Topological Relations in the World of Minimum Bounding Rectangles: A Study with R-trees

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1 Toological Relations in the World of Minimm Bonding Rectangles: A Stdy with R-trees Dimitris Paadias Deartment of Comter Science and Engineering University of California, San Diego CA 0-0, USA dimitris@cs.csd.ed Timos Sellis Deartment of Electrical and Comter Engineering National Technical University of Athens GREECE timos@theseas.nta.gr Yannis Theodoridis Deartment of Electrical and Comter Engineering National Technical University of Athens GREECE theodor@theseas.nta.gr Max J. Egenhofer National Center for Geograhic Information and Analysis University of Maine, Orono ME 0-, USA max@mecan.maine.ed Abstract Recent develoments in satial relations have led to their se in nmeros alications involving satial databases. This aer is concerned with the retrieval of toological relations in Minimm Bonding Rectangle-based data strctres. We stdy the toological information that Minimm Bonding Rectangles convey abot the actal objects they enclose, sing the concet of rojections. Then we aly the reslts to R-trees and their variations, R + -trees and R*-trees in order to minimise disk accesses for eries involving toological relations. We also investigate eries that involve comlex satial conditions in the form of disjnctions and conjnctions and we discss ossible extensions.. INTRODUCTION The reresentation and rocessing of satial relations has gained mch attention in Satial Qery Langages (Paadias and Sellis, ; Egenhofer, ), Image and Mltimedia Databases (Sistla et al., ), Geograhic Alications (Frank, ), Satial Reasoning (Randell et al., ) and Cognitive Science (Glasgow and Paadias, ). Desite the attention that satial relations have attracted in other alication domains, they have not been extensively alied in satial data strctres. So far most of the work on satial access methods has concentrated on the relations disjoint and not_disjoint and on the retrieval of distance information (Rossoolos et al., ). This is not becase other satial relations are nimortant for ractical alications, bt mostly becase of the lack, ntil recently, of definitions for satial relations. In this aer we focs on the retrieval of toological relations, relations that stay invariant nder toological transformations sch as translation, rotation and scaling. In articlar we will deal with toological relations between contigos region objects (the term refers to homogeneosly -dimensional, connected objects with connected bondaries) as defined by the -intersection model (Egenhofer, ). According to this model, each object is reresented in R sace as a oint set which has an interior, a bondary and an exterior. The toological relation between any two objects and is described by the nine intersections of 's interior, bondary and exterior, with the interior, bondary and exterior of (based on the concets of oint-set toology). Ot of different relations that can be distingished by the model, only the following eight are meaningfl for region objects: disjoint, meet, eal, overla, contains (and the converse relation inside) and covers (and the converse covered_by)(see Figre ). disjoint(,) meet(,) overla(,) covered_by(,) covers(,) inside(,) contains(,) Fig. Toological relations eal(,) We call the revios set of toological relations m t in order to distingish them from the set m t ={disjoint, not_disjoint}. The meaning of disjoint is the same in both sets (it imlies that two objects have no common oints), while all the other relations of m t are refinements of not_disjoint. Sometimes in satial data strctres terminology, the term overla (instead of not_disjoint) is sed to denote any configration in which the objects are

2 not disjoint. The relations of m t are airwise disjoint, and they rovide a comlete coverage. Randell et al. () reached the same set of airwise disjoint toological relations following an axiomization based on mereology and exressed in a many-sorted logic. Tests with hman sbjects have shown evidence that the -intersection model has otential for defining cognitively meaningfl satial redicates, a fact that makes the above relations a good candidate for commercial systems (Mark and Egenhofer, ). In fact, the -intersection model has been imlemented with Geograhical Information Systems (GIS); Hadzilacos and Tryfona (), for instance, sed it to exress geograhical constraints, and Mark and Xia () to determine satial relations in ARC/INFO. In addition there are imlementations in commercial systems sch as Intergrah (MGE, ) and Oracle MD (Keighan, ). Frthermore, the relations of m t have been inflential in assessing the consistency of toological information in satial databases (Egenhofer and Sharma, ), ery otimisation strategies (Clementini et al., ) and Satial Reasoning (Grigni et al., ). This aer is concerned with the retrieval of the toological relations of m t sing satial data strctres based on Minimm Bonding Rectangles (MBRs). In articlar we concentrate on R-trees and their variations. Section illstrates the ossible relations between MBRs and describes the corresonding toological relations. Section investigates the toological information that MBRs convey abot the actal objects they enclose with resect to the -intersection model. Section alies the reslts in R-tree-based data strctres and comares the retrieval times. Section is concerned with eries that involve comlex satial conditions. Section discsses extensions that deal with imrecision and Section concldes with comments on ftre work.. MINIMUM BOUNDING RECTANGLES It is a common strategy in satial access methods to store object aroximations and se these aroximations to index the data sace in order to efficiently retrieve the otential objects that satisfy the reslt of a ery. Deending on the alication domain there are several otions in choosing object aroximations. Brinkhoff et al. () comare the se of rotated minimm bonding rectangles, convex hlls, minimm bonding n-corner convexes etc. in satial access methods. In this aer we examine methods based on the traditional aroximation of Minimm Bonding Rectangles. MBRs have been sed extensively to aroximate objects in Satial Data Strctres and Satial Reasoning becase they need only two oints for their reresentation; in articlar, each object is reresented as an ordered air (' l,' ) of oints that corresond to the lower left and the er right oint of the MBR ' that covers (' l stands for the lower and ' for the er oint of the MBR). While MBRs demonstrate some disadvantages when aroximating non-convex or diagonal objects, they are the most commonly sed aroximations in satial alications. The term rimary object/mbr denotes the object/mbr to be located and the term reference object/mbr denotes the object in relation to which the rimary object/mbr is located (in the examles hereafter, the reference objects are grey, while the rimary objects are transarent). Let X and Y be fnctions that give the x and y coordinate of a oint resectively. When we se two oints for the reresentation of the reference object and for the case of one-dimensional sace, the axis is divided into artitions. Three artitions are oen line segments (the interior and the exteriors of line segment ' l ' ) and two artitions are the oints ' l and ' (see Figre ). R l R R l R R l R l R R l R l R0 R l R l R l ' l ' x Fig. Possible relations in D sace If the rimary object is also reresented by two oints ' l and ' ordered on the x axis (X(' l )< X(' )) then the nmber of airwise disjoint relations between the two objects in D sace is. These relations corresond to the relations between time intervals introdced in (Allen, ). The symbols ' l and ' in Figre denote the edge oints (lower and er) for the reference MBR and the characters l and the lower and the er oints of the rimary MBR. In order to stdy the corresondence between MBRs and actal objects with resect to the relations of m t we will aly the revios reslts in D sace sing rojections on the x and y axis. In this case, the constraint for the lower and the er oints of the bonding rectangle is: X(' l ) < X(' ) Y(' l ) < Y(' ) and the nmber of airwise disjoint relations is (the sare of the nmber of relations in D sace). These relations are illstrated in Figre ; they corresond to the highest accracy sing two oints er object, in the sense that they cannot be defined as disjnctions of more "restrictive" relations.

3 R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_0 R i_ R i_ R i_ R _j R _j R _j R _j R _j R _j R _j R _j R _j R 0_j R _j R _j R _j Figre illstrates the corresonding toological relation for the configrations of Figre. For instance, all the configrations of the first row (R _j where j can take any vale from to ) corresond to the disjoint relation, and the total nmber of configrations in which the MBRs are disjoint is. On the other hand, only relation R _ corresonds to the toological relation contains. Eal Contains Inside Covers CoveredBy Disjoint Meet 0 Overla 0 TOTAL 0 0 Fig. Toological relations between MBRs A nmber of satial data strctres based on MBRs have been develoed. The most romising gro incldes R-trees (Gttman, ) and their variations. The R-tree is a height-balanced tree, which consists of intermediate and leaf nodes (R-trees are direct extensions of B-trees in k- Fig. Possible relations between MBRs dimensions). The MBRs of the actal data objects are assmed to be stored in the leaf nodes of the tree. Intermediate nodes are bilt by groing rectangles at the lower level. An intermediate node is associated with some rectangle which encloses all rectangles that corresond to lower level nodes. Each air of nodes may satisfy any of the toological relations of m t. The fact that R-trees ermit overla among node entries sometimes leads to nsccessfl hits on the tree strctre. The R + -tree (Sellis et al., ) and the R * -tree (Beckmann et al., 0) methods have been roosed to address the roblem of erformance degradation cased by the overlaing regions or excessive dead-sace resectively. To avoid this roblem, the R + -tree achieves zero overla among intermediate node entries by allowing artitions to slit nodes. The trade-off is that more sace is reired becase of the dlicate entries and ths the height of the tree may be greater than the original R-tree. On the other hand, the R * -tree ermits overla among nodes, bt tries to minimise it by organising rectangles into nodes sing a more comlex algorithm than the one of the original R-tree.

4 If the only toological relations of interest are the relations of m t, then when the MBRs of two objects are disjoint we can conclde that the objects that they reresent are also disjoint. If the MBRs however share common oints, no conclsion can be drawn abot the toological relation between the objects. For this reason, satial eries involve the following two ste strategy: First a filter ste based on MBRs is sed to raidly eliminate MBRs of objects that cold not ossibly satisfy the ery and select a set of otential candidates. Then dring a refinement ste each candidate is examined (by sing comtational geometry technies) and false hits are detected and eliminated. Brinkhoff et al. () extended the above strategy to inclde a second filter ste with finer aroximations than MBR (e.g. convex hlls) in order to exclde some false hits from the set of candidates. The above technies seed- the retrieval of relations of m t sing R-trees and variations. The resent work bilds on the original two-ste strategy to exlore a larger, and more detailed set of toological relations. Sch an investigation is imortant becase satial eries freently reire the kind of alitative resoltion distingished by m t. We first stdy how MBR aroximations can be sed for the retrieval of toological relations of m t between actal objects and then we aly the reslts in actal imlementations.. TOPOLOGICAL RELATIONS THAT MBRS CONVEY ABOUT THE ACTUAL OBJECTS Since the MBRs are only aroximations of the actal objects, the toological relation between MBRs does not necessarily coincide with the toological relation between the objects. In most cases the MBRs of objects that satisfy a given relation, shold satisfy a nmber of ossible relations with resect to the MBR of the reference object. We will start with relations that involve the retrieval of a small nmber of MBRs and we will gradally move to relations for which a large nmber of MBRs shold be retrieved. In order to answer the ery "find all objects eal to object " we need to retrieve all MBRs that are eal to ', that is all MBRs that satisfy the relation R - with resect to '. Only these MBRs may enclose objects that satisfy the ery. On the other hand, the retrieved MBRs may also enclose objects that satisfy the relations overla, covered_by, covers or meet with resect to (see Figre ). As in the case of m t, a refinement ste is needed when dealing with the relations of m t if the MBRs of the retrieved objects are not disjoint. In a revios aer we have shown how the strategy can be alied for the retrieval of direction relations between extended objects (Paadias et al., ). Fig. Possible relations for objects when the MBRs are eal The relations contains and inside also involve the retrieval of MBRs that satisfy nie configrations. In articlar, the objects that contain can only be in MBRs that contain ' (MBRs that satisfy the relation R - with resect to '), while the objects inside, can only be in MBRs that are inside ' (MBRs that satisfy the relation R _ with resect to '). As in the case of eal, a refinement ste is needed becase: contains(',') disjoint(,) meet(,) overla(,) contains(,) covers(,) and inside(',') disjoint(,) meet(,) overla(,) inside(,) covered_by(,) The rest of the relations involve the retrieval of more than one MBR configrations. For instance, in order to answer the ery "find all objects that cover a given object" we need to retrieve the MBRs that satisfy the relations covers, contains, or eal with resect to the MBR of the reference object. As Figre illstrates, these MBRs satisfy the rojection relations R i-j where i and j in {,,,}. Similarly the retrieval of covered_by involves all MBRs that satisfy the relations R i-j where i,j in {,,,0} with resect to '. R _j R _j R _j R _j R i_ R i_ R i_ R i_ Fig. Configrations of MBRs that yield the relation covers between actal objects The remaining relations involve the retrieval of a large nmber of MBRs. In the case of disjoint, for instance, all the MBRs may enclose objects that are disjoint with the reference object, excet for those that satisfy the relation R i-j where i, in {,,,} and j in {,,,0}, or i

5 R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_ R i_0 R i_ R i_ R i_ R _j R _j R _j R _j R _j R _j R _j R _j R _j R 0_j R _j R _j R _j Fig. Configrations of MBRs that yield the relation disjoint between actal objects in{,,,0} and j in {,,,}. Figre illstrates all configrations of MBRs that enclose objects otentially disjoint with the reference object. The blank configrations in Figre are not to be retrieved when we deal with contigos objects. If, for instance, the MBRs are related by the relation R _, then the MBRs overla and the actal objects also necessarily overla. Figre illstrates sch a configration; in this case a refinement ste is not needed. We followed the same rocedre for the relations meet and overla. Table shows the conclsions that can be drawn if we se the concet of rojections to stdy toological relations. For each relation, the Table illstrates the sbset of the MBR configrations that cold contain objects that satisfy the relation. Fig. Overlaing MBRs for which the actal objects necessarily overla

6 eal(,) contains(,) inside(,) covers(,) covered_by(,) disjoint(,) meet(,) overla(,) Table Toological relations imlemented For some cases the refinement ste can be avoided, that is, the relation between MBRs nambigosly determines the toological relation between the actal objects (e.g., the configration of Figre ). Figre illstrates the configrations for which the refinement ste is not needed (this haens only in some of the cases for disjoint and overla) disjoint overla Fig. Configrations for which a refinement ste is not needed Clementini et al. () stdied the se of MBR aroximations in ery rocessing involving toological relations. Frthermore, they defined a minimal sbset of the nine intersections that can otimally determine the relation between the actal objects taking into accont the freency of the relations. Their findings can be sed dring the refinement ste in order to minimise the cost of the comtation of intersections between otential candidates. In the next section we aly the reslts of sections and to actal imlementations based on R- trees.. IMPLEMENTATION OF TOPOLOGICAL RELATIONS IN R-TREES In order to retrieve the toological relations of m t sing R-trees one needs to define more general relations that will be sed for roagation in the intermediate nodes of the tree strctre. For instance, the intermediate nodes P that cold enclose MBRs ' that meet the MBR ' of the reference object, shold satisfy the more general constraint meet(p,') verla(p,') covers(p,') contains(p,'). Figre 0 illstrates examles of sch configrations. P P ' meet(p,') ' ' covers(p,') ' P ' ' overla(p,') P ' ' contains(p,') Fig. 0 Intermediate nodes that may contain MBRs that meet the MBR of the reference object Following this strategy, the search sace is rned by exclding the intermediate nodes P that do not satisfy the revios constraint. Table resents the relations that shold be satisfied between an intermediate node P and the MBR ' of the reference object, so that the node will be selected for roagation. Notice that the same relation between intermediate nodes and the reference MBR exists for all the levels of the tree strctre. For instance, the intermediate nodes that cold enclose other intermediate nodes P that satisfy the general constraint meet(p,') verla(p,') covers(p,') contains (P,') shold also satisfy the same constraint. This conclsion can be easily extracted from the above table and is alicable to all the toological relations of Table.

7 Relation between MBRs eal(',') contains(',') inside(',') covers(',') Relation between intermediate node P (that may enclose MBRs ') and reference MBR eal(p,') covers(p,') contains(p,') contains(p,') overla(p,') covered_by(p,') inside(p,') eal(p,') covers(p,') contains(p,') covers(p,') contains(p,') covered_by(',') overla(p,') covered_by(p,') eal(p,') covers(p,') contains(p,') disjoint(',') meet(',') disjoint(p,') meet(p,') verla(p,') covers(p,') contains(p,') meet(p,') verla(p,') covers(p,') contains(p,') overla(',') overla(p,') covers(p,') contains(p,') Table Relations for the intermediate nodes Smmarising, the rocessing of a ery of the form "find all objects that satisfy a given toological relation with resect to object " in R-tree-based data strctres involves the following stes:. Comte the MBRs ' that cold enclose objects that satisfy the ery. This maing involves Table.. Find the toological relations that the MBRs ' of the first ste may satisfy with resect to '. This rocedre involves Figre.. Starting from the to node, exclde the intermediate nodes P which cold not enclose MBRs that satisfy the toological relations of the second ste and recrsively search the remaining nodes. This rocedre involves Table.. Follow a refinement ste for the retrieved MBRs, excet for the cases illstrated in Figre. In order to exerimentally antify the erformance of the above algorithm, we created tree strctres by inserting 0,000 MBRs randomly generated. We tested three data files: - the first file contains small MBRs: the size of each rectangle is at most 0,0% of the global area - the second file contains medim MBRs: the size of each rectangle is at most 0,% of the global area - the third file contains large MBRs: the size of each rectangle is at most 0,% of the global area. The search rocedre sed a search file for each data file containing 00 rectangles, also randomly generated, with similar size roerties as the data rectangles. We sed the revios data files for retrieval of toological relations in R-, R + - and R*-trees. In the imlementation of R-trees we selected the adratic-slit algorithm and we set the minimm node caacity to m=0%; in the imlementation of R*-trees we set m=0% while in the imlementation of R + -trees the "minimm nmber of rectangle slits" was selected to be the cost fnction. These settings seem to be the most efficient ones for each method (Beckmann et al., 0; Sellis et al., ). Table illstrates the nmber of hits er search, that is, the nmber of retrieved MBRs for the three files. The nmber of hits er search is inversely roortional to the selectivity of the relation and it is related to the nmber of disk access. Note that the total nmber of MBRs in each colmn is larger than 0,000 becase a MBR may be retrieved for more than one relation. Toological Relation small MBRs medim MBRs large MBRs disjoint,,, meet.0.. overla. 0.. covered_by.0.0. inside eal covers.0.. contains Table Retrieved MBRs er relation for each data file The nmber of disk accesses er search is illstrated grahically in Figre. With the excetion of disjoint, the imrovement in the efficiency of retrieval sing R- trees comared to serial retrieval is immense; this is articlarly tre for small size MBRs where the difference is almost two orders of magnitde. The difference dros as the MBRs become larger. The increase in the size increases the density and, therefore, the ossibility that the reference object is not disjoint with other MBRs or intermediate nodes. The retrieval of disjoint is, as exected, worse than serial retrieval becase this relation reires the retrieval of all the nodes of the tree strctre as derived from the -ste strategy resented above. Clearly, a "real" system wold do a serial retrieval in sch a case instead of sing the tree strctre. Obviosly, toological relations can be retrieved sing the traditional window ery (i.e., not_disjoint relation) sed in satial data strctres. In that case, the distinction between the not_disjoint relation and the actal toological relation to be retrieved can be made dring the refinement ste. The cost of this aroach is roghly eivalent to the cost of meet (according to Table, meet involves the retrieval of almost all MBRs that are not disjoint with the reference MBR). Deending on the relation and the MBR size, or aroach can imrove the erformance by to 0%. Another imortant imrovement is the redction of the MBRs that are selected for the refinement ste. As illstrated in Table, the nmber of hits er search for some relations (e.g., inside, covers) is considerably lower than the hits er The nmber of disk accesses er search sing serial retrieval is eal to 00 for all relations (the size of the data file divided by the age caacity which is eal to 0 entries).

8 disk accesses er search... A R-tree R+-tree R*-tree m o cb i e cv cn toological relation disk accesses er search.... A A A A A A toological relation R-tree R+-tree R*-tree m o cb i e cv cn A A A A A A A A disk accesses er search toological relation R-tree R+-tree R*-tree A A m o cb i e cv cn Small data size Medim data size Large data size Fig. Performance comarison of R-tree variants search for meet (sally below 0%). Therefore the int of the refinement ste is a small sbset of the MBRs that wold be selected sing the window ery. According to Figre, the relations of m t can be groed into three categories with resect to the cost of retrieval. The first gro contains disjoint which is the most exensive relation and shold be rocessed by serial search. The second gro consists of meet, overla, inside and covered_by. The third gro consists of the three relations (eal, covers, contains) which are the least exensive to rocess. The cost difference between the second and the third gro increases as the size of the MBRs becomes larger. Note also that the rimary factor for the retrieval cost is the relation for the intermediate nodes (Table ). The relation inside is more exensive than covers, althogh it retrieves only one ott MBR configration (relation R - with resect to '). This is de to the large nmber of intermediate nodes that cold contain MBRs that satisfy the relation R -. On the other hand, the cost of inside is almost the same as the cost of covered_by, which can be inferred by Table. The comarison of the varios R-tree-based strctres follows, in general, the conclsions drawn in the literatre i.e., the variations R + -trees and R*-trees oterform the original R-trees. For small and medim MBRs, R + -trees erform slightly better than R*-trees. However this advantage is lost if the dlicate entries generate one extra level in the tree strctre, as haened for the large data size of or tests. In sch case the erformance of R + -trees is inferior for the most exensive relations bt remains cometent for the least exensive ones. The irreglar behavior of R + -trees is de to the lack of overla between nodes, which reslts in the ick exclsion of the majority of the intermediate nodes when one of the least exensive relations is retrieved. On the other hand, for the When the data density becomes high it is ossible that all of the entries in a fll node coincide on the same oint of the lane. In sch cases R + trees do not work (Greene, ). This haened for some data sets involving large MBRs dring or tests. exensive relations there is no significant gain and the erformance is worse comared to the other strctres becase of the greater nmber of nodes in the tree. In section we have stdied eries that involve the retrieval of objects that satisfy a toological relation with resect to a reference object. In the next section we extend or reslts for eries that involve comlex satial conditions in the form of disjnctions and conjnctions with resect to one and two reference objects.. COMPLEX QUERIES In some cases the refinement rovided by the relations of m t is not needed. In a cadastral alication, for instance, the difference between inside and covered_by may not be imortant. Consider the ery "find all land arcels in a given area". The land arcels of the reslt shold be inside or covered_by the area, that is, the interretation of in is inside covered_by. We can define toological relations of lower alitative resoltion sing disjnctions of the relations of m t. This is a common strategy in alitative satial knowledge reresentation; it has been sed for direction relations in (Paadias and Sellis, ) and for the above toological relations in (Randell et al., ). Qeries involving low resoltion relations can also be rocessed by the -ste method of section. The only difference is that the set of the MBRs to be retrieved by the first ste is the nion of the MBRs to be retrieved by each of the relations that belong to the disjnction. Frthermore, in some cases the retrieval times do not change. In the revios ery (relation in), the MBRs of the reslt are the same as those that wold be retrieved if the relation of interest were covered_by. This is becase the MBRs to be retrieved for inside constitte a sbset of the MBRs for covered_by (see Table ). Figre illstrates the sbset relations with resect to the ott MBRs.

9 disjoint contain covers overla eal covered_by inside meet Fig. Sbset relations according to the ott MBRs According to Figre the retrieval of the ery "find all objects that satisfy the relation meet or contain or eal or inside " involves the same retrieval time with the ery "find all objects that meet ". On the other hand, eries that involve conjnctions of toological relations with resect to one reference object (e.g., find all objects that are inside and covered_by ) have an emty reslt becase the relations of m t are airwise disjoint. Nevertheless we can erform semantic ery otimisation for eries of the form "find all objects that are inside and overla with ". We can determine that the reslt of this ery is emty if we know that and are disjoint. As Figre illstrates, if is inside and disjoint, it cannot be the case that overlas. Fig. A ery with two reference objects and emty reslt Table illstrates the relations between reference objects for which an emty reslt is retrned withot rnning the ery. When a ery involving two reference objects and is given, the toological relation between the reference objects is examined, and if it is one of the relations of the table, the ott is emty. For the above ery, in addition to disjoint, if and are related by meet, eal, inside or covered_by, the reslt is also emty. Each entry at row r i (, ) and colmn r j (, ) (where r i and r j are relations of m t ) is the comlement of the comosition relation r' i (,) and r j (, ) with resect to m t, (where r' i is the converse relation of r i ). For an extensive discssion abot comosition of toological relations see (Egenhofer, ). If the relation between the reference objects is not one of the relations of the array, then one of the two relations is retrieved sing R- trees and the alifying MBRs ' are filtered with resect to the other reference object. This rocess can be erformed in main memory by checking the relation that each ott MBR of the first ste satisfies with resect to the second reference object. Therefore the nmber of disk accesses deends only on the first relation. The selection of the first relation is based on the size of the reference MBR and on the cost gro that the relation belongs to. The relations covers, contains and eal are referable becase they reire the least disk accesses. If the sizes of the reference MBRs are considerably different, then the smallest reference MBR mst be selected becase the cost of retrieval is roortional to the data size (see Figre ). disjoint(, ) meet(, ) eal(, ) inside(, ) cvrdby(, ) contain(, ) covers(, ) overla(, ) disjoint(, ) meet(, ) eal(, ) inside(, ) cvrdby(, ) contain(, ) covers(, ) overla(, ) --- e ct cv m e i e i cb i ct d e i m e i e i cb e i cb m e i m e i d e i i cb d e i cb m e i e i cb ct cv d m i i cb cv i cb ct i ct cv o e i cb e i cb i cb ct cv e ct cv e ct cv m e i ct cv d m ct cv ct cv d e ct cv i ct cv i cb d m i ct ct cv ct cv m e i i cb cv i cb cv i cb cv Table Conjnctions of relations that yield emty reslts m e i e i cb ct cv o i cb ct i cb cv i cb e ct cv e ct cv i cb ct cv i cb i cb --- e ct cv e ct cv e i cb i ct e ct cv e i cb e i cb ---

10 Althogh the above methods rovide a good fondation for ractical alications, the nderlying theoretical assmtions do not always hold in ractical (GIS) environments. In the next section we discss how or work can be extended to catre real-world imrecision.. NON-CRISP MBRS The se of MBR filters for rocessing toological relations has a very delicate asect, as it relies heavily on the fact that the MBRs are cris reresentations of the objects. A cris MBR has to satisfy two constraints:. the object to be aroximated is flly contained within the rectangle and. each bondary of the MBR coincides with some art of the object s bondary. One may arge that a violation of either condition wold not constitte an MBR in its literal sense (the rectangle wold not be bonding, or the rectangle wold not be minimal); however, even with carefl imlementations of algorithms that create MBRs for satial objects, these constraints can be reglarly violated. This roblem is de to the difficlties of imlementing coordinate-based comter algorithms for geometry that reserve toology. One cannot always reliably derive from a coordinate reresentation toological roerties sch as the coincidence of two oints, or whether a oint close to a line is actally on the line etc. While the first constraint in the definition of cris MBRs can sally be flfilled, e.g., by making the MBRs slightly larger than reired, it is the second constraint (the minimality of rectangles) that introdces the roblem. Sometimes recision is traded for erformance and fast, bt slightly inaccrate, algorithms may be sed to calclate MBRs. Sch algorithms make sre that constraint is flfilled, bt occasionally violate constraint so that some MBRs may actally be larger than necessary. The same may haen de to nmerical inaccracies sch as ronding errors, articlarly, if floating oint arithmetic is sed to reresent the objects coordinates and MBRs are exressed by two integer airs. Therefore, often in imlementations of MBR-based satial access one has to assme some nmerical inaccracies. The -ste algorithm of section can be extended to deal with inaccracies involving slightly larger than cris MBRs. Or extension is based on the concet of concetal neighborhood (Freksa, Egenhofer and Al-Taha ). The concetal neighborhood of the one-dimensional relations of Figre forms a grah, in which each air of directly connected nodes corresonds to a air of relations that are concetal neighbors. Given an initial relation between two objects, if we continosly enlarge the rimary object, we follow the ath indicated by the arrows in Figre a. For instance, if the relation between the objects is R, then extending the rimary object, while keeing the reference object constant, gradally leads to relations R, R, R and R (see Figre ). Likewise, enlarging the reference object changes the relation between the two objects according to the directions of the edges in Figre b. A first-degree concetal neighbor of a relation R i is a relation R j that can be reached from R i via a directed edge in either neighborhood grah. For examle, relation has for first-degree concetal neighbors (relations and if we enlarge the rimary object, and relations and 0 if we enlarge the reference object). On the other hand, relation has one first-degree concetal neighbor, relation which is obtained by enlarging either object. A second-degree concetal neighbor of R i is a relation R j that has at least two first-degree concetal neighbors that are also first-degree neighbors of R i. For examle, the second-degree concetal neighbors of relation comrises the set of relations,,,, and ; they all have at least two common first-degree neighbors with. On the other hand, relation does not have any second-degree neighbors. These reslts can be easily extended to d sace where two relations are k-neighbors if they are k-neighbors in any dimension. R R R R R R R R R R R R R 0 (a) Enlargement of rimary object R R R R R R R 0 R R R R R R (b) Enlargement of reference object Fig. Concetal neighborhoods for relations in d sace We se the notion of concetal neighborhood to overcome the otential roblems that may arise from inaccracy. In addition to the configrations of Table, the candidate MBRs for the first ste may also satisfy the first- and second-degree concetal neighbor relations with resect to the cris MBRs that wold be retrieved. Table shold be sed instead of Table in sch cases. The dark grey rectangles of Table corresond to the cris MBR relations dislayed in Table. The light grey

11 eal contains inside covers rectangles are the additional ones that corresond to first and second neighborhood relations. The extra MBRs to be retrieved increase the retrieval cost, bt assre that all otential objects that satisfy the ery are retrieved even if there exists inaccracy that can reslt in -degree relation deformation. The largest increase in ott MBRs is observed for the relation eal, for which MBR relations have to be considered (in lie of MBR relation in the cris case). On the other hand, the ott MBRs for the relation overla remain constant.. CONCLUSION covered_by disjoint meet overla Table Retrieval sing -neighbors The reresentation and rocessing of toological relations is an imortant toic in a nmber of areas inclding Satial Qery Langages, Image and Mltimedia Databases and Geograhic Information Systems. Desite their imortance, toological relations have not been extensively sed in satial data strctres. In this aer we have focsed on the retrieval of toological relations in MBR-based data strctres. In articlar we have shown how the toological relations disjoint, meet, eal, overla, contains, inside, covers and covered_by (defined by the -intersection model) can be retrieved from R-trees and their variations. First we illstrated the ossible relations between MBRs and we described the corresonding toological relations. Then we stdied the toological information that MBRs convey abot the actal objects they enclose sing the concet of rojections. Finally we alied the reslts in R-tree-based data strctres and we conclded that they are sitable for toological relations, with R + - and R * - trees oterforming the original R-trees for most cases. We also investigated eries that involve comlex satial conditions in the form of disjnctions and conjnctions and we demonstrated how or method can be alied to non-cris MBRs. Althogh we have dealt with contigos regions, ractical alications do not necessarily deal with contigos objects. Geograhic entities, sch as contries with islands, consist of disconnected comonents. The revios reslts have been extended for this case. The only difference is that the nmber of MBRs to be retrieved for some relations increases since the relaxation of the contigity constraint alifies more MBRs as otential candidates. An imlementation of satial relations for non-contigos objects, as well as geograhic examles and detailed descritions can be fond (Paadias and Theodoridis, ) In order to model linear and oint data we need frther extensions becase the toological relations that can be defined, as well as the nmber of ossible rojection relations between MBRs, deend on the tye of objects. Egenhofer (), for instance, defined relations between lines based on the -intersection model, while Paadias and Sellis () have shown that the nmber of different rojections between a region reference object and a line rimary object is. The ideas in this aer can be extended to inclde linear and oint data, objects with holes etc. ACKNOWLEDGEMENTS We wold like to thank Cheryl Jacob for roof-reading this aer and roviding sefl sggestions. Dimitris Paadias was artially sorted by NSF - IRI. Part of this work was done while he was with the Deartment of Geoinformation, Technical University of Vienna. Yannis Theodoridis and Timos Sellis were artially sorted by the Deartment of Research and Technology of Greece (PENED ).

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