of Combinatorial Theory, Series A, Vol.63, pp.43-63, M.L. Yiu an. Mamoulis, Efficient Processing of Top-k Dominating Queries on Multi-Dimensional Data, Proceeings 33r International Conference on Very Large Data Bases VLDB), pp , Vienna, Austria, M.L. Yiu an. Mamoulis, Multi-Dimensional Top-k Dominating Queries, The VLDB Journal, Vol.8, o.3, pp , Z. Zhang, Y. Yang, R. Cai, D. Papaias an A. Tung: Kernel-Base Skyline Carinality Estimation, Proceeings ACM International Conference on Management of Data SIGMOD), pp ,