The International Conference in Knowledge Management (CIKM'94), Gaithersburg, MD, November 1994.

Transcription

1 The Intenational Confeence in Knowledge Management (CIKM'94), Gaithesbug, MD, Novembe 994. Hashing by Poximity to Pocess Duplicates in Spatial Databases Walid G. Aef Matsushita Infomation Technology Laboatoy Two Reseach Way Pinceton, New Jesey Hanan Samet Compute Science Depatment and Cente fo Automation Reseach and Institute fo Advanced Compute Studies The Univesity of Mayland College Pak, Mayland 2042 Abstact In a spatial database, an object may extend abitaily in space. As a esult, many spatial data stuctues (e.g., the quadtee, the cell tee, the R + -tee) epesent an object by patitioning it into multiple, yet simple, pieces, each of which is stoed sepaately inside the data stuctue. Many opeations on these data stuctues ae likely to poduce duplicate esults because of the multiplicity of object pieces. A novel appoach fo duplicate pocessing based on poximity of spatial objects is pesented. This is dieent fom conventional duplicate elimination in database systems because, with spatial databases, dieent pieces of the same object can span multiple buckets of the undelying data stuctue. xample algoithms ae pesented to pefom duplicate pocessing using poximity fo a quadtee epesentation of line segments and abitay ectangles. The complexity of the algoithms is seen to depend on a geometic classication of dieent instances of the spatial objects. By using poximity and the spatial popeties of the objects, the numbe of disk-i/o equests as well as the un-time stoage duing duplicate pocessing can be educed. Intoduction Spatial databases ae usually oganized in data stuctues that povide ecient access and exible manipulation of data. Thee ae seveal ways of epesenting a spatial object inside a data stuctue. Some data stuctues (e.g., the R-tee [9] and the Gid ile [2]) epesent a spatial object by just one entity inside the data stuctue (e.g., by the object's bounding ectangle in the case of the R-tee and by a point in a highe dimensional space in the case of the Gid ile). On the othe hand, anothe family of data stuctues (e.g., the quadtee [0], cell tee [8], and R + -tee [6]) epesent a spatial object by patitioning it into moe than one piece, each of which is stoed sepaately inside the data stuctue. In this pape, we focus on the latte family of data stuctues. We attempt to take advantage of the spatial chaacteistics of objects as a guide to duplicate pocessing (see [5] fo an oveview of techniques fo duplicate pocessing in database systems) that esult fom spatial database opeations. It is woth mentioning that duplicate pocessing in spatial databases is slightly dieent fom conventional duplicate elimination in database systems because with spatial databases, dieent pieces of the same object can span multiple buckets of the undelying data stuctue. o example, in the case of the R + -tee and the quadtee, duplicates aise because the object is patitioned into pieces. We can benet fom knowing that these pieces ae contiguous in space (if this is eally the case). In paticula, we dene an opeation, temed Repot Unique, which manages to pocess duplicates and epot each object in the data stuctue just once, egadless of the multiplicity of the patitions of the object inside the spatial data stuctue. Repot Unique can be viewed as an altenative appoach to hashing (see [] fo a good coveage of hashing techniques) when dealing with spatial data. Repot Unique maintains a dynamic data stuctue, temed the active bode [5], that seves as a epositoy fo the objects cuently being pocessed, temed active objects. The active bode data stuctue esembles a dynamic hash table. When conventional hashing techniques ae used thee is no way of pedicting when an object will not be efeenced again and hence can be safely deleted fom the hash table. Using spatial popeties of the undelying objects, the active bode can gow and shink in constant time. By knowing the extent and poximity of the objects, we can detect when all the pieces compising the object have been pocessed and hence can delete the object enty fom the active bode. This way we can avoid the situation that the hash table gows until it eaches its maximum limit, i.e., O(numbe of objects in the database). In fact, by using the active bode technique we can limit the stoage complexity to be in the ode of the active objects only. A look-up function, also based on spatial poximity, can be used to access enties in the active bode. The look-up function is the spatial analog of a hash function. As we demonstate in the pape, an impotant advantage of poximity-based look-up functions is that as a esult of thei use, buckets in the active bode ae guaanteed not to oveow. In addition, we also show howwe can utilize the poximity infomation to educe the numbe of disk-i/o equests duing duplicate pocessing. The est of the pape is oganized as follows. In Section 2 we descibe the data stuctues that we use fo stoing the undelying spatial database. The emaining sections descibe techniques to pocess duplicates using spatial

2 chaacteistics of the undelying spatial objects. Section descibes an algoithm that uses spatial poximity to eliminate the space equiements that esult fom using othe duplicate elimination techniques (e.g., hash tables). Section 4 demonstates that suppoting the Repot Unique opeation fo a ealistic vaiety of objects equies classifying spatial objects into categoies of dieent complexity. Section 5 shows how these classes aect the pefomance and coectness of the altenative appoaches to Repot Unique. Section 6 contains some concluding emaks. B s 2 The Undelying Spatial Data Stuctues We assume that the spatial objects ae stoed in data stuctues that patition the objects into multiple pieces. We use the quadtee as an example to demonstate ou algoithms fo duplicate pocessing. The quadtee patitions each object into multiple non-ovelapping pieces. Hee, we show how we oganize databases of line segments and ectangles using vaiants of the quadtee. o line segments we use the PMR quadtee [], while fo ectangles we useavaiant of the egion quadtee [0], temed the ectangle quadtee. In the PMR quadtee, a line segment is divided into seveal pieces, whee each piece is associated with the quadtee block that it intesects. igue shows one example of the PMR quadtee. A block in the PMR quadtee is pemitted g d h i igue : An example of a PMR quadtee. to contain a vaiable numbe of line segments. It is constucted by inseting the line segments one-by-one into an initially empty stuctue consisting of one block. ach line segment is inseted into all of the blocks that it intesects o occupies in its entiety. Duing this pocess, the occupancy of each aected block ischecked to see if the insetion caused it to exceed a pedetemined splitting theshold. If the splitting theshold is exceeded, then the block is split once, and only once, into fou blocks of equal size. ach block in the PMR quadtee stoes the object identies of the lines passing though it. The full desciption of the line segments (e.g., the stat and end coodinate values of the end-points) is stoed in what is called the featue table. The index of line l in the featue table is l's object identie. Notice that we cannot simply tavese the featue table and epot the lines that we nd since not all the lines in the featue table have to belong to the data stuctue in question, i.e., that the featue table may be shaed by moe than one PMR quadtee. Rectangles ae stoed in a vaiant of the egion quadtee [0] which we tem a ectangle quadtee. ach ectangle is decomposed into the quadtee blocks that lie inside the ectangle. o example, igue 2a shows the decomposition of ectangle into its constituent quadtee blocks. We assume that the ectangles need not be disjoint. We also adopt the estiction that each quadtee block is completely inside all of the ectangles that ovelap it. In othe wods, the case a c b e f igue 2: The decomposition of ectangle into its constituent quadtee blocks. Block B is decomposed afte ectangle s is inseted. that only pat of a quadtee block lies in a ectangle is not allowed. o example, this estiction means that block B of igue 2a must be decomposed when ectangle s is added to yield the decomposition given in igue 2b. The ectangle quadtee epesentation assumes that all the blocks that ae intenal to a ectangle stoe an object identie which isan oset in a featue table whee the uppe-left and bottomight coodinate values of the ectangle ae stoed. A cental opeation in spatial databases is the window opeation. A window can be in the fom of a ectangle o a polygon. This opeation seves as a building block fo anumbe of queies. Usually, spatial featues span a wide featue space. Howeve, uses ae moe inteested in viewing o queying only potions of the featue space instead of the whole space. xtacting pats of the space to wok with in subsequent opeations is done by windowing. Given a window w, some examples of window-based queies ae: epot all featues inside w, intesect featue f with featue b only inside w, detemine if featue f exists in w, etc. Moe details about window queies and algoithms fo answeing them can be found in []. When a ectangle is clipped (e.g., as a esult of a window opeation), the ectangle quadtee does not update the coodinate values of the the clipped ectangle in the featue table. Notice that we cannot simply tavese the featue table and epot the ectangles that we nd since not all the ectangles in the featue table necessaily belong to the data stuctue in question. Avoiding the xta Space - A ist Attempt In this section we pesent a simple duplicate pocessing algoithm that does not equie additional space (e.g., in contast to the O(n) space equied fo hashing in the case of a database of n objects). We also discuss the limitations of this algoithm. In the emaining sections of the pape, we will addess these limitations and ty to tackle them. We pesent the algoithm fo the case of a database of line segment objects. A simila algoithm can be descibed fo a database of ectangula objects. In ode to exploit the poximity of object pieces we de- ne a test function, say t, such that given a spatial object, say o, with k pieces, p, p 2,..., p k, t hasavalue of false fo only one of the pieces, say p j of o. Application of t to the est of the pieces yields the value of tue. Any function t that satises this citeion can be used as a means of suppessing all pieces of o othe than p j fom epoting o's object identie, and hence can be adopted by the Repot Unique opeation. o example, in the case of hashing,

3 a function t is dened as follows (a is a bit aay with all its elements initialized to have avalue of false, and oid(p i) is the identie of the object to which piece p i belongs): t (p i) = et value a[oid(p i)]; if(not a[oid(p i)]) then a[oid(p i)] tue; etun (et value); We can get id of the O(n) space equiements of the function t in the following way. This gives ise to algoithm Repot Unique descibed below. We tavese the blocks of the PMR quadtee, and fo each block, say B, we pefom the following actions:. o each object identie i stoed in B (that coesponds to piece p i), etieve the line segment desciption, say l i, fom the featue table, and 2. pefom the following testing function t 2 t 2(l i;b) = if(stat point(l i) in B) then etun (false) else etun (tue). If t 2(l i;b) is false, then epot line segment l i. We use igue fo illustation. Notice that fo each line segment l, t 2 is false only when the block containing the stat point oflis pocessed. Application of t 2 to all blocks containing pieces of othe than the uppe-left cone of yields tue (t 2 is efeed to as the point-in-block test). Notice also that the algoithm woks only in the case whee the pieces compising the spatial object ae contiguous. The case whee the pieces of an object ae disjoint (e.g., due to clipping pats of the object by a window opeation, descibed in Section 2) is handled in Section 5. p B l q B 2 B igue : Line l is stoed in blocks B, B 2, and B whee only block B causes the point-in-block test to succeed since the stating point p of l only lies inside B and not in B 2 no in B. The algoithm Repot Unique does not equie any additional space. Howeve, its CPU and disk-i/o costs ae high. The algoithm applies a point-in-block test pe object piece, which equies fou compaisons (each of cost C cpu). In paticula, the point (x; y) is inside block B((bl x;bl y), (u x;u y)), whee bl and u denote the bottom left and uppe ight cones of the block, espectively, i bl x x u x ^ bl y y u y. t 2 is pefomed nk times, whee n is the numbe of objects in the database, and k is the aveage numbe of pieces pe object in the database. Notice that in ode to pefom the point-in-block test (t 2)wemust access the featue table (via the object identi- e) each time we encounte a piece of a line segment. This is necessay in ode to etieve the stat point of the line segment. Assuming that the cost of etieving a segment fom the featue table is C io, then the execution time of the algoithm is 4knC cpu + knc io. The cost tem knc io involves edundant disk-i/o equests. Basically, it epesents the cost of etieving line segments fom the featue table (once fo evey line patition, amounting to k disk equests pe line segment). A bette algoithm would pefom only n such equests, i.e., one equest pe line segment in the database instead of one equest fo each piece of each line segment. Thus, we would like to develop an algoithm that does not need the O(n) exta stoage (o at least having an asymptotic stoage cost less than O(n)), yet one that still pefoms only n disk equests (e.g., n accesses to the featue table). In the following sections, we show howwe can achieve this goal. Howeve, this depends on a close scutiny of the natue of the spatial objects which is the subject of the next section. 4 Object Classication An impotant facto aecting the pefomance of the Repot Unique opeation is the natue of the objects stoed in the database. o example, what is the eect of esticting the lines to be ectilinea, in contast to lines with abitay slopes? As anothe example, suppose that objects ae patially clipped as is the situation afte a window opeation (descibed in Section 2). Is it moe dicult to epot these clipped objects than epoting non-clipped objects? As an illustation of the second example, note that due to the way the PMR quadtee is dened, Repot Unique does not wok popely if the line segments ae patially clipped. This is shown in in igue 4. In paticula, when a line is clipped (e.g., as a esult of a window opeation), the PMR quadtee does not update the stating and ending points of the clipped line in the featue table. This is done since the line may be shaed by othe indexes in the database. In addition, this ensues consistency when potions of line segments ae emoved and then ae added back late in the pocessing as a esult of set opeations [6]. Thus the featue table is not updated and the clipped line segment is stoed implicitly. As a esult, if a line, say l, in igue 4 is clipped so that only s emains and if s does not contain l's stating point, then s will not be epoted by Repot Unique. This case suggests that we have to conside the altenative classes of objects (e.g., clipped objects) when developing algoithms fo Repot Unique since it aects the coectness of the algoithm. Due to space limitations, we estict ou discussion to ectangula objects only. Moe details about the classication and duplicate pocessing algoithms fo line segment objects can be found in [2]. Spatial objects of type ectangle can be classied in the following way (the dieent classes ae given in igue 5): Class- ectangles: a ectangle having no clipped o missing potions as illustated in igue 5a, and temed a egula ectangle. Class-2 ectangles: a ectangle whee only a ectangula potion of it is included and the est of the ectangle is clipped as illustated in igue 5b, and temed a clipped ectangle. Class- ectangles: a ectangle whee seveal potions (holes) may be missing, but still the est of the ectangle is connected, as illustated in igue 5c, and temed a clipped-out ectangle. Class-4 ectangles: a ectangle whee seveal potions (holes) of abitay shape may be missing that esult

4 q p x p y q x q y q p x p y q x q y p l p l B B 2 B B B 2 B featue table clipping window featue table igue 4: Line l as stoed in a PMR quadtee. The esult of clipping l by a window opeation in a PMR quadtee. The oiginal stating point oflin the featue table is neithe in blocks B 2 no in B. Notice that the featue table still stoes point p as pat of the desciption of the clipped line. in the ectangle being decomposed into seveal disjoint pieces, as illustated in igue 5d, and temed a disjoint ectangle. Although disjoint, all the potions of the ectangle efe to, and epesent, just one object of type ectangle. Type- ectangles: ectilinea ectangles, i.e., whose sides ae paallel to the axes (igue a). Type-2 ectangles: abitay ectangles - i.e., ectangles whose othogonal sides have abitay oientations (slopes) (igue b). (c) (d) igue 5: Classication of ectangles Class-, Class- 2, (c) Class-, (d) Class-4. This classication of ectangles is of pactical use. Class- epesents egula ectangles (e.g., bounding boxes of some spatial objects in a map). Class-2 epesents the pats of a ectangle that emain afte a ectangula window opeation (igue 6a). Class- epesents the pats of a ectangle window window window Clipped out egions window clipping polygon (c) clipping polygon igue 6: xample illustating how some of the ectangle classes ae ceated: Class-2, Class-, (c) Class-4. that lie outside one o moe ectangula quey windows (igue 6b). These ectangles esult fom clipping-out egula ectangles against the quey ectangles. Notice that each of the esulting ectangles is still epesentable as a fouconnected egion. Class-4 epesents the pats of a ectangle that lie inside (o outside) one o moe quey windows of abitay shape. o example, igue 6c is the esult of intesecting and clipping a ectangle with a numbe of simple polygons. We futhe classify ectangles of each class into the following types accoding to thei oientation: igue : Two types of ectangles: Type-, Type-2. 5 Duplicate Pocessing fo Rectangula Objects Below, we pesent algoithms fo the Repot Unique poblem that can handle seveal class/type combinations of ectangula objects given in Section 4, and see how these combinations aect the complexity of the algoithms. It is impotant to obseve that ectangles of the undelying database ae allowed to ovelap in space. Let q be the maximum numbe of ovelapping ectangles at any point in space, i.e., fo any block, thee ae at most q ovelapping ectangles. 5. Class- Type- and Class-2 Type- Rectangles The ectangles that coespond to these class/type combinations ae ones whose sides ae paallel to the x and y axes whee some of the ectangles may be clipped (Class-2) but ae still of ectangula fom. Ou goal fo this Class/Type combination is not to pefom any disk I/O to the featue table. In this case, the only infomation at hand ae the ectangle identies in the quadtee blocks. The algoithm taveses the ectangle quadtee blockby-block and maintains the active set of ectangle identi- es. These coespond to the ectangles that intesect the block. A ectangle identie, say id, which coesponds to a ectangle, say, is added to the active set when id is st encounteed duing the tavesal. id is deleted fom the active set once all the quadtee blocks that intesects have been visited. The necessay stoage is bounded by the maximum size of the active set duing the execution of the algoithm. The size of the active set is consideably smalle than n, the numbe of ectangles in the spatial database.

5 In ode to execute the algoithm eciently, we need to oganize the active set of ectangles. The oganization of the active set depends on the opeations that ae to be pefomed on the set. The algoithm pefoms the following thee pimitive opeations on an active set, say A: () test if a ectangle identie exists in A (notice that this is needed in ode to decide whethe a ectangle has aleady been encounteed o if it is encounteed fo the st time in the tavesal), (2) inset a ectangle identie into A, and () delete a ectangle identie fom A. Befoe poviding moe details about the algoithm, obseve that if we ae able to pefom each of the above thee opeations in O() time and maintain the undelying data stuctue that stoes the active set in O() time, then the oveall complexity of the algoithm would be O(nk) time with O(active set size) space. The algoithm visits the blocks of the quadtee in NW, N, SW, S scanning ode. It maintains one basic data stuctue: an active bode [5]. In the context of duplicate pocessing and hashing, the active bode is the spatial analog of a hash table. The active bode epesents the bode between those quadtee blocks that have been pocessed and those that have not. The elements of the active bode fom a \staicase" of vetical and hoizontal edges, moving fom southwest to notheast, as shown by the heavy line in igue 8. ach element of the active bode coesponds to an edge (bode) in the active bode. Initially, the active bode consists of two elements that coespond to the noth and west bodes of the image. When the algoithm teminates, the active bode consists of elements that coespond to the south and east bodes of the entie image. In othe wods, wheneve a node is visited by the algoithm, the elements of the active bode ae updated accodingly to include the bode of this new block. We dene an active ectangle as a ectangle that is patially pocessed, i.e., at least one of the quadtee blocks ovelapping with this ectangle has not been pocessed yet. A ectangle is inactive if eithe all o none of the blocks though which it passes have been pocessed by the tavesal algoithm. Thee is a data bucket associated with each element of the active bode. ach bucket is of capacity q, which is the same as the bucket capacity of the undelying quadtee block. vey element of the active bode stoes in its bucket the set of ectangle identies which coespond to the active ectangles that intesect the potion of the active bode coesponding to this element. Rectangle identies ae added to the active bode when they ae encounteed duing the tavesal, and ae popagated into the active bode until all the pieces of a ectangle have been visited. A ectangle is epoted when it is being deleted fom the active bode. The poblem is how tode- tect that a ectangle has been pocessed in its entiety when thee is no means fo the algoithm to know the coodinate values of the end-points of the ectangle. In ode to achieve this, the algoithm maintains some tempoay infomation in the active bode to help it detemine that the lowe-ight cone of the ectangle has been eached and that the ectangle identie can be epoted. o example, conside the ectangle given in igue 2a. We use igue 8 to illustate the pocess of epoting its pesence. Heavy shading is used to indicate that the block has aleady been tavesed. When block A is encounteed duing the tavesal (igue 8a), 's identie (associated with A) is inseted into the active bode. When block B is visited (igue 8b), 's identie is popagated into the active bode. Upon encounteing a westen o southen bounday of a ectangle fo the st time (e.g., when pocessing block C in igue 8c) a special make symbol, say e, is associated with the active bode element that led to the detection of this fact (e.g., the active bode element that is adjacent to block C). Notice that e is popagated into the active bode when block D is pocessed (igue 8d). As anothe example, we obseve that a special make symbol, say s is added to the active bode element contiguous to block (igue 8e) and is popagated into the active bode elements to the east of, when thei adjacent blocks in the database ae pocessed. At the time block is pocessed (igue 8f), the active bode elements to the noth and west of will contain the special makes e and s, espectively. This situation seves to indicate that we ae though. At this point, both e and s ae deleted fom the active bode and 's identie is epoted. Notice that is epoted once all the blocks compising have been visited by the tavesal pocedue. o example, in igue 8f, at the time block is visited, we ae sue that all the blocks inside ae aleady pocessed by the algoithm. This is because a NW, N, SW, S tavesal ode is admissible [4]. The complexity of the algoithm can be evaluated as follows. When a block in the quadtee that coesponds to a leaf node is pocessed, the potion of the active bode that is adjacent to the block must be located. In [], a technique is descibed that enables the blocks to be located in the active bode in constant time. This is achieved by tavesing (and updating) the active bode along with the quadtee tavesal while passing and stacking pointes to guaantee that the exact location in the active bode is available (O() time) wheneve needed, so that seaching in the active bode is entiely avoided. The eade is efeed to [] fo futhe details. Once the appopiate active bode element islo- cated, testing this element fo the existence of a ectangle as well as insetion and deletion of a ectangle can each be done in O(q) time which is eally a constant oo(). As demonstated in this section, ou algoithm does not need to pefom any disk I/O equests to access the featue table. As a esult, the time complexity of this algoithm is O(nkq) which is popotional to the numbe of object pieces in the database that ae encounteed duing the tavesal. Notice that the facto q is included because all the pocedues that access the active bode pefom in O(q) time, as descibed in the intoduction of Section 5. The space complexity iso(active ectangles), whee the active ectangles ae the ones that intesect the active bode at any given point. The size of the active set in the wost case is O(qT) whee q is the maximum numbe of objects that can be stoed in a quadtee block befoe it oveows (i.e., the bucket size) and T is the width (e.g., pixels) of the space compising the undelying spatial database. In pactice, it is expected that the size of the active set is consideably smalle than O(qT). Obseve that it is guaanteed that the active bode buckets will neve oveow since thee is one-to-one coespondence between elements of the active bode and blocks of the undelying quadtee. Since each block can hold up to q ectangles, then at wost each bucket will hold the same numbe of ectangles and hence cannot oveow. Notice that the algoithm does not depend on the actual coodinate values of the ectangle since it does not access the enties in the featue table and hence it woks coectly even if ectangles ae clipped due to some intesection with a ectangula window (Class-2 ectangles).

6 active bode active bode active bode A B C D A B C D A B Ce D (c) active bode active bode active bode A B Ce De A B C De A B C D s e e s s (d) (e) (f) = visited block = block of the ectangle igue 8: When block A is pocessed is inseted into the active bode. is popagated into the active bode when block B is pocessed. (c) When block C is pocessed a special make symbol e is inseted into the active bode. (d) e is popagated into the active bode when block D is pocessed. (e) When block is pocessed a special make symbol s is inseted into the active bode. (f) is epoted at the time block is pocessed and both e and s ae deleted fom the active bode. 5.2 Class- Type- and Class-4 Type- Rectangles Class- and Class-4 imply that the ectangles may have notches on them o may even consist of disjoint pieces. o these classes, we cannot avoid accessing thei enties in the featue table in ode to know the eal extent of the ectangles while tavesing them. Rectangle identies ae popagated in the active bode (egadless of whethe we ae tavesing a pat of the ectangle o a clipped out potion of the ectangle) until the lowe-ight cone of the ectangle is visited by the tavesal. In this case, the ectangle is epoted and deleted fom the active bode. This implies that the algoithm has to issue some disk I/O equests to the featue table. The algoithm poceeds as follows:. Tavese the spatial database as well as the active bode simultaneously in the NW, N, SW, S ode. 2. Maintain in the active bode: the identies of the ectangles that intesect the active bode o whose pieces have not yet been pocessed in thei entiety by the algoithm, and the coodinate values of the end-points of the ectangles in.. When the block containing the lowe-ight cone of ectangle is eached, epot the identie of, and delete 's infomation fom the active bode. igue 9 illustates the locations at which a disk I/O equest is issued and when a ectangle is epoted fo seveal possible ectangles in Class- and Class-4. Notice that fo Class- ectangles one, o moe disk I/O equests can be issued. o example igues 9a-c and e have only one disk I/O equest, igue 9d has two disk I/O equests as the uppe-left cone of the ectangle is clipped-out, while igue 9f has many disk I/O equests. o Class-4 ectangles, moe than one disk I/O equest is possible (e.g., igue 9g) depending on the numbe of blocks in one of the fou line segments of the window bounday that has a N-SW oientation and is closest to the oigin of the undelying space. Notice that fo Class-4 ectangles, if the uppe-left cone of the ectangle is not clipped out, then only one disk I/O equest is issued egadless of the numbe of disjoint pieces of the ectangle (e.g., igue 9h). Since the uppe-left cone of the ectangle is pocessed st, a disk I/O equest is issued to the featue table and the coodinate values of the end-points of the ectangle ae made known to the active bode. This helps avoid any futhe disk I/O equests. Obseve that this is not the case when the uppe-left cone of the ectangle is clipped out (e.g., igue 9f). igues 9f and 9g illustate the cases whee the maximum numbe of disk I/O equests fo Class- and Class-4 ectangles can be geneated, espectively. Let k w be the numbe of blocks in the edge of the window with a N{SW oientation that is closest to the oigin of the undelying space. igue 0 illustates the wost case numbe of disk I/O equests that ae equied with lines of this oientation. Obseve that a disk I/O equest occus with evey othe piece in the object. Theefoe, d kw e disk I/O equests ae 2 issued. Then, the maximum numbe of disk I/O equests to the featue table fo Class- and Class-4 is d kw e. 2

7 (c) (d) = disk I/O equest igue 9: -(c) When the ectangle is st encounteed a disk I/O equest is issued to etieve the ectangle's coodinate values. (d) Two disk equests ae issued wheneve, the uppe-left cone of the ectangle is clipped out (Class- ). (e) Only one disk equest is issued since the uppe-left cone of the ectangle is not clipped out (Class-). (f) The wost-case fo Class- ectangles occus when many quey windows clip out pats of the ectangle such that the uppeleft piece of the emaining potion of the ectangle foms a line segment of N-SW oientation. (g) Redundant disk equests ae issued when the uppe-left piece of the ectangle is pocessed since the uppe-left cone of the ectangle is clipped out (Class-4). (h) Only one disk equest is issued although the ectangle is patitioned into moe than one disjoint piece (Class-4). Notice that the uppe-left cone of the ectangle is not clipped out. It is woth mentioning that the task of uniquely epoting spatial objects is dieent fom that of connected component labeling [, 4]. In the latte task we ae inteested in assigning a unique identie to each fou- (o eight-) connected egion in the undelying space. The pincipal dieence is that, geneally speaking, the inputs of the two tasks ae diffeent. Moe specically, in Repot Unique we can always pefom a disk I/O equest to the featue table and etieve the object's exact desciption. This infomation is not made available to the connected component labeling algoithms. In fact, the main pupose of the connected component labeling algoithms is to build the object's exact desciption fom the local infomation given in the undelying space. As a esult of the dieence in the natue of the type of input and the goals of the two tasks, thei complexities ae (e) = disk I/O equest igue 0: The maximum numbe of disk equests that can aise fo a line with N-SW oientation. (f) (g) (h) dieent. o example, igue demonstates the wostcase complexity fo the connected component labeling algoithm (measued in tems of the new labels that have to be intoduced). On the othe hand, the shape of the e- B C 6 A igue : An 8 8 image that esults in geneating a maximum numbe of equivalence pais fo the connected component labeling algoithm. gion in igue does not epesent the wost case disk I/O complexity fo Repot Unique. In paticula, only two disk I/O equests will be issued (i.e., when blocks A and B ae visited). The wost case numbe of disk I/O equests fo Repot Unique fo clipped-out ectangles (Class-) is given in igue 9f. Notice that having a saw-tooth-like shape (e.g., as in igue ) does not change the wost case since at block B of igue, a disk I/O equest is issued to etieve the exact desciption of the ectangle fom the featue table, and this desciption will be popagated in the active bode in the easten and southen diections. This coves all the sawtooths to the east and south of block B, theeby obviating the need fo additional disk I/O equests (e.g., at block C). 5. Rectangles of Type-2 In ode to ensue that ectangles of abitay oientations ae epoted only once, we maintain thei ectilinea bounding box in the active bode (igue 2a) and epot the ectangle identie once the lowe-ight cone of the bounding box is coveed by the tavesal. As descibed in Section 5.2 = disk I/O equest igue 2: A bounding box is stoed in the active bode fo ectangles of Type-2. Seveal disk equests occu when pocessing the line segment closest to the uppe-left cone. (also in igues 9f, 9g, and 0) the numbe of disk I/O equests to the featue table is popotional to d kw e, whee 2 k w is the numbe of blocks in the line segment of the window bounday with a N-SW oientation that is closest to the oigin of the undelying space (igue 2b). Notice that once this set of disk I/O equests is issued, the algoithm does not need to access the featue table again since the

8 popagation of the ectangle identie and coodinate values in the active bode will help avoid any futhe disk I/O equests to the featue table. 6 Conclusion and utue Reseach Table 6 gives a summay of ou esults. It shows the dieent complexities of the algoithms developed fo the vaious class/type object combinations. As we can see, duplicate Type- Type-2 Class- O(knq) avg. cpu, no i/o O(knq) avg. cpu, d kw 2, en i/o Class-2 O(knq) avg. cpu, no i/o O(knq) avg. cpu, d kw 2, en i/o Class- O(knq) avg. cpu, d kw 2 en i/o O(knq) avg. cpu, d kw 2 en i/o Class-4 O(knq) avg. cpu, d kw 2 en i/o O(knq) avg. cpu, d kw 2 Table : en i/o Summay of time and space complexities of Repot Unique fo class/type combinations of ectangle objects. pocessing using the spatial chaacteistics of objects is a challenging poblem. In paticula, using the spatial chaacteistics of the objects enables us to adapt techniques used with hashing. o example, we wee able to educe the numbe of disk I/O equests (e.g., in compaison to O(nk) disk I/O equests when using hashing), sometimes to 0 o O(n) which is a signicant impovement. In addition, by using the extent and poximity of spatial objects we wee able to detect when an object is no longe needed in the active bode (analogous to a hash table), and hence we wee able to bound the size of the table. This was achieved by deleting the objects that ae entiely pocessed by the algoithm. utue wok involves developing bette algoithms fo othe class/type combinations, classifying spatial objects besides line segments and ectangles as well as developing algoithms fo them, and consideing othe spatial data stuctues that patition objects besides the quadtee vaiants. Acknowledgements The st autho conducted most of this eseach while at The Univesity of Mayland, College Pak. The suppot of the National Science oundation unde Gant IRI is gatefully acknowledged. Refeences [] W. G. Aef and H. Samet. cient pocessing of window queies in the pyamid data stuctue. In Poceedings of the 9th. ACM SIGACT-SIGMOD- SIGART Symposium on Pinciples of Database Systems (PODS), pages 265{22, Nashville, TN, Apil 990. (also in Poceedings of the ifth Bazilian Symposium on Databases, Rio de Janeio, Bazil, Apil 990, 5{ 26). [2] W. G. Aef and H. Samet. Uniquely epoting spatial objects: Yet anothe opeation fo compaing spatial data stuctues. In Poceedings of the 5th Intenational Symposium on Spatial Data Handling, pages 8{89, Chaleston, SC, August 992. [] M. B. Dillencout and H. Samet. Using topological sweep to extact the boundaies of egions in maps epesented by egion quadtees. Submitted fo publication, 99. [4] M.B. Dillencout, H. Samet, and M. Tamminen. A geneal appoach to connected-component labeling fo abitay image epesentations. Jounal of the ACM, 9(2):25{280, Apil 992. [5] R. lmasi and S. B. Navathe. undamentals of Database Systems. Benjamin/Cummings, Redwood City, CA, 989. [6] C. aloutsos, T. Sellis, and N. Roussopoulos. Analysis of object oiented spatial access methods. In Poceedings of the 98 ACM SIGMOD Intenational Confeence on Management of Data, pages 426{49, San ancisco, May 98. [] Michael J. olk and Bill Zoellick. ile Stuctues - Second dition. Addison-Wesley, Reading, MA, 992. [8] O. Gunthe. The design of the cell tee: an objectoiented index stuctue fo geometic databases. In Poceedings of the ifth I Intenational Confeence on Data ngineeing, pages 598{605, Los Angeles, ebuay 989. [9] A. Guttman. R-tees: A dynamic index stuctue fo spatial seaching. In Poceedings of the 984 ACM SIGMOD Intenational Confeence on Management of Data, pages 4{5, Boston, June 984. [0] A Klinge. Pattens and seach statistics. In J. S. Rustagi, edito, Optimizing Methods in Statistics, pages 0{. Academic Pess, New Yok, 9. [] R. C. Nelson and H. Samet. A consistent hieachical epesentation fo vecto data. Compute Gaphics, 20(4):9{206, August 986. (also Poceedings of the SIGGRAPH'86 Confeence, Dallas, August 986). [2] H. Nievegelt, H. Hintebege, and K. C. Sevcik. The gid le: an adaptable, symmetic multikey le stuctue. ACM Tansactions on Database Systems, 9():8{, Mach 984. [] A. Rosenfeld and A. C. Kak. Digital Pictue Pocessing. Academic Pess, New Yok, second edition, 982. [4] H. Samet. Connected component labeling using quadtees. Jounal of the ACM, 28():48{50, July 98. [5] H. Samet. Applications of Spatial Data Stuctues: Compute Gaphics, Image Pocessing, and GIS. Addison-Wesley, Reading, MA, 990. [6] H. Samet. The Design and Analysis of Spatial Data Stuctues. Addison-Wesley, Reading, MA, 990.