BI-TEMPORAL INDEXING

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1 BI-TEMPORAL INDEXING Mirella M. Moro Uniersidade Federal do Rio Grande do Sul Poro Alegre, RS, Brazil hp:// Vassilis J. Tsoras Uniersiy of California, Rierside Rierside, CA 92521, USA hp:// SYNONYMS Bi-emporal Access Mehods DEFINITION A bi-emporal index is a daa srucure ha suppors boh emporal ime dimensions, namely, ransacionime (he ime when a fac is sored in he daabase) and alid-ime (he ime when a fac becomes alid in realiy). The characerisics of he ime dimensions suppored imply arious properies ha he bi-emporal index should hae o be efficien. As radiional indices, he performance of a emporal index is described by hree coss: (i) sorage cos (i.e., he number of pages he index occupies on he disk), (ii) updae cos (he number of pages accessed o perform an updae on he index; for example when adding, deleing or updaing a record), and (iii) query cos (he number of pages accessed for he index o answer a query). HISTORICAL BACKGROUND Mos of he early work on emporal indexing has concenraed on proiding soluions for ransacion-ime daabases. A basic propery of ransacion-ime is ha i always increases. Each newly recorded piece of daa is ime-samped wih a new, larger, ransacion ime. The immediae implicaion of his propery is ha preious ransacion imes canno be changed. Hence, a ransacion-ime daabase can rollback o, or answer queries for, any of is preious saes. On he oher hand, a alid-ime daabase mainains he enire emporal behaior of an enerprise as bes known now. I sores he curren knowledge abou he enerprise's pas, curren or een fuure behaior. If errors are discoered abou his emporal behaior, hey are correced by modifying he daabase. In general, if he knowledge abou he enerprise is updaed, he new knowledge modifies he exising one. When a correcion or an updae is applied, preious alues are no reained. I is hus no possible o iew he daabase as i was before he correcion/updae. By supporing boh alid and ransacion ime, a bi-emporal daabase combines he feaures of he oher emporal daabase ypes. While i keeps is pas saes, i also suppors changes anywhere in he alid ime domain. Hence, he oerlapping and persisen mehodologies proposed for ransacion-ime indexing (for deails see chaper on Transacion-Time Indexing) can be applied [6, 9, 5]. The difference wih ransacion-ime indexing is ha he underlying access mehod should be able o dynamically manage inerals (like an R-ree, a quad-ree ec.). For a wors-case comparison of emporal access mehods, he reader is referred o [7]. SCIENTIFIC FUNDAMENTALS When considering emporal indexing, i is imporan o realize ha he alid and ransacion ime dimensions are orhogonal [8]. While in arious scenarios i may be assumed ha daa abou a fac is enered in he daabase a he same ime as when i happens in he real world (i.e., alid and ransacion ime coincide), in pracice, here are many applicaions where his assumpion does no hold. For example, daa records abou he sales ha occurred during a gien day are recorded in he daabase a he end of he day (when bach processing of all daa colleced during he day is performed). Moreoer, a recorded alid ime may represen a laer ime insan han he ransacion ime when i was recorded. For example, a conrac may be alid for an ineral ha is laer han he (ransacion) ime when his informaion was enered in he daabase. The aboe properies are criical in he design of a bi-emporal access mehod since he suppor of boh alid and ransacion ime affecs direcly he way records are creaed or updaed. Noe ha he erm ineral is used here o mean a conex subse of he ime domain" (and no a "direced duraion"). This concep has also been named a period ; in his discussion howeer, only he erm ineral is used.

2 The reader is referred o he enry on Transacion-Time Indexing, in which a ransacion ime daabase was absraced as an eoling collecion of objecs; updaes arrie in increasing ransacion-ime order and are always applied on he laes sae of his se. In oher words, preious saes canno be changed. Thus a ransacion-ime daabase represens and sores he daabase aciiy; objecs are associaed wih inerals based on his daabase aciiy. In conras, in he chaper on Valid-Time Indexing, a alid-ime daabase was absraced as an eoling collecion of ineral-objecs, where each ineral represens he alidiy ineral of an objec. The allowable changes in his enironmen are he addiion/deleion/ modificaion of an ineral-objec. A difference wih he ransacion-ime absracion is ha he collecion's eoluion (pas saes) is no kep. Noe ha when considering he alid ime dimension, changes do no necessarily come in increasing ime order; raher hey can affec any ineral in he collecion. This implies ha a alid-ime daabase can correc errors in preiously recorded daa. Howeer, only a single daa sae is kep, he one resuling afer he correcion is applied. A bi-emporal daabase has he characerisics of boh approaches. Is absracion mainains he eoluion (hrough he suppor of ransacion-ime) of a dynamic collecion of (alid-ime) ineral-objecs. Figure 1 offers a concepual iew of a bi-emporal daabase. Insead of mainaining a single collecion of ineralobjecs (as a alid-ime daabase does) a bi-emporal daabase mainains a sequence of such collecions C( i ) indexed by ransacion-ime. Assume ha each ineral I represens he alidiy ineral of a conrac in a company. In his enironmen, he user can represen how he knowledge abou company conracs eoled. In Figure 1, he -axis (-axis) corresponds o ransacion (alid) imes. A ransacion ime 1, he daabase sars wih ineral-objecs and. A 2, a new ineral-objec I z is recorded, ec. A 5 he alidime ineral of objec is modified o a new lengh. When an ineral-objec I j is insered in he daabase a ransacion-ime, a record is creaed wih he objec's surrogae (conrac_no I j ), a alid-ime ineral (conrac duraion), and an iniial ransacion-ime ineral [, UC). When an objec is insered, i is no ye known if i (eer) will be updaed. Therefore, he righ endpoin of he ransacion-ime ineral is filled wih he ariable UC (Unil Changed), which will be changed o anoher ransacion ime if his objec is laer updaed. For example, he record for ineralobjec I z has ransacion-ime ineral [ 2, 4 ), because i was insered in he daabase a ransacion-ime 2 and was deleed a 4. Noe ha he collecions C( 3 ) and C( 4 ) correspond o he collecions C a and C b of Figure 1 in he Valid-Time Indexing chaper, assuming ha a ransacion-ime 4 he erroneous conrac I z was deleed from he daabase. C( 1 ) C( 2 ) C( 3 ) C( 4 ) C( 5 ) I z I z Figure 1. A bi-emporal daabase. Based on he aboe discussion, an index for a bi-emporal daabase should: (i) sore pas saes, (ii) suppor addiion/deleion/modificaion changes on he ineral-objecs of is curren logical sae, and (iii) efficienly access and query he ineral-objecs on any sae. Figure 1 summarizes he differences among he arious daabase ypes. Each collecion C( i ) can be hough of on is own, as a separae alid-ime daabase. A alid-ime daabase differs from a bi-emporal

3 daabase since i keeps only one collecion of ineral-objecs (he laes). A ransacion-ime daabase differs from a bi-emporal daabase in ha i mainains he hisory of an eoling se of plain-objecs insead of ineral-objecs. A ransacion-ime daabase differs from a conenional (non-emporal) daabase in ha i also keeps is pas saes insead of only he laes sae. Finally, he difference beween a alid-ime and a conenional daabase is ha he former keeps ineral-objecs (and hese inerals can be queried). There are hree approaches ha can be used for indexing bi-emporal daabases. Approach 1: The firs one is o hae each bi-emporal objec represened by a bounding recangle creaed by he objec's alid and ransacion-ime inerals, and o sore i in a conenional mulidimensional srucure like he R-ree. While his approach has he adanage of using a single index o suppor boh ime dimensions, he characerisics of ransacion-ime creae a serious oerlapping problem [5]. A bi-emporal objec wih alid-ime ineral I ha is insered in he daabase a ransacion ime, is represened by a recangle wih a ransacion-ime ineral of he form [, UC). All bi-emporal objecs ha hae no been deleed (in he ransacion sense) will share he common ransacion-ime endpoin UC (which in a ypical implemenaion, could be represened by he larges possible ransacion ime). Furhermore, inerals ha remain unchanged will creae long (in he ransacion-ime axis) recangles, a reason for furher oerlapping. A simple bi-emporal query ha asks for all alid ime inerals ha a ransacion ime i conained alid ime j, corresponds o finding all recangles ha conain poin ( i, j ). Figure 2 illusraes he bounding-recangle approach; only he alid and ransacion axis are shown. A 5, he alid-ime ineral is modified (enlarged). As a resul, he iniial recangle for ends a 5, and a new enlarged recangle is insered ranging from 5 o UC. I 2 ( i, j ) UC I Figure 2. The bounding-recangle approach for bi-emporal objecs. Approach 2: To aoid oerlapping, he use of wo R-rees has also been proposed [5]. When a biemporal objec wih alid-ime ineral I is added in he daabase a ransacion-ime, i is insered a he fron R-ree. This ree keeps bi-emporal objecs whose righ ransacion endpoin is unknown. If a biemporal objec is laer deleed a some ime ' >, i is physically deleed from he fron R-ree and insered as a recangle of heigh I and widh from o ' in he back R-ree. The back R-ree keeps biemporal objecs wih known ransacion-ime ineral. A any gien ime, all bi-emporal objecs sored in he fron R-ree share he propery ha hey are alie in he ransacion-ime sense. The emporal informaion of eery such objec is hus represened simply by a erical (alid-ime) ineral ha cus he ransacion axis a he ransacion-ime when his objec was insered in he daabase. Inserions in he fron R-ree objecs are in increasing ransacion ime while physical deleions can happen anywhere on he ransacion axis.

4 In Figure 3, he wo R-rees mehodology for bi-emporal daa is diided according o wheher heir righ ransacion endpoin is known. The scenario of Figure 2 is presened here (i.e., afer ime 5 has elapsed). The query is hen ranslaed ino an ineral inersecion and a poin enclosure problem. A simple biemporal query ha asks for all alid ime inerals which conained alid ime j a ransacion ime i, is answered wih wo searches. The back R-ree is searched for all recangles ha conain poin ( i, j ). The fron R-ree is searched for all erical inerals ha inersec a horizonal ineral H ha sars from he beginning of ransacion ime and exends unil poin i a heigh j. (0, j ) H I 3 ( 3, 2 ) ( i, j ) ( 3, 1 ) 3 The fron R-ree I 2 ( i, j ) The back R-ree Figure 3. The wo R-ree mehodology for bi-emporal daa. When an R-ree is used o index bi-emporal daa, oerlapping may also incur if he alid-ime inerals exend o he eer-increasing now. One approach could be o use he larges possible alid-ime imesamp o represen he ariable now. In [2] he problem of addressing boh he now and UC ariables is addressed by using bounding recangles/regions ha increase as he ime proceeds. A ariaion of he R-ree, he GR-ree is presened. The index leaf nodes capure he exac geomery of he bi-emporal regions of daa. Bi-emporal regions can be saic or growing, recangles or sair-shapes. Two ersions of he GR-ree are explored, one using minimum bounding recangles in non-leaf nodes, and one using minimum bounding regions in non-leaf nodes. Deails appear in [2]. Approach 3: Anoher approach o address bi-emporal problems is o use he noion of parial persisence [4,1]. This soluion emanaes from he absracion of a bi-emporal daabase as a sequence of collecions C() (in Figure 1) and has wo seps. Firs, a good index is chosen o represen each C(). This index mus suppor dynamic addiion/deleion of (alid-ime) ineral-objecs. Second, his index is made parially persisen. The collecion of queries suppored by he ineral index srucure implies which queries are answered by he bi-emporal srucure. Using his approach, he Bi-emporal R-ree ha akes an R-ree and makes i parially persisen was inroduced in [5]. Similar o he ransacion-ime daabases, one can use he oerlapping approach [3] o creae an index for bi-emporal daabases. I is necessary o use an index ha can handle he alid-ime inerals and an oerlapping approach o proide he ransacion-ime suppor. Muli-dimensional indexes can be used for supporing inerals. For example, an R-ree or a quad-ree. The Oerlapping-R-ree was proposed in [6],

5 where an R-ree mainains he alid ime inerals a each ransacion ime insan. As inerals are added/deleed or updaed, oerlapping is used o share common pahs in he relean R-rees. Likewise, [9] proposes he use of quad-rees (which can also be used for spaioemporal queries). There are wo adanages in iewing a bi-emporal query as a parial persisence or oerlapping problem. Firs, he alid-ime requiremens are disassociaed from he ransacion-ime ones. More specifically, he alid ime suppor is proided from he properies of he R-ree while he ransacion ime suppor is achieed by making his srucure parially persisen or oerlapping. Concepually, his mehodology proides fas access o he C() of ineres on which he alid-ime query is hen performed. Second, changes are always applied o he mos curren sae of he srucure and las unil updaed (if eer) a a laer ransacion ime, hus aoiding he explici represenaion of ariable UC. Considering he wo approaches, oerlapping has he adanage of simpler implemenaion, while he parial-persisence approach aoids he possible logarihmic space oerhead. KEY APPLICATIONS The imporance of emporal indexing emanaes from he many applicaions ha mainain emporal daa. The eer increasing naure of ime imposes he need for many applicaions o sore large amouns of emporal daa. Accessing such daa specialized indexing echniques is necessary. Temporal indexing has offered many such soluions ha enable fas access. CROSS REFERENCES Temporal Daabases, Transacion-Time Indexing, Valid-Time Indexing, B+-ree, R-ree RECOMMENDED READING [1] B. Becker, S. Gschwind, T. Ohler, B. Seeger, and P. Widmayer (1996). An Asympoically Opimal Muliersion B- Tree. VLDB J. 5(4): [2] R. Bliujue, C.S. Jensen, S. Salenis and G. Sliinskas (1998). R-Tree Based Indexing of Now-Relaie Biemporal Daa. VLDB Conference, pp: [3] F. W. Buron, M.M. Hunbach, and J.G. Kollias (1985). Muliple Generaion Tex Files Using Oerlapping Tree Srucures. Compu. J. 28(4): [4] J.R. Driscoll, N. Sarnak, D.D. Sleaor, and R.E. Tarjan (1989). Making Daa Srucures Persisen. J. Compu. Sys. Sci. 38(1): [5] A. Kumar, V.J. Tsoras, and C. Falousos (1998). Designing Access Mehods for Biemporal Daabases. IEEE Transacions on Knowledge Daa Engineering 10(1): [6] M.A. Nascimeno and J.R.O. Sila (1998). Towards hisorical R-rees. Proceedings of SAC, pages [7] B. Salzberg and V.J. Tsoras (1999). A Comparison of Access Mehods for Time-Eoling Daa. ACM Compuing Sureys, 31(2): [8] R.T. Snodgrass and I. Ahn (1986). Temporal Daabases. IEEE Compuer 19(9): [9] T. Tzouramanis, M. Vassilakopoulos and Y. Manolopoulos (2000). Oerlapping Linear Quadrees and Spaio- Temporal Query Processing. Compu. J. 43(4):

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