Transaction-Consistent Global Checkpoints in a Distributed Database System

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1 Proceedngs of the World Congress on Engneerng 2008 Vol I Transacton-Consstent Global Checkponts n a Dstrbuted Database System Jang Wu, D. Manvannan and Bhavan Thurasngham Abstract Checkpontng and rollback recovery are well-known technques for handlng falures n dstrbuted database systems. In ths paper, we establsh the necessary and suffcent condtons for the checkponts on a set of data tems to be part of a transacton-consstent global checkpont of the dstrbuted database. Ths can throw lght on desgnng effcent, non-ntrusve checkpontng technques and transparent recovery technques for dstrbuted database systems. Keywords: Checkpontng, Recovery, Fault-tolerance, Dstrbuted databases 1 Introducton It s a common practce to take checkponts of a database from tme to tme, and restore the database to the most recent checkpont when a falure occurs. In order to mantan the atomcty (explaned n Secton 2) of transactons, t s desrable that a checkpont records a state of the database whch reflects the effect of a set of completed transactons and not the results of partally executed transactons. Such a checkpont of the database s called a transacton-consstent checkpont[1]. A straghtforward way to take a transacton-consstent checkpont of a dstrbuted database s to block all newly submtted transactons and wat tll all the currently executng transactons fnsh and then take the checkpont. Such a checkpont s guaranteed to be transacton-consstent, but ths approach s not practcal, snce blockng newlysubmtted transacton wll dramatcally ncrease transacton response tme and ths s not acceptable for the users of the database. A more effcent way would be to save (checkpont) the state of each data tem ndependently and perodcally wthout blockng any transacton. How- Ths materal s based n part upon work supported by the US Natonal scence Foundaton under Grant No. IIS and the US Department of Treasury Award #T Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the authors and do not necessarly reflect the vews of the Natonal Scence Foundaton or the Department of Treasury. Jang Wu and D. Manvannan are wth the Department of Computer Scence, Unversty of Kentucky, Lexngton, KY Tel/Fax: / Emal:{wuang,man}@cs.uky.edu. Bhavan Thurasngham s wth the Department of Computer Scence, Unversty of Texas at Dallas Tel/Fax: /2349 Emal:bhavan.thurasngham@utdallas.edu. ever f each data tem s checkponted ndependently and perodcally, the checkponts of a data tem may not be part of any transacton-consstent global checkpont of the database and hence are useless. In ths paper, we address ths ssue and establsh the necessary and suffcent condtons for a checkpont of a data tem (or the checkponts of a set of data tems) to be part of a transacton-consstent global checkpont of the database. Ths result would be useful for constructng a transacton-consstent global checkpont ncrementally from the checkponts of each ndvdual data tem. Ths s because, by applyng ths condton, we can start from an useful checkpont of a data tem and then can ncrementally add checkponts of other data tems untl we get a transacton-consstent checkpont of the database. Ths can also throw lght on desgnng effcent non-ntrusve dstrbuted checkpontng algorthms. 1.1 Motvaton and Obectves In a dstrbuted system, to mnmze the lost computaton due to falures, the state of the processes nvolved n a dstrbuted computaton are perodcally checkponted. When one or processes nvolved n a dstrbuted computaton fals, the processes are restarted from a prevously saved consstent global checkpont. When process are ndependently checkponted, the checkponts taken may not be part of any consstent global checkpont and hence are useless [2]. Netzer and Xu establshed the necessary and suffcent condtons for the checkpont of a process to be useful. Netzer and Xu [2] ntroduced the noton of zgzag paths between checkponts of processes nvolved n the computaton. They proved that a checkpont of a process s useful f and only f there s no zgzag path from that checkpont to tself. In ths paper, we generalze ths result to database systems. 1.2 Organzaton of the paper The remander of ths paper s organzed as follows. In Secton 2 we ntroduce the necessary background requred for understandng the paper. Secton 3 dscusses related works. In Secton 4 we present the necessary and suffcent condtons for a set of checkponts on a set of data tems to belong to a transacton consstent global checkpont and prove ts correctness. Secton 5 concludes the

2 Proceedngs of the World Congress on Engneerng 2008 Vol I paper and also presents the future work. 2 Background 2.1 System model We consder a model of dstrbuted database system smlar to the model n [1]. In ths model, a dstrbuted database system conssts of a set of data tems resdng at varous stes. Stes can exchange nformaton va messages transmtted on a communcaton network, whch s assumed to be relable. The data tems of the database are accessed by transactons and the transactons are controlled by transacton managers (TM) that resde on these stes. The TM s responsble for the proper schedulng of transactons usng approprate concurrency control algorthms n such a way that the ntegrty of the database s mantaned. In addton, the data tems at each ste are controlled by a data manager (DM). Each DM, whch s responsble for controllng access to tems at ts ste. Each data tem s checkponted by a transacton perodcally. Before a transacton takes a checkpont of a data tem t obtans an exclusve lock on the data tem so no other transacton can be accessng that data tem whle t s checkponted. The state of a data tem changes when a transacton accesses that data tem for a wrte operaton. In order to guarantee the ntegrty and effcency of transacton processng, four requrements referred to as ACID [4] must be mantaned. Atomcty: Each transacton s executed n ts entrety, or not at all executed. Consstency preservaton: Executon of a transacton n solaton (that s, wth no other transacton executng concurrently) preserves the consstency of the database. Isolaton: Even though multple transactons may execute concurrently, the system guarantees that, for every par of transactons T and T, t appears to T that ether T fnshed executon before T started, or T started executon after T fnshed. Thus, each transacton s unaware of other transactons executng concurrently n the system. Durablty: After a transacton completes successfully, the changes t has made to the database persst, even f there are system falures. In order to mantan ACID requrements and acheve maxmum performance, a proper schedule of transactons need to be arranged n whch the operatons of varous transactons are nterleaved as much as possble. Gven a schedule, a drected graph, referred to as precedence graph [5] or seralzaton graph [4], can be constructed to llustrate the procedure of all the transactons runnng n the database system. The seralzaton graph serves as an mportant tool to analyze transacton processng n the dstrbuted database systems. Each checkpont on a data tem s assgned a unque sequence number. We assume that the database conssts a set of n data tems X = {x 1 n}. In addton, we denote by C k the checkpont on x wth sequence number k. The set of all checkponts on data tem x s denoted by C = {C k k : k 0}. The ntal state of data tem x s represented by checkpont C 0 and a vrtual checkpont C vrtual represents the last state obtaned after termnaton of all transactons accessng data tem x. A data tem s checkponted only after the state of the data tem changes. That s, after a data tem s checkponted, t s not checkponted agan untl at least one other transacton has accessed and changed the data tem. Let T = {T 1 m} be a set of transactons that access the database system. In order to analyze the relatonshp between checkponts of varous data tems, we assume that each checkpont of a data tem x s taken by a specal transacton, called checkpontng transacton. We denote by T C k takes checkpont C k atomcty of transactons, T C k the checkpontng transacton that of data tem x. In order to mantan s the local transacton whch s requred to be scheduled to access a data tem when there are no other transactons accessng the data tem. The set of checkpontng transactons that produce the checkponts C s denoted by T C and the set of all checkpontng transactons n the system s denoted by T C. A global checkpont of the database s a set S = {C k 1 n} of local checkponts consstng of one checkpont for each data tem. The set of checkpontng transactons that produce the global checkpont S s denoted by S T = {T C k 1 n}. We use C k and T C k nterchangeably. Sometmes, when we say a checkpont of a data tem we mean the checkpontng transacton whch takes that checkpont. Each regular transacton s a partally ordered set of read and/or wrte operatons (operatons are partally ordered because two adacent read operatons n a transacton are not comparable). A checkpontng transacton conssts of only one operaton (namely the checkpontng operaton), an operaton that s smlar to a wrte operaton whch requres mutually exclusve access to the data tem. Let R (x ) (respectvely, W (x )) denote the read (wrte) operaton of T on data tem x X and O C k the checkpontng operaton of T C k (x ) denote on data tem x. A schedule ε over T T C s a famly of dsont sets of partally ordered operatons of transactons n T T C on the data tems (one set for each data tem) [1]. Let ε(x ) consst of all read, wrte and checkpontng operatons

3 Proceedngs of the World Congress on Engneerng 2008 Vol I on x of all transactons n T T C. We denote by < x the partal order nduced by all read, wrte, and checkpontng operatons on x by the schedule ε over T T C. Gven a schedule ε over T T C, we defne the relaton < T between transactons n T T C wth respect to a schedule ε as follows: 1) T < T T ( ) ( x k X : (R (x k ) < xk W (x k )) (W (x k ) < xk W (x k )) (W (x k ) < xk R (x k ))). 2) T < T T C k O C k 3) T C k (W (x ) < x (x )) (R (x ) < x O C k < T T (O k C (x ) < x W (x )) (O C k (x )). (x ) < x R (x )). A schedule s seral f the operatons belongng to each transacton appear together n the schedule [4]. A schedule ε s seralzable f the schedule has the effect equvalent to a schedule produced when transactons are run serally n some order. The concurrency control algorthm ensures that a schedule of transactons runnng n the dstrbuted database system s seralzable. One mportant knd of seralzaton, called conflct seralzablty (CSR) [4] s consdered n ths paper. An executon ε CSR ff the relaton < T s acyclc. A seralzaton order of a set of transactons wth respect to a schedule ε over T s defned as a lnear orderng of all the transactons such that f T < T T (ether T or T could be checkpontng transacton), then T must appear before T n the orderng. If ε CSR, there must exst a seralzaton order for ε over T that s compatble wth < T. Formal defnton of a transacton consstent global checkpont follows [1]: Defnton 1 A global checkpont of a dstrbuted database system s sad to be transacton-consstent (trconsstent or smply consstent, for short) wth respect to a set of transactons T f there exsts a seralzaton order (whch s a sequence of transactons) σ 1 σ 2 for an executon ε CSR of T such that the data tem states represented by the global checkpont s the same as those read by a read-only transacton T CP after all transactons n σ 1 have fnshed executon and before any transacton n σ 2 has started executon. If the concurrency control algorthm guarantees an executon ε CSR, then the relaton < T nduces a drected acyclc graph (Dag) on T T C and conversely. We call ths graph the global seralzaton graph wth respect to the schedule ε of T T C. For each data tem, the transactons accessng that data tem nduce a component of the global seralzaton graph. The local seralzaton graph nduced by the transactons n T T C accessng data tem x s denoted by G x (V x, E x ); the vertex set V x = {T k T k C T k T has accessed data tem x ; C k s the checkpont of x taken by local checkpont transacton T k C where } and the edge set E x = {E TT x E TC x 1: E TT x = {(T, T ) T, T V x ; T < T T }. 2: E TC x 3: E CT x = {(T, T k C ) T, T k C Ex CT }, V x ; T < T T k C }. = {(T k C, T ) T, T k C V x ; T k C < T T }. By mergng the local seralzaton graphs G x (V x, E x ), we can construct the global seralzaton graph G(V, E) where V = and x X E = x X V x E x We use the followng notatons throughout the paper: T + T ff there s a path from transacton T to T (T and/or T could be a checkpontng transacton). T T ff there s an edge from T to T (T or T could be a checkpontng transacton). Let σ 1 T and σ 2 T be such that σ 1 σ2 = φ. Then by σ 1 S T σ 2 wth respect to the seralzaton order nduced by conflct-seralzable executon ε over T, we mean that each checkpontng transacton n S T s executed after every transacton n σ 1 has been executed and before any transacton n σ 2 has started executon. In partcular, f σ 1 σ2 = T, then the set of checkponts S taken by S T s tr-consstent ff σ 1 S T σ 2. Next, we make the followng observatons: Observaton 1 For any checkpontng transacton T k C, must snce t accesses the data tem x exclusvely, T C k have a path n the local seralzaton graph ether to or from any transacton T that has accessed x. Observaton 2 For any checkpontng transacton T k C, snce t accesses the data tem x exclusvely, f T + T k C and T k C + T, then n the local seralzaton graph nduced by the operatons n T T C on the data tem x any path from T to T must pass through T C k Observaton 3 In the local seralzaton graph nduced by the operatons n T T C on the data tem x, for any checkpontng transacton T k C and two other transactons T and T that have accessed x, the followng holds:.

4 Proceedngs of the World Congress on Engneerng 2008 Vol I 1. If T k C + T and there exsts T + T wthout any checkpont along the path n the local seralzaton graph, then T k C + T. 2. Smlarly, f T + T k C and there exsts a path T + T from T to T wthout any checkpont along the path n the local seralzaton graph, then T + T k C. Observatons 1 and 2 are trval. Observaton 3 holds because n case 1, suppose T k C + T s not true, then T + T k C from Observaton 1. Snce T k C + T, from Observaton 2, every path n the local seralzaton graph from T to T must pass through T k C from Obser- vaton 2, whch contradcts our assumpton that there exsts a path T + T wthout any checkpontng transactons along the path. Smlar argument can be used to prove the correctness of case 2 n Observaton 3. We make use of these Observatons n the proof of the two mportant theorems n Secton 4. 3 Related Work The checkpontng algorthms for dstrbuted database systems can be classfed as log-orented and dumporented. In log-orented approach, perodcally a dump of the database s taken and also a marker s saved at approprate places n the log. When a falure occurs, the latest dump s restored and the operatons on the log after the dump was taken s appled to the dump untl the marker s reached to restore the system to a consstent state. In ths approach, proper postonng of the marker n the log wll produce a consstent global checkpont. In the dump-orented approach, checkpontng s referred to as the process of savng the state of all data tems n the database (or takng a dump of the database) n such a way that t represents a transacton-consstent checkpont of the database. The algorthms proposed n [6, 7, 8, 9] take ths approach. The basc dea behnd the algorthm n [6] s to dvde the transactons nto two groups: those before or after the checkpontng process. Ths algorthm s non-ntrusve but requres a copy of the database stored temporarly. Ths temporary copy s used by transactons that could not be decded whether or not they belong to any of the two groups when the checkpontng process s gong on. Pu [7] uses color method (whte and black) to dstngush data tems that have started checkpontng wth data tems that have not. Transactons accessng both whte and black data tems have to be aborted or delayed n order to mantan consstency, whch ncreases transacton response tme. Plarsk et al. [8] consder checkponts as checkpont transactons, one for each data tem. In addton, a checkpont number (CPN) s assocated wth each checkpont. By comparng the CPN, forced checkponts on data tems are taken n order to mantan consstency. The prevous two algorthms are coordnated algorthms, n whch one process ntates and coordnates the checkpontng actvty. The algorthm proposed by Baldon et al. [9] uses a non-coordnated approach, n whch no process ntates checkpontng and each data tem s checkponted ndependently. Lke the algorthm of Plarsk et al. [8], checkpont numbers are used to synchronze checkpontng process and forced checkponts are taken to mantan consstency. Ths algorthm s fully dstrbuted but may may suffer from large checkpontng overhead due to forced checkponts. 4 Necessary and Suffcent Condton Frst, we establsh the necessary and suffcent condtons for a set of checkponts, one from each data tem to form a transacton-consstent global checkpont. Theorem 1 Let T = {T 1,, T m } be a set of transactons accessng the database consstng of n data tems X = {x 1,,x n }. Assume that each data tem s checkponted by a checkpontng transacton that runs at the ste contanng the data tem. Let S = {C k 1 n} be a set of checkponts, one for each data tem and let S T = {T k C 1 n} be the set of checkpontng transactons that produce S. Let ε be a schedule over T. Then S s a tr-consstent global checkpont ff there s no path between any two checkpontng transactons S T n the global seralzaton graph correspondng to the schedule ε. Proof: For lack of space we omt the proof of the theorem. Theorem 1 s useful for verfyng whether a gven global checkpont s transacton-consstent. However, ths theorem does not help n constructng a transactonconsstent global checkpont ncrementally. Ths s because, f there s no path between two checkponts of two dfferent data tems, t does not mean that these two checkponts together can be part of a transactonconsstent global checkpont. Therefore, addtonal restrctons need to be added n order to be able to extend a gven set of checkponts to a transacton-consstent global checkpont. As mentoned earler, our goal s to come up wth the necessary and suffcent condtons for a set of checkponts to be part of a transacton-consstent global checkpont. Next, we ntroduce some termnology for developng the necessary and suffcent condtons for a set of checkponts to be part of a transacton-consstent global checkpont. Netzer and Xu [2] ntroduced the concept of zgzag paths between checkponts of a dstrbuted computaton and used t to establsh the necessary and suffcent condtons for a set of checkponts of a dstrbuted computaton to be

5 Proceedngs of the World Congress on Engneerng 2008 Vol I part of a consstent global checkpont. We generalze ther defnton of zgzag paths to checkponts n dstrbuted database systems and use t for establshng the necessary and suffcent condtons for a set of checkponts of a dstrbuted database to be part of a transacton-consstent global checkpont. Defnton 2 Let C k be a checkpont taken by the check-, and let C k be another check-. We say a pontng transacton T C k pont taken by checkpontng transacton T C k to T C k f there exsts a set zgzag path exsts from T k C of transactons T = {T 1, T 2,, T v } T such that a) T 1 T s a transacton such that T k C the global seralzaton graph; T 1 n b) for any T k T (1 k < v), T k+1 T (1 < (k + 1) v) s a transacton such that 1: T k T k+1 (we call such an edge as reverse edge); or 2: T k T k+1 or (T k T C kw w T k+1 for some w); and T C kw w c) T v T s a transacton such that T v T k C ; Note that a path n the global seralzaton graph s also a zgzag path but not conversely. Next, we establsh the necessary and suffcent condton formally. Theorem 2 A set S of checkponts, each checkpont of whch s from a dfferent data tem, can belong to the same consstent global checkpont wth respect to a seralzable schedule of a set of transactons ff no checkpont n S has a zgzag path to any checkpont (ncludng tself) n S n the global seralzaton graph correspondng to that schedule. Proof: (If-Part:) Suppose no checkpont n S has a zgzag path to any checkpont (ncludng tself) n S. We construct a consstent global checkpont S that contans the checkponts n S plus one checkpont for each data tem not represented n S as follows: For each data tem that has no checkpont n S and that has a checkpont wth a zgzag path to a member of S, we nclude n S ts frst checkpont that has no zgzag path to any checkpont n S. Such a checkpont s guaranteed to exst because the vrtual checkpont of a data tem does not have an outgong zgzag path. For each data tem that has no checkpont n S and that has no checkpont wth zgzag path to a member of S, we nclude ts ntal checkpont (t s also the frst checkpont that has no zgzag path to any member of S and there cannot be a zgzag path from any checkpont n S to ths ntal checkpont). We clam that S s a tr-consstent global checkpont. From Theorem 1, t s suffcent to prove that there s no path between any two checkponts of S n the global seralzaton graph. Suppose there s a path from a checkpont A S to a checkpont B S. Assume that the checkpont A was taken on data tem x and checkpont B was taken on data tem on x. Case 1: A, B S. Ths condton mples that a zgzag path exsts from A to B, contradctng the assumpton that no zgzag path exsts between any two checkponts n S. Case 2: A S S and B S. Ths contradcts the way S S s constructed (checkponts n S S are chosen n such a way that no zgzag path exsts to any member of S from those checkponts). Case 3: A S and B S S. B cannot be an ntal checkpont, snce no checkpont can have a path to an ntal checkpont. Then by the choce of B, B must be the frst checkpont on x that has no zgzag path to any member of S. The checkpont precedng B on x, say D, must have a zgzag path to some member of S, say E. Let T u be the transacton (that accessed x and created the edge T u B) that les on the zgzag path from A to B. Smlarly, let T v be a transacton (that accessed x and created the edge D T v ) that les on the zgzag path from D to E. Note that such transactons exst because B and D are checkponts of data tem x. In addton, accordng to our assumpton that checkpont s taken only after the state of the data tem has been changed by transacton, let us suppose such transacton s T w. Clam: There exsts a zgzag path from A to E n the global seralzaton graph. Proof of the clam: Based on Observaton 1, we have ether T u + D or D + T u n local seralzaton graph correspondng to data tem x. If T u + D, due to the exstence of zgzag path from D to E, we get a zgzag path from A to E through T u, D and T v. and hence the clam. On the other hand, f D + T u, then because of the edge from T u to B, we know that T u must happen between D and B n the local seralzaton graph. Smlarly, we can show that T v must also happen between D and B n the local seralzaton graph. Meanwhle snce T w happen between D and B, we conclude that any path between T u, T v and T w must have no checkponts along the path snce D and B are two adacent checkponts n the local seralzaton graph of x. Snce T w nvolves wrte

6 Proceedngs of the World Congress on Engneerng 2008 Vol I operatons, whch s smlar to checkpontng operatons, all the observatons for checkpontng transactons can be used for T w. From Observaton 1, T w must have a path to or from both T u and T v, therefore four cases arse: 1) If T u + T w + T v, the zgzag path from A to T u, the path T u + T w + T v and the zgzag path from T v to E construct the zgzag path from A to E. 2) If T v + T w + T u, the zgzag path from A to T u, the reverse path T v + T w + T u and the zgzag path from T v to E construct the zgzag path from A to E. 3) If T w + T u and T w + T v, the zgzag path from A to T u, the reverse path from T u to T w, the path from T w to T v and the zgzag path from T v to E construct the zgzag path from A to E. 4) If T u + T w and T v + T w, the zgzag path from A to T u, the path from T u to T w, the reverse path from T w to T v and the zgzag path from T v to E construct the zgzag path from A to E. Snce local seralzaton graph s a component of the global seralzaton graph, any local path can fnd ts correspondence n the global seralzaton graph. Therefore the zgzag path from A to E must can be found n the global seralzaton graph. Therefore no matter what the stuaton s, we can always fnd a zgzag path from from A to E, whch a contradcton to our assumptons. Ths s a contradcton to the assumpton that no zgzag path exsts between any two checkponts n S. Case 4: A S S and B S S. As n case 3, B must be the frst checkpont on x that has no zgzag path to any member of S. Then the checkpont that precedes B on data tem x, say D, must have a zgzag path to some member of S, say E. Then, as n case 3, there exsts a zgzag path from A to E. Ths contradcts the choce of A where A s the frst checkpont on data tem x wth no zgzag path to any member of S. Therefore S, contanng S, s a tr-consstent global checkpont. (Only-f Part:) Conversely, suppose there exsts a zgzag path between two checkponts n S (ncludng zgzag cycle), then we show that they cannot belong to the same transacton-consstent global checkpont. Assume that a zgzag path exsts from A to B (A could be B) and along such a path, the length of consecutve reverse edges s at most w. We use nducton on w to show that A and B cannot belong to the same consstent global checkpont. Base case (w = 0): If the length of consecutve reverse edges s at most zero, the zgzag path from A to B s n fact a path from A to B. Then, from Theorem 1, A and B can not belong to the same consstent global checkpont. Base case (w = 1): Suppose the length of consecutve reverse edges along the zgzag path from A to B s at most one. Let the consecutve reverse edges wth length equal to one from A to B be T 1,1 T 2,1,, T 1,u T 2,u. Suppose those reverse edges are components of local seralzaton graph correspondng to data tems x 1,1,, x 1,u. Clam: x 1,1,, x 1,u can not all be equal to x where A takes place. Proof of clam: Suppose x 1,1,, x 1,u are all equal to x. Then A, T 1,1, T 2,1,, T 1,u, T 2,u are transactons accessng x (note that we use A and the checkpontng transacton that takes the checkpont A nterchangeably). From Observaton 1, the followng two cases arse: 1) A + T 2,u. If ths s the case, a path A + B va T 2,u exsts and hence A and B can not be part of a transacton-consstent global checkpont, by Theorem 1. 2) T 2,u + A. Snce T 2,u T 1,u, we must have T 1,u + A from Observaton 3. If ths s the case, when we consder the reverse edge T 1,u 1 T 2,u 1, the followng two sub-cases arse: 2.1) A + T 2,u 1. In ths case, a cycle T 1,u + A + T 2,u 1 + T 1,u from T 1,u to tself exsts. However, a cycle can not exst f the schedule of T T C CSR. 2.2) T 2,u 1 + A. Snce T 2,u 1 T 1,u 1, we must have T 1,u 1 + A from Observaton 3. If ths s the case, we need to consder the prevous reverse edge T 1,u 2 T 2,u 2 n the zgzag path and make a smlar argument wth that edge. Proceedng lke ths, we wll end up wth a path T 1,1 + A; snce A + T 1,1, we have A + A,.e., A s on a cycle whch s a contradcton to the assumpton that the schedule of T T C CSR s seralzable. So, our assumpton that x 1,1,, x 1,u are all equal to x s wrong and hence the proof of the clam. Usng arguments smlar to the one above, we can show that x 1,1,, x 1,u can not all be x. So far, we have proved that there must exst a data tem assocated wth a reverse edge that s dfferent from both x and x. Let us assume such a data tem s x 1,p wth assocated reverse edge as T 1,p T 2,p. Next we prove our clam that A and B can not belong to a transacton-consstent global checkpont. On data tem x 1,1 that both T 1,1 and T 2,1 have accessed, no checkpont taken after T 1,1, say D 1, can be combned

7 Proceedngs of the World Congress on Engneerng 2008 Vol I wth A to form a consstent global checkpont due to the path A + D 1 (from Theorem 1). Therefore, on x 1,1 we can only use some checkpont C 1 taken before T 2,1 accessed x 1,1 to construct a transacton-consstent global checkpont contanng A. Usng smlar argument, on x 1,2, whch both T 1,2 and T 2,2 have accessed, any checkpont taken after T 1,2 accessed, say D 2, can not be combned wth C 1 to form a consstent global checkpont due to the path from C 1 + D 2. So we have to use some checkpont C 2 on x 1,2, whch was taken before T 2,2 accessed x 1,2. Smlarly, on x 1,p, whch both T 1,p and T 2,p have accessed, we have to use some checkpont C p, whch was taken before T 2,p to construct a transactonconsstent global checkpont contanng A. On the other hand, on data tem x 1,u that both T 1,u and T 2,u have accessed, no checkpont taken before T 2,u, say C u, can be combned wth B to construct a transacton-consstent global checkpont due to the path path C u + B. Therefore, on x 1,u, we can only use some checkpont D u taken after T 1,u accessed x 1,u to construct a transacton-consstent consstent global checkpont contanng B. Smlarly, on x 1,u 1, whch both T 1,u 1 and T 2,u 1 have accessed, any checkpont taken before T 2,u 1, say C u 1 can not be combned wth D u to construct a transacton-consstent global checkpont contanng due to the path path C u 1 + D u. So we have to use some checkpont D u 1 on x 1,u 1 that was taken after T 1,u 1 accessed x 1,u 1. Proceedng lke ths, on x 1,p, whch both T 1,p and T 2,p have accessed, we have to use some checkpont D p, whch s taken after T 1,p accessed x 1,p, to construct a transacton-consstent global checkpont contanng B. Thus, for data tem x 1,p, we can only use a checkpont taken before T 1,p and T 2,p have accessed x 1,p to construct a transacton-consstent global checkpont contanng A; on the other hand, we can only use a checkpont taken after T 1,p and T 2,p have accessed to construct a transactonconsstent global checkpont contanng B. So, for data tem x 1,p, there n no checkpont that can be combned wth both A and B to construct a transacton-consstent global checkpont. Ths proves the Theorem n the base case w = 1. Next, assume that f there s a zgzag path from A to B whch contans consecutve reverse edges wth length at most k, then A and B together cannot belong to a transacton-consstent global checkpont. We prove that a zgzag path from A to B whch contans consecutve reverse edges of lengths at most k + 1 can not belong to a transacton-consstent global checkpont. Suppose the seres of consecutve reverse edges along the zgzag path from A to B are: T 1,1 T u1,1 (u 1 k + 1); T 1,2 T u2,2 (u 2 k + 1); T 1,v T uv,v (u v k + 1). Thus, on the zgzag path from A to B that we consder, we have consecutve reverse edges of lengths u 1,, u v, (u k + 1 ). Each of these reverse edges should come from the local seralzaton graph of a data tem. Suppose the reverse edges are edges of local seralzaton graphs correspondng to data tems x 1,1,, x u1 1,1,...,..., x 1,v,, x uv 1,v respectvely. Frst, we show that at least one of the data tems x 1,1,, x u1 1,1...,... x 1,v,, x uv 1,v s not equal to x (recall that A s the checkpont of data tem x ). Suppose x 1,1,, x u1 1,1...,..., x 1,v,, x uv 1,v are all same as x. Then A, T 1,1,, T u1,1...,... T 1,v,, T uv,v are transactons accessng x. Based on Observaton 1, two cases arse: 1) A + T uv,v. If ths s the case, a path A B va T uv,v exsts, and hence A and B together cannot be part of a transacton-consstent global checkpont by Theorem 1. 2) T uv,v + A. Because of the seres of reverse edges T 1,v T uv,v on x, from Observaton 3, we have T 1,v + A. Then, when we consder the seres of reverse edges T 1,v 1 T uv 1,v 1, the followng two sub-cases arse: 2.1) A + T uv 1,v 1. In ths case, a cycle exsts T 1,v + A + T uv 1,v 1 + T 1,v, whch s a contradcton to the fact that the schedule of T T C CSR. 2.2) T uv 1,v 1 + A. Because of the seres of reverse edges T 1,v 1 T uv 1,v 1 on x, based on Observaton 3, we have T 1,v 1 + A. In ths case, we need to consder the prevous sequence of reverse edges T 1,v 2 T uv 2,v 2 and repeat the analyss smlar to case 2.1) and 2.2). Contnung ths process, we wll end up wth a cycle n the seralzaton graph whch s a contradcton to the fact that T T C CSR. Ths means that our assumpton that x 1,1,, x u1 1,1...,... x 1,v,, x uv 1,v are all x s wrong. Usng smlar arguments, we can show that not all the data tems x 1,1,, x u1 1,1...,... x 1,v,, x uv 1,v can be equal to x. Suppose x 1,1,, x u1 1,1...,... x 1,v,, x uv 1,v are all x. So far we have proved that there must exst a data tem assocated wth at least one reverse edge n the zgzag path from A to B that s dfferent from both x and x. suppose such a data tem s x g,p assocated wth the reverse edge T g,p T g+1,p whch s one of the reverse edges n the sequence of reverse edges T 1,p T up,p. Next, we prove that A and B can not be part of a transacton-consstent global checkpont.

8 Proceedngs of the World Congress on Engneerng 2008 Vol I On data tem x 1,1 that both T 1,1 and T 2,1 have accessed, no checkpont D 1, taken after T 1,1 has accessed x 1,1, can be combned wth A to construct a consstent global checkpont because there s a path from A to D 1. Therefore we can only use some checkpont C 1, taken before T 2,1 on x 1,1 to construct a consstent global checkpont contanng A. On x 1,2, whch both T 1,2 and T 2,2 have accessed, no checkpont taken after T 12, say D 2, can be combned wth C 1 to form a consstent global checkpont because there s a zgzag path from C 1 to D 2 wth consecutve reverse edges of length at most k. So we have to use some checkpont C 2 on x 1,2, whch was taken before T 2,2 accessed x 1,2. Proceedng lke ths, on data tem x g,p, whch was accessed by the transactons T g,p and T g+1,p, no checkpont D p taken after both T g,p and T g+1,p, have accessed can be combned wth C p 1, to construct a consstent global checkpont due to the exstence of the zgzag path of consecutve reverse edges of length at most k. So we have to use some checkpont C p whch was taken before T g,p and T g+1,p have accessed x g,p. On the other hand, on x 1,v, whch both T 1,v and T 2,v have accessed, no checkpont C v that was taken before T 2,v accessed x 1,v combned wth B to construct a trconsstent global checkpont because C v has a zgzag path to B wth consecutve reverse edges of length at most k. Therefore, on x 1,v, we have to use some checkpont D v, that was taken after T 1,v accessed x 1,v. On x 1,v 1, whch both T 1,v 1 and T 2,v 1 have accessed, we can not use any checkpont C v 1 that was taken before T 2,v 1 to construct a consstent global checkpont contanng D v due to the exstence of a zgzag path wth consecutve reverse edges of length at most k So, we have to use some checkpont D v 1 on x 1,v 1 that was taken after T 1,v 1 accessed. Proceedng lke ths, on x g,p, we have to use some checkpont D p that was taken after T g,p has accessed to construct a tr-consstent global checkpont contanng D p 1. Thus, on the data tem x g,p, whch s dfferent from both x and x, no checkpont that was taken before T g,p accessed can be used to construct a tr-consstent global checkpont contanng A and no checkpont taken after T g,p accessed can be used to construct a tr-consstent global checkpont contanng B. Thus, x g,p does not have any checkpont that can be combned wth both A and B to construct a tr-consstent global checkpont. Therefore A and B can not belong to a consstent global checkpont. Ths proves the theorem. Proof: Follows from the Theorem by takng S as the set contanng the checkpont. 5 Concluson Checkpontng has been tradtonally used for handlng falures n dstrbuted database systems. If each data tem s ndependently checkponted, the checkponts taken may not be useful for constructng a transactonconsstent global checkpont of the entre database. In ths paper, We have presented the necessary and suffcent condton for a set of checkponts of a set of data tems n the database to be part of a transacton-consstent global checkpont of the dstrbuted database. References [1] S. Plarsk and T. Kameda, Checkpontng for dstrbuted databases: startng from the bascs, IEEE Transactons on Parallel and Dstrbuted Systems, vol. 3, pp , [2] R. H. B. Netzer and J. Xu, Necessary and suffcent condton for consstent global snapshots, IEEE Trans. Soft. Eng., vol. 6, pp , [3] S. H. Son, An algorthm for non-nterferng checkponts and ts practcalty n dstrbuted database systems, Informaton Systems, vol. 14, no. 5, pp , [4] A. Slberschatz, H. F. Korth, and S. Sudarshan, Database system concepts, [5] M. Snghal and N. G. Shvaratr, Advanced concepts n operatng systems, [6] S. H. Son and A. K. Agrawala, Dstrbuted checkpontng for globally consstent states of databases, IEEE Transactons on Software Engneerng, vol. 15, no. 10, pp , October [7] C. Pu, On-the-fly, ncremental, consstent readng of entre databases, n Proceedngs of the 11th Conference on Very Large Database. Morgan Kaufman Pubs. (Los Altos, CA), Stockholm, 1985, pp [8] S. Plarsk and T. Kameda, A novel checkpontng scheme for dstrbuted database systems, n Proc. ACM SIGACT-SIGMOD-SIGART Symp. on Prncples of Database Sys., Nashvlle, TN, [9] R. Baldon, F. Quagla, and M. Raynal, Consstent checkpontng for transacton systems, The Computer Journal, vol. 44, no. 2, Corollary 1 A checkpont of a data tem n a dstrbuted database can be part of a tr-consstent global checkpont of the database ff t does not le on a zgzag cycle.

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