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Topological operators: a relaxed query processing approach

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Abstract

Relaxation and approximation techniques have been proposed as approaches for improving the quality of query results, in terms of completeness and accuracy, in environments where the user may not be able to specify the query in a complete and exact way, since data are quite heterogeneous or she may not know all the characteristics of data at hand. This problem, mainly addressed for relational and XML data, is nowadays quite relevant also for geo-spatial data, due to their increasing usage in highly critical decisional processes. Among geo-spatial queries, those based on spatial and more precisely topological relations are currently used in an increasing number of applications. As far as we know, no approach has been proposed so far for relaxing queries based on topological predicates when they return an empty or insufficient answer, in order to improve result quality and user satisfaction. In this paper, we consider this problem and we present a general relaxation strategy for, possibly multi-domain, topological selection and join queries. Two specific semantics are also provided: the first applies the minimum amount of relaxation in order to get an acceptable answer; the second relaxes the given query of a certain fixed amount, depending on the considered topological predicate. Index-based processing algorithms, for efficiently executing relaxed queries based on the proposed semantics, are also presented and a specific topological similarity function, to be used for relaxation purposes, is proposed. Experimental results show that the overhead given by query relaxation is acceptable.

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Notes

  1. See Table 1 for the exact meaning of topological relations.

  2. PG Lig stands for the polygon representing Liguria—see Fig. 1.

  3. We assume that the feature type Pr (Province) has a property called region representing the administrative region a province belongs to.

  4. As we will see later, there is no need to define compatibility rules for entries of different types since the algorithms we are going to present only compute topological similarity intervals between entries of the same type.

  5. PG Lig stands for the polygon representing Liguria—see Fig. 1.

  6. Notice that, when taking as input the objects corresponding to the MBRs, INR can be interpreted as a sort of relaxed evaluation.

  7. For join, we obtained even better results, since computing join using sequential scans is very inefficient; thus, for space constraints, we have not reported these results in this paper.

  8. We exclude also touch since its relaxation with v no_dj  = 0.5 includes disjoint.

  9. We consider in instead of equal, differently from what has been done for selection, since equal in this case is a too strong selective relation, thus it is not suitable to highlight the differences between the proposed techniques.

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Author information

Authors and Affiliations

  1. Dipartimento di Informatica, University of Verona, Strada le Grazie 15, 37134, Verona, Italy

    Alberto Belussi

  2. Dipartimento di Informatica e Scienze dell’Informazione, University of Genoa, Via Dodecaneso 35, 16146, Genoa, Italy

    Barbara Catania & Paola Podestà

Authors
  1. Alberto Belussi
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  2. Barbara Catania
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  3. Paola Podestà
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Corresponding author

Correspondence to Barbara Catania.

Appendices

Appendix A: An example of topological similarity function

Table 6 Similarity values for topological relationships defined in Table 1
Full size table

Appendix B: Compatibility tables

Table 7 summarizes selection compatibility rules for leaf and intermediate entries, for objects of any dimension. The first column of the table contains the relation θ satisfied by e a .r and O.r; the second column points out the dimension of O.r, since both c() and co() may change when MBRs degenerates to points (in very particular cases, MBR can also degenerate to a vertical or horizontal line; for sake of readability we did not report these cases in the tables); the fourth and the fifth columns contain c(θ, dim(ft), dim(O)) and co(θ, dim(ft), dim(O)), respectively, for any pair of object dimension (dim(ft), dim(O)), pointed out in the third column.

Table 7 Compatibility sets for selection operator
Full size table

Compatibility rules for join operator are shown in Table 8. With respect to Table 7, the second column points out the dimension of (d(e a .r), d(e b .r)), since both ci() and co() may change when MBRs degenerates to points.

Table 8 Compatibility sets for join operator
Full size table

For sake of readability in table heading we replace dim() with d(). The notation\({\cal T}_{x,y}\) has been introduced in Section 3. Empty compatibility sets always correspond tosituations in which MBRs of non object entries degenerate to points or point objectsare compared.

Appendix C: Query processing algorithm for relaxed selection operator based on BF semantics

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Belussi, A., Catania, B. & Podestà, P. Topological operators: a relaxed query processing approach. Geoinformatica 16, 67–110 (2012). https://doi.org/10.1007/s10707-011-0124-9

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