A Boolean Query Processing with a Result Cache in Mediator Systems

Transcription

1 A Boolea Query Processig with a Result Cache i Mediator Systems Jae-heo Cheog ad Sag-goo Lee * Departmet of Computer Sciece Seoul Natioal Uiversity Sa 56-1 Shillim-dog Kwaak-gu, Seoul Korea {cjh, sglee}cygus.su.ac.kr Abstract A mediator system is a kid of a meta-search egie that provides a seamlessly itegrated search service for diverse search egies (collectios). Sice collectios of a mediator system are geographically distributed, its performace is maily iflueced by the data trasmissio time betwee the mediator ad its collectios. Existig mediator systems employ a result cache that is composed of the results of previously issued queries to reduce this trasmissio time. However, these systems do ot support a geeral Boolea query model but oly simple oes such as a cojuctive query, a sigle keyword query, ad or so. I this paper, we propose a ew method to efficietly process geeral Boolea queries usig a result cache for mediator systems. Also preseted is a way to sematically partitio the give result cache i order to reduce the complexity of iferece. 1. Itroductio The huge umber of iformatio sources ad search egies o the Iteret makes itegrated search services, or meta-search services, almost ievitable. The mediatio architecture [1] is a atural choice for a meta-search system, i which a mediator provides users with seamless access to iformatio from distributed ad heterogeeous resources. Sice the iformatio sources ad search egies that a mediator system deals with are geographically distributed over the etwork, the data trasmissio time betwee the mediator ad its sources is a determiig factor i overall performace. I order to reduce the size of the query results that have to be trasferred, most mediators make use of the results of previous queries, typically stored i a result cache. Curret methods of query processig that utilize the result cache support oly simple query models that do ot allow geeral Boolea queries. * Author s work was supported i part by research fud provided by the Korea Research Foudatio, Support for Faculty Research Abroad Cache represetatio ad iferece based o the geeral Boolea model gives rise to complicated problems. First of all, it is ot practical for a result cache to be equipped with a keyword idex, such as a iverted file, sice it caot be kow i advace what keywords will occur i a query. Thus, if a query cotais at least oe keyword ot i the result cache, the etire cache must be searched exhaustively for proper sub-results that satisfy the curret query. Aother complicatio arises whe iformatio sources do ot support the NOT operator. To retrieve the portio of the result for a query Q that is ot already i the result cache C (we call this a miss result), a ew query Q AND NOT C (the miss query) would be set to the sources. If, however, some of the sources do ot support the NOT operator, the mediator caot retrieve those results directly. We propose a ew method to solve these problems ad to efficietly process geeral Boolea queries usig a result cache. We preset a method that utilizes the result cache to miimize the result size that must be retrieved from the sources for a keyword-based geeral Boolea query. The miss results ca be computed effectively eve whe the target collectios do ot support the NOT operator. Also preseted is a way to sematically partitio the give result cache i order to reduce the complexity of iferece. I the ext sectio, we preset other works related to the curret topic. A basic cache-based query-processig model is preseted i sectio 3. I sectio 4, we propose our base method of cache-based query processig. I sectio 5, we preset a method to sematically partitio the curret result cache ad how to process geeral Boolea queries with a partitioed result cache. Performace issues are cosidered i sectio 6. We coclude the paper i sectio 7 with a summary of our cotributios ad a brief discussio o future directios.. Related works There have bee a umber of articles o sematic data cachig [, 3, 4, 5, 6]. I [], the detailed method of rewritig a iitial query usig a cached query is

2 described. However a sigle database eviromet is assumed ad the algorithm for fidig a query match (a hit query i our cotext) is icomplete. A query optimizatio method is proposed i [3], which uses cached queries i a mediator system amed HERMES. This system, however, deals with oly simple coditios of the form c 1 θ c (where θ is ay of <,, or ). Cache replacemet strategies based o rececy ad sematic distace of cached queries are cosidered i [4], while cosistecy problems of cojuctive queries are dealt with i [5]. The query approximatio method preseted i [6] also cosiders oly cojuctive queries. Perhaps the work most closely related to sematic cachig i the database literature is [7], which deals with materialized views. This work, however, deals oly with cojuctive queries i SQL. A scheme for rewritig queries uder the Boolea query model cosiderig the capabilities of iformatio sources is preseted i [8]. The rewritig method is based o the exact-predicate matchig. I [9], a query history based virtual idex (QVI) is proposed. It is based o the observatio that queries are repeated over time ad users. However, oly simple keyword queries like Retrieve documets that cotai Database were cosidered. We propose to exted this basic idea to a keyword-based geeral Boolea retrieval model. 3. Prelimiaries I this sectio, we preset some basic cocepts ecessary to build our ew method. We adopt the keyword-based Boolea query model, where a query, for istace, Retrieve documets that cotai both keywords digital ad library ca be represeted by a Boolea expressio digital AND library. For otatioal coveiece, we use the symbol,, ad to deote the Boolea operators AND, OR, ad NOT, respectively Query model I a keyword-based Boolea query model, queries are defied recursively as follow: Defiitio 1: Query 1. A keyword i a query.. If A is a query, the ( A) is also a query. 3. If A ad B are queries, the (A B) ad (A B) are also queries. 4. All queries are geerated by applyig the above rules. O the Iteret, the target of a query is the set of documets. A documet ca be defied as a cojuctio of keywords that occur i it. For example, a documet that cotais keywords such as database, iformatio retrieval, ad mediator ca be represeted by database iformatio retrieval mediator. For a give query Q, the query result of Q, deoted by [Q], is defied as a set of documets that satisfy Q. I geeral, a result of a query varies as a documet database chages. However, the chage of a documet database s state is ot domiated by updatig or deletig existig documets but by appedig ew documets i the iformatio retrieval eviromet o the Iteret. So, we ca assume that the state of a documet database is static without loss of geerality. Example 1: Suppose a set of documets is {t 1, t t 3, t 1 t 3 t 5, t 4 t 6, t 5 t 6 }. The, the result of a query (t 1 t ) (t 3 t 5 ) is {t t 3, t 1 t 3 t 5 }. If two differet queries have the same result, we say that these two queries are equivalet. A query is a subquery of aother query if the result of the former is fully cotaied i that of the latter. To simplify our query model, we assume that all keywords are idepedet of each other. 3.. Cache-based query processig model A result cache is defied as the collectio of previously issued queries ad their results. Sice queries are represeted by Boolea expressios, a result cache is represeted by the disjuctio of previously posed queries. The cotets of a result cache are the uio of previous results. Defiitio : Result Cache Suppose Q 1, Q,, Q are queries that have bee issued by users up to this poit. The curret result cache, deoted by C, is defied as the disjuctio of previous queries (Q 1 Q Q ). Example : Suppose queries that have bee issued util ow are (t 1 t ), (t 3 t 5 ), (t t 3 ), ad (t 1 t 3 t 5 ), the the curret result cache is represeted by (t 1 t ) (t 3 t 5 ) (t t 3 ) (t 1 t 3 t 5 ). Whe a ew query is submitted, it ca be decomposed ito two sub-queries such as a hit query ad a miss query. While a hit query ca be aswered from the curret result cache, a miss query must be trasferred to the appropriate collectios to get aswers. Defiitio 3: Hit Query (HQ) For a give query Q, if a query Q C subsumes (Q C), the Q C is called a hit query for Q ad C. Q C is called the optimal hit query (OHQ) if Q C is equivalet to Q C.

3 Defiitio 4: Miss Query (MQ) For a give query Q, if a query Q C subsumes (Q C), Q C is called a miss query for Q ad C. Q C is called the optimal miss query (OMQ) if Q C is equivalet to Q C. If the result of Q C is ot empty ad Q is ot equivalet to C, the the origial query Q ca be both a hit query ad a miss query for Q ad C by itself. If Q is equivalet to C, the optimal hit query for Q ad C is equivalet to Q. Ad if Q C is empty, the optimal miss query for Q ad C is equivalet to Q. Example 3: Suppose a query Q is t (t 1 t 3 ) ad the curret cache C is t 1 (t t 3 ). The, the OHQ is Q C (t 1 t ) (t 1 t 3 ) (t t 3 ) ad the OMQ is Q C (t ( t 1 t 3 )). The procedure to process queries usig a result cache is composed of 3 steps. Step 1: Query Decompositio Whe a query is posed, the mediator decomposes the query ito a HQ ad a MQ. I geeral, the iitial query ca be used as both HQ ad MQ. Step : Retrieve Results The mediator retrieves cached results from the cache by applyig HQ, ad u-cached results by sedig MQ to the appropriate search egies. If we use a OHQ, we ca reduce the time to retrieve data from the cache. Usig a OMQ ca miimize the size of the results that should be trasferred from the search egies. Step 3: Itegratio of Results & Cache Update The fial result ca be geerated by itegratig the results from the cache ad those from the data sources. The cache is updated with the results retrieved from the data sources Problem statemet We should have reasoable ad practical solutios i order to realize the above procedure. First of all, if we wat to miimize the whole computatio time, we should decompose the iitial query ito a OHQ ad a OMQ. Ufortuately, however, it might be costly if the decompositio process employs a geeral logic iferece. I additio, splittig up a query ito two parts leads to more complicated queries that might be too expesive to process. Secod, if the size of a cache is large, the time required to get aswers from the cache ca be quite log. Third, sice it is commo that miss queries cotai oe or more NOT operators ad there might exist some search egies that caot directly aswer those miss queries because they do ot support the NOT operator. To those search egies, the mediator should sed a query that subsumes the miss query ad does ot cotai ay NOT operator. Ufortuately, however, it is expesive ad complicated to compute those subsumig queries. 4. Cache-based Query Processig We propose a ew descriptio method based o the propositioal logic for queries ad a result cache i this sectio. We also propose algorithms to process queries with a result cache Ufolded Disjuctive Normal Form (UDNF) We defie Ufolded Disjuctive Normal Form (UDNF) to be a variatio of the DNF where every cojuct is pair wise disjoit or usatisfiable. This meas that the itersectio of the result sets of ay two cojucts is always empty. Defiitio 5: UDNF A formula F is said to be i a ufolded disjuctive ormal form if ad oly if F has the form of F F 1 F F, where every F 1, F is pair wise disjoit. If a result cache is trasformed to its UDNF, its result is partitioed ito several disjoit sub-results represeted by cojucts respectively. There exist may ways to trasform a Boolea expressio to its UDNF. I this paper, we represet UDNF by meas of miterm. A miterm for a give set of keywords is defied as a cojuctio i which every keyword occurs exactly oce, either i its positive or egative form [10]. For istace, if the set of keywords is {t 1, t, t 3 } the the set of miterms is {t 1 t t 3, t 1 t t 3, t 1 t t 3, t 1 t t 3, t 1 t t 3, t 1 t t 3, t 1 t t 3, t 1 t t 3 }. Sice ay Boolea expressios ca be equivaletly trasformed ito a disjuctio of selected miterms ad miterms are pair wise disjoit, a Boolea expressio ca be equivaletly trasformed to a disjuctio of miterms ad it is a UDNF. Example 4: A formula t 1 t for the set of keywords {t 1, t } is equivalet to (t 1 t ) (t 1 t ) ( t 1 t ), which is a disjuctio of miterms. 4.. Geeratio of UDNF I this sectio, we itroduce a method to geerate a UDNF for a give Boolea expressio. Algorithm 1 describes a ormal method to geerate a UDNF of a iitial query. NormGeUDNF(A, B) returs a UDNF of B where A is the uiversal set of keywords.

4 Ufortuately, however, it is ot feasible to predefie a whole set of miterms i a keyword-based query model. Therefore, the result cache is represeted by miterms over the set of keywords that have occurred i previous queries. Upo processig of a ew query, the set of keywords is icremetally updated with ew keywords from that query. Both the cotet ad the represetatio of the cache are also updated. Algorithm describes a icremetal method to geerate a UDNF of a iitial query. IcGeUDNF(a, C) updates the curret UDNF, deoted by C, with a ew keyword a. It is well kow that ormalizatio of a Boolea expressio is very expesive ad ofte leads to a expoetial icrease with the size of the expressio. To solve this problem i this paper, a represetatio of the curret result cache is sematically partitioed to several sub-represetatios so that the size of the represetatio might be sigificatly reduced. Sectio 5 covers this solutio. Algorithm 1. Normal Geeratio of UDNF NormGeUDNF(A, B) Iput A: set of keywords B: target Boolea expressio Output UDNF of B cosiderig A BEGIN 1. Covert B to the correspodig postfix form.. U : set of miterms that ca be geerated by A 3. for every literal l i of B 4. covert l i to the disjuctio of appropriate miterms of U 5. ed for 6. R miterms of the first left literal 6. for every Boolea operator op of B 7. lr: miterms of the secod argumet of op 8. if op is AND the 9. R R lr 10. else op is OR the 11. R R lr 1. ed if 13. ed for 14. retur R END Algorithm. Icremetal Geeratio of UDNF IcGeUDNF(a, C) Iput C: curret UDNF a: ew keyword Output ew UDNF of C BEGIN 1. for every cojuct c i of C. c i c i a 3. C C c i 4. c i c i a 5. C C c i 6. ed for 7. retur C END 4.3. Decompositio of a query to OHQ ad OMQ ad retrieval of results I this subsectio, we itroduce a method to decompose a query ito two sub-queries such as OHQ ad OMQ. A miterm satisfies the followig trivial properties. Property 1: Let {m 1, m,..., m } be the uiversal set of miterms defied by a set of keywords. Suppose Q is a Boolea expressio ad represeted by m 1 m... m k where k. If a documet satisfies Q, it satisfies oly oe m i (1 i k) satisfies it but ot the other m i s. Property : If Q is a Boolea expressio, the Q UDNF is logically equivalet to the disjuctio of miterms that does ot occur i Q UDNF. Accordig to above two properties, a mediator ca easily decompose a query to its OHQ ad OMQ. Suppose the curret query is Q ad the curret result cache is C. Ad suppose Q UDNF ad C UDNF are UDNFs of Q ad C respectively. I the first place, Q C is equivalet to the set of commo miterms of Q UDNF ad C UDNF, sice other miterms that occur either i Q UDNF or C UDNF caot satisfy Q C accordig to property 1. Secodly, Q C is equivalet to the set of miterms that occurs i Q UDNF but ot i C UDNF, sice C is equivalet to C UDNF ad C UDNF is logically equivalet to the disjuctio of miterms that does ot occur i C UDNF accordig to property. Cosequetly, OHQ ad OMQ ca be computed by simply comparig miterms of Q UDNF ad C UDNF. (See figure 1) t 1 t Result Cache t 3 (t 1 t t 3 ) (t 1 t t 3 ) t t 1 Optimal Hit Query (t 1 t t 3 ) t 3 t Curret Query t 3 (t 1 t t 3 ) (t 1 t t 3 ) ( t 1 t t 3 ) t t 1 t 1 t 3 Optimal Miss Query ( t 1 t t 3 ) (t 1 t t 3 ) Figure 1. Compute OHQ ad OMQ Algorithm 3 describes the procedure to geerate OHQ ad OMQ of the curret query for the curret result cache.

5 Algorithm 3: Geeratio of OHQ ad OMQ GeOHMQ(C,Q) Iput C: UDNF of the curret cache Q: curret query Output OHQ ad OMQ BEGIN 1. for every keyword a of Q that does ot occur i C. C IcGeUDNF(a, C) 3. ed for 4. Q NormGeUDNF(all keywords of Q, Q) 5. for every keyword a of C that does ot occur i Q 6. Q IcGeUDNF(a, Q) 7. ed for 8. OMQ Q 9. for every cojuct q of Q 10. for every cojuct c of C 11. if c implies q the 1. OHQ OHQ c 13 OMQ OHQ c 14. ed if 15. ed for 16. ed for 17. retur OHQ ad OMQ END Example 5: Suppose a result cache is represeted by (t 1 t ) (t 1 t ) ad the curret query is (t 1 t 3 ) where t i is a keyword. Sice the query cotais a keyword that does ot occur i the curret cache represetatio, the cache expressio is updated to (t 1 t t 3 ) (t 1 t t 3 ) (t 1 t t 3 ) (t 1 t t 3 ) by applyig IcGeUDNF(t 3, (t 1 t ) (t 1 t )). Ad the query is trasformed to (t 1 t t 3 ) (t 1 t t 3 ) by applyig IcGeUDNF(t, (t 1 t 3 )). Cosequetly, (t 1 t t 3 ) (t 1 t t 3 ) is a OHQ ad the OMQ is ull NOT Operator The followig theorem makes it possible to retrieve the results of the OMQ without usig egatios. This is useful whe the target data source does ot support the NOT operator. Lemma 1: Suppose Q m i m j (i j) where m i ad m j are miterms defied o the set of keywords. Let i (p i ) be the sub-cojuct of egative (positive, resp.) literals of m i ad A( i ) be the set of atoms i i. If A( i ) A( j ) ad A( j ) A( i ), the Q is logically equivalet to (p i p j ) ( i j ). Proof) Sice m i p i i ad m j p j j, m i m j (p i i ) (p j j ). Ad (p i p j ) ( i j ) (p i i ) (p j j ) (p i j ) (p j i ). For p i j (or p j i ), there exist two cases. (1) A( i ) A( j ) I this case, there should exist at least oe keyword that occurs i p i i o-egated form ad i j i egated form. Cosequetly, p i j is always false. () A( i ) A( j ) Sice both A( i ) A( j ) ad A( j ) A( i ) are ot empty, there exists at least oe keyword that occurs i p i i o-egated form ad i j i egated form. Therefore, p i j is false like i the first case. From (1) ad (), every p i j (or p j i ) should be false. Therefore, (p i p j ) ( i j ) is equivalet to (p i i ) (p j j ). Sice m i is (p i i ) ad m j is (p j j ), Q is equivalet to (p i p j ) ( i j ). Let N(m i ) be the umber of literals that occur i m i. The, we ca state the followig lemma. Lemma : Suppose Q m i m j (i j) where m i ad m j are miterms defied o the same set of keywords. If A( i ) A( j ) ad N( i ) N( j ) 1, the Q is logically equivalet to (p i p j ) ( i j ). Proof) (p i p j ) ( i j ) m i m j (p i j ) (p j i ). Sice A( i ) A( j ), p j i is false but p i j is ot. Sice N( i ) N( j ) 1, the oly keyword that does ot occur i p i j occurs i m i i egated form ad i m j i oegated form. Let that keyword be a. The, p i j is logically equivalet to (p i j a) (p i j a). Sice (p i j a) ad (p i j a) are idetical with m i ad m j, respectively, Q is logically equivalet to (p i p j ) ( i j ). The followig theorem presets a ecessary coditio for computig a subsumig query of a OMQ. Theorem 1: Let Q UDNF be the UDNF of a give query Q. Formally, Q UDNF ca be defied as follows: Q UDNF m 1 m m where m i is a miterm defied uder a set of keywords. For ay two miterms m i ad m j (i j), if either A( i ) A( j ) ad A( j ) A( i ) or A( i ) A( j ) ad N( i ) N( j ) 1, the Q is logically equivalet to (p 1 p... p ) ( 1... ). Proof) Q (p 1 1 ) (p )... (p ). Ad (p 1 p... p ) ( 1... ) (p 1 1 ) (p )... (p ) (p 1 )... (p )... (p ). For those i j such that A( i ) A( j ) ad A( j ) A( i ), p i j ad p j i are false accordig to lemma 1. Ad for i j such that A( i ) A( j ) ad N( i ) N( j ) 1, p j i is false ad p i j is logically equivalet to p i i accordig to lemma. Cosequetly, Q is logically equivalet to (p 1 p... p ) ( 1... ).

6 If a OMQ satisfies the coditios metioed i theorem 1, a ew query that subsumes the OMQ ad does ot cotai ay NOT operator ca be geerated easily. Example 5: Suppose a OMQ is (a b c) ( a b c) (a b c). Sice it satisfies coditios of theorem 1, it is logically equivalet to ((a c) (b c) (a b)) ( a b c). The first part, ((a c) (b c) (a b)), is set to the data collectios ad the secod part, ( a b c), is used to filter out the extraeous results i the itegratio stage. A query that cotais a miterm that does ot satisfy the coditios of theorem 1 should be processed differetly. Sice there exist may ways to deal with this situatio, we illustrate a simple oe i the followig example. Example 6: Suppose a OMQ is (a b c d) ( a b c d) (a b c d) ( a b c d) (a b c d). This OMQ as a whole does ot satisfy the coditios of theorem 1. We partitio this OMQ ito two sub-queries. The first sub-query is composed of the first four miterms ad the remaider is the secod sub-query. The, we ca apply theorem 3 to each sub-query separately. (See figure ) Defiitio 6: Optimal Represetatio A represetatio of a result cache is called optimal if it ca fid OHQ ad OMQ for all queries by simply comparig miterms. Example 6: Suppose the uiversal set of keywords is {t 1, t, t 3 } ad the curret result cache is represeted by (t 1 t t 3 ) (t 1 t t 3 ) ( t 1 t t 3 ). The, it is a optimal represetatio. I geeral, if a result cache is represeted by a set of miterms defied by uiversal set of keywords, it is always optimal. However, it is impractical sice the umber of keywords ca be very large. To overcome this problem, we itroduce a curretly optimal represetatio. Defiitio 7: Curretly Optimal Represetatio If a represetatio of a result cache ca simply fid OHQ ad OMQ for the curret query, it is called a curretly optimal represetatio for the query. Every optimal represetatio is curretly optimal for all queries. Example 7: Suppose the curret query is (t 1 t t 3 ). If the curret result cache is represeted by (t 1 t ) ( t 1 t ) is ot curretly optimal. But if the same result cache if represeted by (t 1 t t 3 ) ( t 1 t t 3 ) (t 1 t t 3 ) ( t 1 t t 3 ), it is curretly optimal. (a b c d) (a b c d) ( a b c d) ( a b c d) ((a c) (a b d) (b c) b) (( b d) c a ( a c d)) a b c d (a c d) ( b) Whe a ew query is issued, the curret represetatio of a result cache ca be trasformed to a ew oe icremetally by applyig algorithm so that it is curretly optimal. However, the complexity of this algorithm is O( ) where is the umber of curret keywords, which is impractical. sed (a c) (a b d) (b c) b to data sources filter out the result with (( b d) c a ( a c d)) itegratio sed (a c d) to data sources filter out the result with ( b) Figure. Decompositio of a OMQ 5. Sematic partitioig of a result cache I this sectio, we itroduce a method to sematically partitio a represetatio of a result cache Optimal represetatio 5.. Sematic partitioig If we partitio the set of keywords of a result cache ito several sub-sets, the total umber of miterms eeded to describe a result cache is sigificatly reduced. For istace, suppose the umber of keywords is 30. If we wat to represet a result cache at a time, we eed 30 1,073,741,84 miterms. However, if we partitio the set of keywords ito 5 sub-sets, the each set cotais oly 6 keywords ad 6 64 miterms are eeded respectively. Let the curret represetatio of a result cache C be R C (K) where K is the set of keywords that metioed i queries util the. R C (K) is described by a disjuctio of miterms defied by K. For istace, suppose K is {t 1, t }, the R C (K) is described by (t 1 t ) ( t 1 t ). Some data i a result cache are idetified by (t 1 t ) ad the others by ( t 1 t ). Suppose K is partitioed ito K 1, K,..., K where K equals to K 1 K... K. The, R C (K) ca also be

7 partitioed ito R C (K 1 ), R C (K ),..., R C (K ). Ad R C (K) is equivalet to R C (K 1 ) R C (K )... R C (K ). Whe a represetatio R C (K) is partitioed ito R C (K 1 ) R C (K )... R C (K ), R C (K i ) ad R C (K j ) (i j) may ot be ecessarily disjoit with each other. (See figure 3) R C (K) sectio to process geeral Boolea queries usig a sematically partitioed cache. I the first place, methods to process sigle ad double keyword queries are preseted. The, preseted is a method to process geeral Boolea queries. Method 1: Sigle-keyword queries K 1 R C (K 1 ) K R C (K )... K R C (K ) First of all, we cosider a sigle keyword query. There are two possible ways to process a sigle keyword query with a partitioed cache. If there exists at least oe subrepresetatio i which the keyword of a query occurs, the the query ca be processed i ormal way proposed i this paper. We defie this way as the ormal processig way. If the keyword does ot occur i ay subrepresetatio, the the query ca be processed by extedig ay oe of sub-represetatios. This is defied as the exteded processig way. Result Cache Figure 3. Sematic Partitioig of a Result Cache There exist may ways that a represetatio of a result cache is partitioed ito some sub-represetatios. For example, it ca be doe whe a metrics such as the umber of cojucts holds a specific coditio. Suppose the curret result cache is represeted by (t 1 t t 3 ) ( t 1 t t 3 ) (t 1 t t 3 ) ( t 1 t t 3 ) (t 1 t t 3 ) ad the umber of cojucts are restricted ot exceedig 3. The, the curret represetatio should be simply partitioed ito two sub represetatios such as (t 1 t t 3 ) ( t 1 t t 3 ) (t 1 t t 3 ) ad ( t 1 t t 3 ) (t 1 t t 3 ). The detailed ways to partitio a result cache sematically is left to a future work. Defiitio8: Partially Optimal Represetatio Suppose the curret query is trasformed to its CNF (Cojuctive Normal Form), amely, d 1 d... d m. d i (1 i m). A partitio of R C (K), R C (K 1 ) R C (K )... R C (K ), is called partially optimal if there exists at least oe partitio R C (K j )(1 j ) that is curretly optimal for at least oe of all disjucts, d i s. If a partitio of R C (K) is partially optimal, a mediator ca fid OHQ ad OMQ for the part or the curret query by simply comparig miterms. Sice the others ca be used as a filter coditio ad a query cotais at most 3 or 4 keywords simultaeously, a partially optimal represetatio is a practical solutio. 5.. Query processig with a partitioed cache I this sectio, we itroduce a method based o the partially optimal represetatio metioed previous Method : Disjuctive queries with double keywords For a disjuctive query with two keywords, we ca calculate its results by applyig followig properties. Let OHQ(q, C) ad OMQ(q, C) be a optimal hit query ad a optimal miss query of q for a cache C respectively. The, OHQ(t 1 t, C 1 C ) ad OMQ(t 1 t, C 1 C ) are logically equivalet to OHQ(t 1 t, C 1 ) OHQ(t 1 t, C ) ad OMQ(t 1 t, C 1 ) OMQ(t 1 t, C ) respectively. Therefore, the result of (t 1 t ) ca be aswered by [OHQ(t 1 t, C 1 )] [OHQ(t 1 t, C )] [OMQ(t 1 t, C 1 )] [OMQ(t 1 t, C )]. I this case, there are possible three sub-cases. Case 1: A keyword t i occurs i a K i. I this case, we ca process the query directly by meas of above property. Case : Oe keyword occurs i a K i but ot the other Suppose t 1 occurs i K 1 but t does ot occur i ay sub-represetatio. The, OHQ(t 1, C 1 ) ad OMQ(t 1, C 1 ) ca be calculated i ormal processig way. OHQ(t, C i ) ad OMQ(t, C i ) ca be calculated i exteded processig way. Case 3: Both of two keywords do ot occur i ay subcache. Every OHQ ad OMQ ca be calculated i a exteded processig way. Method 3: Cojuctive queries with double keywords For a cojuctive query, we oly have to calculate the results of oe (we call this a hit keyword) of two keywords ad filter out its results with the other keyword.

8 If there exists at least oe keyword that occurs i a subrepresetatio, we select that keyword as a hit keyword. The result of a hit keyword ca be calculated i ormal processig way. If both of two keywords do ot occur i ay sub-represetatio, we select a radom keyword as a hit keyword. I this case, the result of a hit query ca be calculated i exteded processig way. Geeral queries A query is trasformed to its CNF (Cojuctive Normal Form). Let Q be a query ad Q CNF be its CNF. Accordig to method 3, we select ay oe disjuct from Q CNF ad process it by applyig method. The, we filter out its results with other disjucts. The efficiecy of this method is domiated by which disjuct might be selected. Sice the mai goal of cache-based query processig is to reduce the size of results that should be trasferred from collectios, we select a disjuct so that the umber of keywords ot i a cache should be miimal. 6. Performace Aalysis To prove the efficiecy of our method, we use the followig otatio: Table 1. Performace Metrics Metrics Descriptio the expected time to retrieve a documet from a cache td the average time to decompose a query ito OHQ ad OMQ(cotaied i ) S the expected time to retrieve a documet from both a cache ad a search egie ts the expected time to retrieve a documet from a search egie Q the umber of distict keywords i a query mq the umber of distict keywords that is i a query but ot i the cache C the umber of distict keywords i the cache the umber of distict keywords that is i the cache but ot i a query dc the average umber of documets i a cojuct of the cache N the umber of documets of the result for a give query T(Q) the estimated time to get a documet of the result for a give query Q First of all, we assume that dc ad N are greater tha or equal to 3. This assumptio is ot oly reasoable but also practical sice it is commo that the umber of results for a query is large. Sice k + k) ts is the time to retrieve k results from a cache ad (N k) from a search egie, the expected time to retrieve whole results from both a cache ad a search egie should be 1 1 N N k 0 ( k + k) ts) calculated as follows: 1 S N N 1 k 0 1 N N 1 k 0 ( k + k) ts) k + N 1 k 0. Therefore, S ca be k) ts N N N k N ts ts k N + k 0 k 0 k 0 1 N( N 1) N( N 1) ts N( N 1) ts + N 1 N( N 1) + N( N 1) ts N N( N 1)( + ts) N Theorem : For all queries, > td. 1 Proof) From algorithm 3, we ca say the followig. C+ mq Q Q+ C+ mq+ Q+ td I order to retrieve data from the cache, we should divide each partitio of the cache ito sub-partitios by meas of ewly-occurred atoms i a query. The, compare the cojucts of the cache with those of the query. Sice the average umber of data objects i a cojuct of the cache is dc ad the umber of atoms i the query ad the C+ mq cache is C + mq, it takes dc to retrieve data C + mq+ Q+ from the cache. It also takes to compare the cojucts of the cache with those of the query. Cosequetly, we ca say the followig. C+ mq C+ mq+ Q+ dc + Therefore, C+ mq Q+ 1 td ( dc 1) (1+ ) Sice C + mq Q + 1 I fact, > td ca be trivially proved sice we assumed that td is cotaied i. However, we preset a proof to show this assumptio is feasible.

9 1 td ( dc > 0( QdC> 3) > td ) C+ mq Accordig to theorem, it is clear that retrievig results from the cache is more expesive tha decomposig a query ito OHQ ad OMQ. I other words, it is ot too expesive to decompose a query. I geeral, ts is much larger tha sice ts cotais a etwork overhead ad a query should be set to may diverse data sources. If this assumptio is agreed o, we ca state the followig theorem. Theorem 3: For all queries, ts > S >. Proof) I) ts > S N( N 1)( + ts) S N N + N) N N + N)( ) + N N > 0( Q N > 0, ts > ) II) S > N( N 1)( + ts) S N N N + ( ) N N N + N) ts ( a ) N ( QtS a, a> 1) N ( 1)( a 1) ) N N ((N 1)(a 1) ) should be greater tha zero to make S >. Sice N 3 ad N 1, a should be greater tha to make (N 1)(a 1) >. Sice it is commo that the time to retrieve data from a search egie is much greater tha that to retrieve data from a cache, we ca reasoably ad practically assume that a is always greater tha. Therefore, from I ad II, ts > S >. [10] has reported that o the average 88% of the total queries uses terms that have bee used already. So, we ca state that a keyword repeatedly occurs i a ew query with probability p. Theorem 4: For a give query Q, T(Q) is always less tha ts. Proof) Q ca be trasformed a correspodig DNF like C 1 C... C where C i is (t i1 t i... t im ), t ij is a keyword. There exist three cases i processig C 1 C... C. (1) The result of C 1 C... C ca be retrieved oly from the cache. I this case, we do t have to sed C 1 C... C to ay search egies. I this case, every C i should be aswered from the cache. We ca show that the probability that C i might be aswered from the cache is (1 (1 p) m ). Therefore, the whole probability of this case should be (1 (1 p) m ). () The result of C 1 C... C should be retrieved oly from search egies, sice the cache does ot cotai ay results for C 1 C... C. I this case, we should get aswers of every C i from search egies oly. We ca show that the probability that C i might be aswered oly form search egies is (1 p) m. Therefore, (1 p) m is the probability of this case. (3) If the cache ca aswer C 1 C... C partially, we ca get aswers of C 1 C... C from both the cache ad search egies. The probability of this case is 1 (1 p) m (1 (1 p) m ), sice the sum of these three probabilities should be 1. Cosequetly, T(Q) is (1 (1 p) m ) + (1 p) m ts + (1 (1 p) m (1 (1 p) m ) ) S. Let (1 p) m be q the, 0 < q < 1 ad T(Q) is (1 + q ts + (1 q (1 ) S. T ( Q) (1 q (1 q (1 ) S Sice ts > S >, ts ad S ca be replaced by a ad b respectively where a > b > 1. Therefore, T( Q) a (1 + b q (( a b) ( a b) q (( a b)(1 q ( Qa> b> 1, q< 1) > 0 ts > T (Q) a q + b(1 + ( b 1)(1 ) ) + ( b 1)(1 ) b Accordig to theorem 4, our method ca process ay Boolea queries more efficietly tha others that do ot use the cached results.

10 7. Coclusio I this paper, we studied the maagemet of a result cache uder the keyword-based Boolea query model i the mediator cotext. I order to improve the query processig performace of a mediator system, it is essetial to efficietly recogize which part of a give query ca be aswered from a result cache ad which should be set to target collectios. I this paper, we proposed a efficiet method that allows this by adoptig the geeral Boolea query model i represetig the query ad the result cache. Curretly, we are extedig the work to query models based o first order logic to allow attribute-based Boolea queries. 8. Refereces [1] G. Wiederhold, Mediators i the Architecture of Future Iformatio Systems, IEEE Computer, Mar. 199, pp [] C. M. Che ad N. Roussopoulos, The Implemetatio ad Performace Evaluatio of the ADMS Query Optimizer: Itegratig Query Result Cachig ad Matchig. EDBT 1994, pp [3] S. Adali, K. S. Cada, Y. Papakostatiou, ad V. S. Shubrahmaia, Query Cachig ad Optimizatio i Distributed Mediator Systems, proceedigs of ACM SIGMOD Iteratioal Coferece o Maagemet of Data, Motreal, Caada, Jue 1996, pp [4] S. Dar M. J. Frakli, B. T. Josso, D. Srivastava, ad M. Ta, Sematic Data Cachig ad Replacemet, proceedigs of the d VLDB coferece, 1996, pp [5] A. M. Keller ad J. Basu, A Predicate-based Cachig Scheme for Cliet-Server Database Architectures, the VLDB joural, Vol. 5, No. 1, 1996, pp [6] D. Miraker, M. Taylor, ad A. Padmaaba, A Tractable Query Cache By Approximatio, Techical Report, MCC, [7] S. Abiteboul ad O. M. Duschka, Complexity of Aswerig Queries usig Materialized View, proceedigs of ACM SIGMOD-SIGACT-SIGART Symposium o Priciples of Database Systems, Seattle, WA, Jue [8] K. C. Chag, H. Garcia-Molia, ad A. Paepcke, Boolea Query Mappig Across Heterogeeous Iformatio Sources, IEEE Trasactios o Kowledge & Data Egieerig, Vol. 8, No. 4, 1996, pp [9] D.-g. Kim ad S.-g. Lee, QVI: Query-based Virtual Idex for Distributed Iformatio Retrieval System, proceedigs of ISCA 13 th Iteratioal Coferece CATA, 1998, pp [10] M. M. Mao, Digital Logic ad Computer Desig, Pretice-Hall, Eglewook Cliffs, NJ, 1979.