Approximate Flexible Queries Using Hausdorff Distance

Approximate Flexible Queries Using Hausdorff Distance A. Aggoune1, A.Bouramoul2, M-K. Kholladi2 1

Computer Science Department, University 8 may 45 of Guelma, B.P. 410, Guelma 24000, Algeria [email protected]

2

Computer Science Department, Misc Laboratory, University of Mentouri Constantine. B.P. 325, Constantine 25017, Algeria [email protected], [email protected]

Abstract-- Approximate flexible queries have emerged as an effective approach in the process of interrogation of the databases, and more especially for dealing the problem of empty answers. Most work in this domain, however, has so far been limited in its applicability and capacity to treat the problem of empty answers. In this paper, we propose the use of Hausdorff distance as an effective tool for approximate flexible queries in the context of treatment the Empty Answers Problem (EAP). Our approach is based on measuring semantic proximity between flexible queries and using this measure to provide approximate answers to failed queries (queries with empty answers). We develop novel approach for approximate flexible queries by using the Hausdorff distance applied between fuzzy predicates compounded these queries. This, of course, guarantees extremely non empty answers with fast response times. Key words-- Information retrieval, Data base, Flexible queries, Semantic proximity, Empty answers, Hausdorff distance.

I.

INTRODUCTION

Nowadays, there is more and more interest in using the World Wide Web, especially, for searching and retrieving information over large databases that are available "on-line". Exploiting Web-based information sources is non-trivial because the user has no direct access to the data. The most well-known problem approached in this field is the "empty answer problem", that is, the problem of providing the user with some alternative data when there is no data fitting his query. Current works for treat this problem are based on the mechanism of reformulation of the initial user's query can be a good alternative to improve the information selectivity [19]. More recently, in [20], another solution has been proposed. It is based a particular of domain ontology to improve the

performance of information retrieval systems. The relaxation paradigm [3]-[11] is one of the basic cooperative techniques used in most of such approaches. In [5], another solution has based on tolerance relation modeled by a parametric relative proximity relation. These works can be classified into two families: the approaches guided by query, and approaches guided by a "workload". The first family, is based on the technique of relaxation and reformulation which consists in modifying the initial query either by the modification of the conditions of the query of the user, or by adding new conditions, or uses a summary operation or by a calculation of distributions of data in the database examined [2]-[3]-[4]-[5]-[6]-[7]-[1]. In the second family of approaches guided by a "Workload", several studies have been developed, which are generally based on cooperative answers and approximation queries, using a set of queries previously evaluated by the system "workload" and the application of a measure of similarity between this workload and the EA to query [8]-[9]-[10]- [11]-[12][13]-[14]-[15]-[16]. Among the approaches to this family: approach based synopses type samples, histogram and approach based wavelet. The evaluation results between these two families of approaches [1]-[2]-[8], clearly show that the approaches guided by a Workload of past queries give a better result compared to the first family of approaches. Indeed, as part of our work, we are interesting in the work of the second family of approaches, when using a Hausdorff distance for measuring the semantic proximity between queries. This paper is structured as follows: the next section recalls the flexible queries on the one hand and Hausdorff distance on the other hand. In Section 3, we present our measuring semantic proximity between flexible queries. Section 4 shows how a semantic proximity can be used for approximate extraction process. In Section 5, we illustrate the search

algorithms of our solution and we conclude in Section 6 with a reminder of the main contributions of the study and outline some future works. II. PRELIMINARY NOTIONS A. Flexible queries Flexible queries (or fuzzy queries) [1]-[2] are queries in which user preferences can be expressed. They also improve the ability of expression of query languages and the needs of users. The attributes of these queries are no longer in principle "all or nothing" but may be more or less satisfied. Several works have been proposed in the literature to introduce flexibility in extraction process. The majority of these works use the formalism of fuzzy sets to model preferences and linguistic terms and to evaluate predicates with such terms [4]. Predicates called gradual (or fuzzy) whose result is a degree of satisfaction, as young and well-paid, are described using fuzzy sets. These fuzzy criteria can be combined with operators of conjunction or disjunction or medium expressing the effects of compensation between criteria. A fuzzy predicate can also compare two attributes using not only the usual operators (equality, superiority, etc..), But also gradual operators such as "more or less equal" or "significantly higher". It is possible to change (in the weakening or strengthening) the meaning of a predicate using a modifier that is usually associated with an adverb (as "very," "somewhat," "relatively" "really"). For example, "very expensive" is more restrictive than "expensive" and "fairly high" is less demanding than "high" [5]. The results of flexible query are then qualified depending on their relevance to the selection criteria and can be ordered by degree of relevance that indicates how the condition is satisfied [6]. B. Hausdorff distance In the following, we recall the principle of the Hausdorff distance and we review the main approaches that can be followed to compute such a distance. 1) Crisp Set: Consider two subsets A and B of a space U (equipped with a metric). The most popular scalar extension of distance between A and B is the Hausdorff distance defined as [5]-[7]: d H (A, B) = max {H(A, B), H(B, A)},

(1)

Where H(A, B) stands for the directed Hausdorff distance from A to B. d(u, B) and We have H(A, B) = Sup u∈A d(u, B) = Inf v∈B d(u, v). The expression d(u, v) stands for a standard distance (such as Euclidean distance). Formula (1) can be written in the following condensed

form: d H (A, B) = max {sup u∈A inf v∈B d(u, v), sup v∈B inf u∈A d(u, v)}. (1') The idea that governs this distance is the following: for each element in A look for the closest element in B, then check for the element in A for which the distance to the closest element in B is maximal. The same is done exchanging B and A and the longest distance of the two component is kept. Intuitively, if the Hausdorff distance is δ, then every point of A must be within a distance δ of some point of B and vice versa. Note that d H is a metric and, in particular, the following statement holds: d H (A, B) = 0 if and only if A = B. Usually, the following equalities are assumed to be true d H (A, ∅) = d H (∅, B) = +∞ and d H (∅, ∅) = 0. Example 1. Let A = [a 1 , a 2 ] and B = [b 1 , b 2 ] be two regular intervals and let d(u, v) = |u − v|. Then, it easy to check that d H (A, B) = max(|a 1 − b 1 |, |a 2 − b 2 |). 2)Fuzzy Sets: The Hausdorff distance between fuzzy sets can be either fuzzy or scalar. Hereafter, we only focus on the scalar version. For the fuzzy evaluation, more details are available in [7] [4]. Dubois and Prade's Approach: In [7], the Hausdorff distance was generalized to fuzzy sets in the following way. Let F and G be two fuzzy sets on U, for any r ∈ R+, let D r (F) defined by µ D (F) (u) = sup v ∈ U {µ F (v) / d(u, v) ≤ r} r (2) Here D r (F), the dilation of F by r, is the result of applying to all points of F a local max operation within a region of radius r. Now, the generalized Hausdorff distance writes: d H (F, G) = inf {r ∈ R+ / F ⊆ D r (G) ∧ G ⊆ D r (F)} (3) Can easily see that the dilation is only "horizontal", then, we may never be able to cover the other set and hence d H (F, G) cannot be defined. Thus, two fuzzy sets must have the same supremum for the Hausdorff distance between them to exist; this is a serious drawback of this definition. Chaudhuri and Rosenfeld's Approach: Another definition of the Hausdorff distance between two fuzzy sets is proposed in [4]. This definition is more general and is valid in the case of two fuzzy sets with unequal maximum memberships. More details about this case are available in [4]. In the following, we consider only fuzzy sets with the same supremum. Let F and G be two discrete fuzzy sets. Let T = {t 1 , t 2, …, tm} the set of all the distinct membership values of F and G. The Hausdorff distance between F and G is defined by the following expression: ∑ti dH (Fti, Gti) (4) d H 2 (F, G) = ∑ti

Where F ti (resp. G ti ) stands for the t i -level cut of F (resp. G). d H 2 (F, G) can be seen as a membershipweighted average of the crisp Hausdorff distances between the level sets of the two fuzzy sets. In case of continuous fuzzy sets, formula (4) is modified in the following form [7]: d H 2 (F, G) =

∫t dH (Ft, Gt) dt ∫t dt

(5)

Example 2. Let now U represent the numeric universe of discourse of the variable "age" of a person. Let also F = "about thirty" and G = "between_26_and_28" two fuzzy sets on U defined by the following two trapezoidal membership functions (t.m.f): F = (30, 30, 3, 3) and G = (26, 28, 1, 1). Now, we evaluate the distance between F and G using formula (5). First, let us precise that F α and G α are regular intervals and can be expressed as follows: F α = [3α + 27, 33 − 3α]; G α = [α + 25, 29 − α]. Then, one can easily check that d H 2 (F, G) = 1 2∫0 t max(( t + 25 ) − ( 3t + 27 ) , ( 29 − t ) − ( 33 − 3t ) )dt = 7/2 III. MEASURING PROXIMITY Taking into account the strong intuitive connection between proximity and distance, and by using the Hausdorff distance measure, we can be estimate to what extent two flexible queries are close, semantically speaking.

proximity measure. The simplest conversion function can be obtained by setting s = 1 and t = 1. B. Compound Queries Let D be a relational database containing n attributes A 1 , A 2 ,…, A n with D(A i ) being the domain of values pertaining to A i . We make the assumption that D(A i ) is closed and bounded (resp. finite) if A i is a continuous (resp. discrete) attribute. Let also Q be a flexible compound query of the form P 1 ∧…∧ P k where the symbol '∧' stands for the connector 'and' (which is interpreted by the 'min' operator) and P i (i=1,k) is a fuzzy predicate pertaining to the attribute A i . Let also Q' be a flexible query of the form Q' = P' 1 ∧…∧ P' s where P' j (j=1,s) is a fuzzy predicate pertaining to the attribute A j . To evaluate the semantic proximity between the two compound queries Q and Q', we distinguish three cases [17]-[18]: Case 1: Q and Q' cover the same attributes exactly. Case 2: Q' covers all the attributes specified in Q. Case 3: Q' does not cover all the attributes specified in Q. We use the notation Σ Q to denote the set of answers to the query Q. Case 1: Q and Q' cover the same attributes exactly. Hence, Q' = P' 1 ∧…∧ P' k with (P i , P' i ) ∈ D(A i ), for i=1,k, which means that both P i and P' i are predicates that constrain the same attribute A i . In this case, Prox(Q, Q') is evaluated in a three-step procedure: For each pair (P i , P' i ), i=1,k, compute Dist(P i , P' i ) = d H 2 (P i , P' i ); 2) For each pair (P i , P' i ), i=1,k, compute Prox(P i , P' i ) using, for instance, formula (5); 3) Then, compute Prox(Q, Q') = min i=1, k Prox(P i , P' i ). 1)

A. Single Predicate Queries (SPQ) Let Q = P and Q' = P' be two SPQ where P and P' are gradual predicates represented by means of fuzzy sets (of course, P and P' are pertaining to the same attribute, say A). To evaluate which extent Q and Q' are close, semantically speaking, we make use of the Hausdorff distance index between the fuzzy predicates involved in those two queries. Then, we write: [17][18] (6) Dist(Q, Q') = d H 2 (P, P'), With Dist(Q, Q') stands for a distance measure between Q and Q'. Let now Prox(Q, Q') denote a proximity measure between Q and Q'. Prox(Q, Q') based on the distance Dist(Q, Q') can be obtained in two steps: • Normalizing Dist(Q, Q') with a function f norm that reduces the range to the interval [0, 1]. • Under this assumption, Prox(Q, Q') can be defined in the following way: Pr ox(Q ,Q' ) = 1 − f norm (Dist (Q ,Q' ))

(7)

The well known expressions of f norm are [4]: f norm (x) = min(1, x) and f norm (x) = x/(1+x). Another approach to defining a proximity measure is to use a conversion function on the distance measure. For instance, Prox(Q, Q') can be defined by: t Prox(Q, Q')=1+ Dist (Q, Q') s

(8)

The positive constants s and t adjust the size of the

Case 2: Q' covers all the attributes specified in Q. Then, Q' can be transformed into the following form Q' = P'1 ∧  ∧ P' k ∧ P' k +1 ∧  ∧ P' s with (P i , P' i )∈ D(A i ), for i=1,k. Attributes A j (for j=k+1,s) are not specified in Q. To estimate the proximity between Q and Q', we can apply two strategies: (S1) increased of Q: the idea is to complete Q with constraints on the missing attributes (ie, j for j = k +1 to s). Since the user does not specify any constraints on these attributes, all values of their fields are allowed. Thus, Q can be written: Q = P 1 ∧ ... ∧ P k ∧ D(A k+1 ) ∧… ∧ D(A s ). (S2) Weakening of Q': the idea here is to focus solely on the attributes specified by the user and to measure the closeness of their constraints specified in Q and those present in Q'. Superfluous attributes present in Q' can be deleted. Thus, Q' is rewritten as follows: Q’ R = P’ 1 ∧ ... ∧ P’ k It is easy to see that Q' R is a variant of relaxed Q'. The following relation Σ Q' ⊆ Σ Q’R is always true (ie, the responses of Q' are also of Q' R ). This strategy will take any interest in the processing of EA to queries, since the objective is to provide approximate solutions and alternatives to these queries.

Thus, if the measure Prox (Q, Q' R ) is high (or above a certain threshold), it is perfectly acceptable to offer the answers of Q' as approximate answers to the query Q if its evaluation does produced no response (i.e., Σ Q = ∅). The Strategy S2 can measure the proximity between the constraints of the attributes present in both queries Q and Q'. This proximity can be characterized by local interests who in practice are shown above. As for the strategy S1, it calculates a global proximity between Q and Q'. All those same attributes not specified in Q are taken into account in this calculation. As part of the problem of interest, this property presents a significant disadvantage. Indeed, suppose that Q: "Salary= around 3 € " is an empty query answer. Let Q': "Age = 27 years ∧ Salary = [2.8, 3.2]". The fact reflect the closeness of the value of age 27 and D (Age) (ie Prox (27, D (Age)), With D (Age) = [0, 100], significantly weakening the global proximity (based on min operator) between Q and Q'. This could lead to the rejection of Q' and therefore his answers so that they might find interesting because they approximately satisfy the criterion on the attribute "Salary ".

Case 3: Q' does not cover all the attributes specified in Q. Then, Q' can be transformed into the following form Q' = P' 1 ∧ ... ∧ P' b ∧ P' k+1 ∧… ∧ P' s , with b < k and (P i , P' i ) ∈ D(A i ), for i = 1 to b, and the predicats P j (for j = k+1 to s) are not specified in Q. To calculate Prox(Q, Q') is: • Added to Q' the domaines for missing attributs specified in Q. Q' is wiritten so: Q' = P' 1 ∧ ... ∧ P' b ∧ D(A b+1 )∧…∧ D(A k ) ∧ P' k+1 ∧… ∧ P' s ; • Applying one of two strategies described in case 2. IV. APPROACH PROCESS

FOR

APPROXIMATE EXTRACTION

In this section, we show how the measuring proximity introduced between queries can be a good alternative to improve the information selectivity by using our approximate extraction process. A. Problem Formulation Let Q be a flexible query. In this case, Σ Q contains the items of the database that somewhat satisfy the fuzzy requirements involved in Q. Formally, Σ Q = {t ∈ D / µ Q (t) > 0}, where t stands for a database tuple. Definition. We say that Q results in empty answers if Σ Q = ∅. [2] This means that no data in the database somewhat satisfies the fuzzy conditions involved in Q. In the following, we propose an alternative approach to deal with this problem. The approach proposed leverages the previous queries that have been evaluated by the system and have produced non-empty answers on the one hand and our measuring semantic proximity on the other hand. B. Principle of the Approach Assume that we have at our disposal a past query workload W(D). Let Q = P 1 ∧…∧ P k be a flexible

query with empty answer. The intuition of how we intend to use the workload W(D) in answering Q is the following. The workload may perhaps reveal that, in the past, some queries that have been processed and produced a set of non-empty answers are close to Q, semantically speaking. Thus, it is more convenient to propose as a response to Q the answers of the query that is the closest one to Q than "nothing". To achieve this, we proceed in the following way: 1) Partitioning the workload W(D) in 03 subsets treated the three cases previously cited: W = (D, Q)={Q'/ Q'∈W(D) ∧ |D(Q')∩D(Q)|=k}, W > (D, Q) = {Q'/ Q' ∈ W(D) ∧ |D(Q') ∩ D(Q)|> k}, and W < (D, Q)={Q' / Q' ∈ W(D) ∧ |D(Q') ∩ D(Q)|< k}, 2) (a)- If W = (D, Q)≠ ∅, for element Q'∈ W=(D, Q), estimate the proximity Prox(Q, Q'). Ranking in descending order of queries. (b)- Else if W > (D, Q) ≠ ∅, then we consider W > (D, Q) and apply (a). (c)- Else, we consider W < (D, Q) and apply (a). 3) Choose the closest query Q app in Q, and affect its answers a same degree of membership, which takes as its value, the proximity measured between Q and Q app named Prox (Q, Q app ). . V. SEARCH ALGORITHMS The algorithm of processing the EAP in case of SPQ takes as input the user query Q, the subset W Q (D) is equal to the beginning Workload W(D). The first step (lines 1 to 8) of this algorithm is to determine if the query Q has empty response (ΣQ = Ø) or not (ΣQ ≠ Ø). In the latter case, the algorithm determines if Q exist or not in the Workload W Q (D). To decide if to add this query to the W Q (D) and show his answers, or directly display the results (if where Q is already in W Q (D)). The second step (lines 9-16) is affected when EAP is detected. In this context, the first step to do is to search queries from the WQ (D) are exactly the same attribute specified in Q (to build the set W = (D, Q)), this is achieved by the procedure Filter ( WQ (D)), which modifies the elements of the WQ (D), keeping only the non zero proximity queries, then it ranks the W Q (D) (line 12) by the procedure RankOrder; in descending order according to the degree of proximity. Then, selects the closest query of Q named Q App (line 13) and we display its approximate answers to the Q, taking as degree of membership value of proximity Prox (Q, Q App ). Input: Q: initiale query /* Q:=P */ W Q (D): workload 1: Begin 2: execute Q; 3: if ∑ Q ≠Ø then 4: Begin 5: if Q∉ W Q (D)then W Q (D):=W Q (D)∪{Q};

6: end; 7: Return ∑Q; 8: end 9: else /* ∑ Q =Ø */ 10: Begin 11: Filter (W Q (D)); 12: RankOrder(W Q (D)); 13: Q App = First(W Q (D)); 14: Return ∑Q App ; 15: end; 16: end Output: ∑Q ou bien ∑Q App ALGORITHM 1. TREATMENT OF EAP TO SPQ

Following, we define, the different procedures used in our algorithm:

A. Procedure to Filter WQ (D) Given the Workload W Q (D) to determine if their queries have the same attribute of Q, by executing the following algorithm: Procedure Filter (var set E) 1. begin 2. let Aux = E; E = ∅; 3. while Aux ≠ ∅ do 4. begin 5. Q' = First(Aux); 6. compute Prox(Q, Q'); 7. if Prox(Q, Q') > 0 then 8. E = E ∪ {Q'}; 9. Aux = Aux − {Q'}; 10. end while 11. end. ALGORITHM 2. FILTER OF WORKLOAD

B. Procedure to estimate the semantic proximity: The measure of semantic proximity between queries is based on the use of Hausdorff distance by applying the formula (8), whose algorithm is as follows: Procedure Prox (Q = P, Q' = P') 1. begin 2. compute Dist(P, P'); 3. compute Prox(P, P'); 4. Prox(Q, Q') = Prox(P, P'); 5. return Prox(Q, Q'); 6. end ALGORITHM 3. ESTIMATION OF SEMANTIC PROXIMITY (SPQ)

C. Procedure to ranking: As we said in the algorithm 1, it realizes a rank in descending order of queries W Q (D) according to the semantic proximity associated. We chose the selection rank for its simplicity, the algorithm is as follows: Procedure RankOrder(var set E)

1: Begin 2: for i=1 to |E|-1 DO 3: for j=i to |E| DO 4: begin 5: if Prox(Q,Q i )k then /*second case */ 21: begin 22: case2=true; 23: Q' Rel = P’ 1 ∧ P’ 2 ∧…∧P’ k ; 24: compute Prox(Q, Q' Rel ); 25: if Prox(Q, Q' Rel )>0 then E = E ∪ {Q'}; 26: end; 27: Aux1 = Aux1 − {Q'}; 28: end; 29: if case2=true then exit 30: else /*third case */ 31: begin 32: Q'= First (Aux2); 33: Q’’=Increased (Q'); 34: Q’’=Weakening (Q"); 35: compute Prox(Q, Q"); 36: if Prox(Q, Q’’)>0 then E = E ∪ { Q’}; 37: Aux2 = Aux2 − {Q’}; 38: end; 39: end;

[8]

40: end. ALGORITHM 5. FILTER OF WORKLOAD (COMPOUND QUERIES)

[9]

The changes in the procedure Prox (Q, Q') are defined in the following algorithm:

[10]

Procedure Prox (Q = P 1 ∧…∧ P k , Q' = P' 1 ∧…∧ P' k ) 1. begin 2. for i = 1 to k do 3. compute Dist(P i , P' i ); 4. compute Prox(P i , P' i ); 5. end for 6. compute 7. Prox(Q, Q') = min i=1, k Prox(P i , P' i ); 8. return Prox(Q, Q'); 9. end

[11]

[15]

ALGORITHM 6. ESTIMATION (COMPOUND QUERY)

[16]

OF

SEMANTIC

[2]

[3]

[4]

[5]

[6] [7]

[13]

[14]

PROXIMITY

VI. CONCLUSION Introducing semantic proximity in extraction process has improved the expressive power of the queries and contributed to finding more answers. Nevertheless some problems still happen in the framework of fuzzy querying. In this paper, a problem that could arise when users attempt to exploit Web large databases has been addressed. We have shown how they can be automatically approached. The key tool of the proposed approaches is a measuring semantic proximity between queries. The main advantage of our proposal is the fact that it operates only on the conditions involved in the user query without adding new conditions or performing any summarizing operation nor using the data distribution of the queried database. Such approaches can be useful to construct intelligent information retrieval systems that provide the user with cooperative answers. One direction of future works concern the implementation step and the test of the efficiency of the approach on some large practical examples.

[1]

[12]

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