Availability of Coding Based Replication Schemes. Gagan Agrawal. University of Maryland. College Park, MD 20742

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1 Availability of Coding Based Replication Schemes Gagan Agrawal Department of Computer Science University of Maryland College Park, MD Abstract Data is often replicated in distributed systems to improve availability and performance. This replication is expensive in terms of disk storage requirements since the existing schemes require full les to be stored at each site. The coding scheme suggested by Rabin can be used to reduce the storage requirements in the replication based systems. In this paper, we evaluate the availability of a coding based replication scheme where simple voting is used to maintain correctness of replicated data. We show that the storage requirement to maintain the data with a given availability is reduced signicantly. We further present how some of the extensions of the voting scheme can be modied to manage this coding based replication. Again, we evaluate the availability of these and study the reduction in the storage space requirements achieved. 1 Introduction In a distributed system, data can be replicated to provide fault-tolerance against site failures and network partitions and to improve performance. This data replication requires a replica control protocol to maintain consistency of the data. One such method for replica control is the weighted voting scheme suggested by Giord [6]. In this algorithm, each node is assigned a number of votes. If N is the total number of votes assigned to all the nodes, then a quorum of r votes is required to do a read operation, and a quorum of w votes is required to perform a write operation. These quorum values are such that, r + w > N and 2w > N. This ensures write-write mutual exclusion and read-write mutual exclusion. A particular case of weighted voting is simple voting in which each node is assigned exactly one vote. With weighted voting, a node which wants to perform an operation on data, rst sends a request to all the nodes in the system. These nodes reply with their version numbers (denoting the number of successful updates made). When the requester node gets r(w) votes, it can perform the read (write) operation. For read, it reads the data from a node with the highest of the version numbers given by the nodes that have replied. For a write, the requester node determines the highest version number, and writes on all the nodes that constitute the write quorum. All of these nodes then have the version number as one more than the highest version number. The quorum conditions ensure that the node with the highest of the version numbers in a read/write quorum has the latest copy of the data. Two major extensions to the voting are dynamic voting [7] and voting with witnesses [10]. In dynamic voting, the quorum (for both read and write operations) needs to include only the majority of the sites that participated in the last update rather than the majority of all the sites. In [7], it has been shown that this approach improves upon the availability oered by the weighted voting scheme. In voting with witnesses, some of the sites may keep witnesses instead of regular copy of the le. A witness stores the version number but no data. It therefore, takes very little storage space. In [10], it has been shown that replacing some of the regular copies with witnesses may reduce the storage requirement with only a minor reduction in the availability oered. However, the reduction in storage space requirements is not signicant since if the number of sites with witnesses increases, the availability starts getting aected substantially. A major drawback of replication based schemes is the high degree of disk storage requirement. Typically in a distributed system, given the values of the reliability of nodes and the desired availability, the le may have to be replicated at a certain number of nodes to achieve that availability. If the le is replicated at N nodes, the disk storage requirement increases N fold (using witnesses may reduce it to some extent). The coding scheme suggested by Rabin [11] can be used to reduce the storage requirements of replicated systems [1, 8]. This coding has also been used for reliable storage of les in single node systems [3]. In this coding a le F is encoded and broken into n parts,

2 each of size jf j=m (n > m). Only one such part of the le, called Coded Partial File (CPF), is stored at each node. The le can be reconstructed by any m of such n CPFs. With the data being stored in the coded form, the read and write algorithms need to redesigned. In [8], we have shown how the simple voting algorithm can be modied to manage such coding based replication. In this paper, we evaluate the availability of a coding based replication scheme in which simple voting is used to maintain the correctness of replicated data. We show that the storage space required to maintain data with certain availability can be reduced signicantly with this coding. We further present how the extensions to the voting scheme (dynamic voting and voting with witnesses) can be modied to maintain correctness of data in coding based replication. Again, we evaluate the availability of these schemes and study how the storage space requirements are reduced. In general, we establish how Rabin's Coding can be used to signicantly reduce the storage requirements in replicated systems, irrespective of the replica control scheme being used. The rest of the paper is organized as follows. In section 2, we see how the coding scheme suggested by Rabin can be used to store replicated data and how the simple voting protocol changes to manage this replication. In section 3, we analyze the availability of this simple voting with coding. In section 4, we present how dynamic voting can be modied to maintain correctness when coding is used to store replicated data and study the reductions in the storage space achieved. In section 5, we do the same exercise for voting with witnesses. 2 Using Coding for Replicated Files In the coding scheme suggested by Rabin [11], a le F of size jf j is broken into n parts of size jf j=m such that any m of these n parts are sucient to reconstruct the le, where n m. Splitting and recombining of les is done by using n vectors in m dimensional vector space, such that any m of these n are linearly independent. Details of how these les can be split and recombined can be seen from [11, 8]. We will refer to such a system of coding as an (m; n) system. In [8], we have shown how this coding can be used to store replicated data in distributed systems. If the data is to be replicated at N nodes, then instead of storing full le at each node (and increasing the disk storage N fold), we can store the replicated data using this coding. With coding, we may need to store only one CPF at each node in which case the disk storage is reduced by a factor of m. We will refer to a system with (m; n) coding and N sites participating in the replication as an (m; n; N) system. When coding is used to store replicated data, proper read and write mechanisms are required such that data consistency is maintained and high availability is supported. Here, we briey describe the modi- cations required to voting scheme [6] to manage this replication. In our (m; n; N) system, we assume that n N. Exactly one CPF is stored at each node. We consider only the simple voting, in which each node is assigned one vote each. To study the quorum requirements, we shall use the following denitions. Denition 1: Minimum Sucient Quorum (MSQ) for read operations is dened as the smallest number of votes which may allow a read operation to complete successfully. We denote this by r 0. Denition 2: Maximum Necessary Quorum (MNQ) for read operation is dened as the largest number of votes which may be required to successfully complete a read operation. We denote this by r 00. The write quorum is denoted by w 00. The quorum requirements for (m,n,n) system with simple voting are then as follows: w 00 + r 00 N + m w 00 r 00 w 00 + r 0 > N r 0 m n w 00 The rst restriction ensures that the write quorum and MNQ for read operations intersect at at least m CPFs in the system. The MSQ for read operation should intersect with the write quorum at least one node. Also, r' needs to be at least m since the use of coding scheme requires m CPFs to reconstruct the original le. The write operation needs to update at least w" distinct CPFs in the system. If in a write operation involving k sites (k n 0 ) only l(l < n 0 ) distinct CPFs are present, then the write operation completely rewrites some duplicate CPFs in the participating set such that, at the end of the write operation, the participating set consists of at least n 0 distinct CPFs. This method of updating CPFs is called dynamic redistribution of CPFs. Note that the purpose of dynamic redistribution is to give high resiliency at lower values of n [8]. The read operation proceeds as follows. The site initiating the operation (initiator) rst collects r 0

3 votes. It then checks if there are at least m distinct CPFs with the latest update. If so, it can read from any m distinct CPFs with latest version number. 3 Availability The most important performance metric of any replication based scheme is the availability that it offers. The measure of read (write) availability that we use is the probability (in steady state) of read (write) quorum being available in the system. In the previous section, we presented how Rabin's coding scheme can be used to store replicated data. If only one CPF is stored at each node, instead of a full le, then clearly, the disk storage requirement have been reduced. But, the quorum requirements have also changed and hence the availability oered may also be less. However, what we may be interested in is the disk storage space required to get the same amount of availability. The storage space factor (denoted by l) of a (m; n; N) system is given by N=m. A storage factor of l means that the overall storage required in the system for a particular le is l times the size of the original le. To be able to compute the reductions in the storage space requirements, we need to compute the availability oered by a given (m,n,n) system. 3.1 System Model To be able to compute the availability oered, we make the following assumptions about our system. We assume that only the nodes in the system fail and recover and the communication links never fail. To make our analysis feasible through stochastic models, we assume that failure (recovery) at each up (down) site arrives as an independent exponential process with rate (). We also assume that any update operation updates exactly w" distinct CPFs in the system. Now, consider an (m,n,n) system with quorum requirements selected as described in section 3. For computing the read availability, the main concern is to determine the probability of nding m distinct CPFs with the latest update when k, r 0 k < r 00 sites are available. In general, this will depend upon the failures and recoveries that have taken place since the last update in the system. To keep our analysis simple, we will like to avoid this extra parameter of the relative frequency of updates and failures (and recoveries). We, therefore, do our analysis separately under two extreme assumptions. In rst we say that failures and recoveries in the system are much frequent then the updates. We call this infrequent update assumption. We expect this to give a lower bound on the availability oered by the our system. The other extreme assumption that we use for our analysis is that updates are much frequent than failures and recoveries in the system. We refer to this assumption as the frequent update assumption. This assumption is actually true for many applications, otherwise, frequent polling can be used to make this true [7]. However, we do admit that, for certain applications, this assumption may give higher estimate of the availability oered by our system. In the next two subsections, we analyze the availability of our system under these two extreme assumptions. 3.2 Infrequent Update Assumption In the infrequent update assumption, the failures and recovery in the system are much frequent then the updates. Hence, when a read request arrives, the probability of any node in the system being up is the same irrespective of whether it was updated or not during the last update operation. Consider any node of the system in the steady state. If the rates of arrival of failures and recoveries in the system are and respectively, then the probability of a given node being up at certain time is given by. We denote this probability by p. The probability + of k out of N nodes being up at any time is given N as k p k (1 p) Nk. If less than r' nodes are up at any time, the read operation shall not be possible. Alternatively, if r" or more nodes are up, then the read operation will denitely be possible. If k(r 0 k < r 00 ) sites are up, then the probability of having at least m sites with the latest updates (denoted by P (N; w 00 ; k)) is given as P k P (N; w 00 ; k) = j=m w 00 Nw 00 j kj = P k j=kr +1 w 00 Nw 00 0 j kj Note that in the system, w" nodes have the latest update in the system and N - w" nodes do not have the latest update. The number of ways of choosing k nodes out of N such that exactly j of these were updates during the last update and the other k-j were not is w 00 Nw 00 j kj. The read operation will be possible if m or more of the w" sites with the latest update are present. Also, if k sites are up and if w 00 + r 0 = N + 1, then at least k r sites with the latest update are present. (Hence the summation in the denominator is from k r 0 + 1). The availability of read operations can then be stated as A r = P N N j=r 00 j P r 00 1 N j=r 0 j p j (1 p) Nj + p j (1 p) Nj P (N; w 00 ; j)

4 From the above we note that the availability is independent of n, as long as n is at least w Frequent Update Assumption In this subsection, we analyze the availability offered by our system under the assumption that the updates are much frequent than the failures and recoveries. Hence, as soon as any failure or recovery takes place in the system, an update takes place with that topology. We perform the analysis under this assumption using a stochastic model. The exact model that we use is Generalized Stochastic Petri Nets (GSPN) [9]. We use the Stochastic Petri Net Package (SPNP) [4] to solve the GSPN we use for our analysis. can be drawn from them are from plotting graphs at the various A marking is a set of two-tuples describing the number of tokens at each place in the net. The number of tokens in a place \P" in a marking is given by mark(\p"). The events in the system are called transitions, ring of transitions changes the marking. There are arcs drawn from places to transitions, called inarcs and from transitions to places, called outarcs. An inarc from place P to transition T is denoted by (P { T) and similarly an outarc from transition T to place P is denoted by (T { P). Each arc a has certain multiplicity, (denoted by mul(a)), with can be dened in terms of the marking existing before the transition res (the default is 1). When a transition res, the number of tokens at each of the places having an inarc to that transition decreases by the multiplicity of that arc. Similarly, number of tokens at each place having an outarc from that transition increases by the multiplicity of that arc. For a transition to re, each of the places having an inarc to that transition must have the number of tokens at least equal to the multiplicity of the corresponding inarc. The generalized Petri nets allow the ring of the transitions to be either immediate or with an exponentially distributed time. The parameter of this exponential process for a transition T can be dened in terms of the marking and is denoted by rt(t). Also, enabling conditions of transitions (en(t)) can be dened in terms of the marking. The default is true. Transitions can be given priorities (denoted by pr(t)). If many transitions are enabled in any marking, then no transition with a priority less than the highest priority among these transitions can re. Default priority is zero. Our system, with the frequent update assumption, can easily be modeled with this GSPN. Each node in the system can in one of three states. It may be operational and having the latest update (\UP"), it may be \UP" T1 T2 6 6 pr(t6) = 1 T3 T4 \DOWN" T6 mul(\old" { T6) = mark(\old") mul(\t6 { \UP") = mark(\old") rt(t1) = mark(\up") rt(t2) = mark(\old") rt(t3) = mark(\down") rt(t4) = (w" - mark(\up")) rt(t5) = (N - w" - mark(\old")) - T5 - en(t3) = mark(\up") + mark(\old") w" - 1 en(t4) = mark(\up") + mark(\old") < w" - 1 en(t5) = en(t4) en(t6) = mark(\up") + mark(\old") w" Figure 1: GSPN for simple voting with Coding \OLD" up and may not have the latest update (i.e. has an old copy) (\OLD") or it may be down (\DOWN"). The GSPN therefore has three places, each having a number of tokens equal to the number of nodes in that state. The transitions used are as follows. Any up node may fail, i.e. will become down (T1). Old nodes may also fail (T2). Because of the frequent update assumption, if the total number of nodes up (i.e. \UP" or \OLD") is at least w", then no node can be in the \OLD" state. Hence, there is an immediate transition from \OLD" to \UP" (T6), with the enabling condition that mark(\up") + mark(\old") should be at least w". The multiplicity of both the inarc to and outarc from this transition is equal to mark(\old"). Now, consider the transitions from \DOWN". When a node recovers, then if after this recovery if the total number of operational nodes will be at least w", then the recovered node will directly go to \UP" (T3) (because of the frequent update assumption). However, if this is not the case, then this down node may go to \UP" if it has the last update (T4) otherwise it will go to \DOWN" (T5). Since the number of sites in last update were w", transition T4 takes place with the rate (w" - mark(\up")). Similarly, transition T5 takes place with the rate (N - w" - mark(\down"). 6

5 The system will be available for read operations in a marking if mark(\up") + mark(\old") is at least r' and mark(\up") is at least m. 3.4 Results In this subsection we discuss the results obtained from the analysis with frequent and infrequent update assumptions. The vote assignment used for analysis is, r 00 = N + m w 00 and r 0 = max(r 00 w 00 = N+4 2 m + 1; m). The purpose of above vote assignments is to keep the value of w" same for any m ( m = 1..4) for a given N. Note that we use read availability as the metric for evaluating dierent systems. Our analysis shows that the dierence in the availability calculated under frequent update assumption and infrequent update assumption is insignicant, especially when the number of nodes participating in the replication is quite high. To study the reductions in the storage space, we consider the storage space requirements (l) for achieving certain availability in a given system. Note that m = 1 implies simple voting without coding (i.e. with full le replication). In gure 2, we consider the storage space requirements for dierent values of availability, when p is xed at These values have been taken from the analysis with infrequent update assumption only, we had noted earlier that there isn't much dierence between the results obtained with infrequent and frequent update assumptions. The storage space required to achieve certain availability has been reduced signicantly, except when the availability desired is quite low. In gure 3, we study the l required at different values of p, when availability required is 0:999. When p is high, the l required decreases with increase in m. However, exceptions can be noted at relatively lower values of p. In general, we note that use of coding can reduce the storage requirements signicantly. Moreover, the reduction in storage space achieved increases with increasing m. Some exceptions do happen when either p is small or N is small. These can be explained as follows. When N is close to m, then the probability of nding m distinct CPFs with dierent updates becomes low, hence the availability oered is reduced. Similarly, when p is low, the probability of system being in a state with less than r 00 operational nodes is quite high, hence increasing m reduces availability signicantly. 4 Dynamic Voting In this section we present how the dynamic voting scheme can be modied to manage this coding based replication. We then evaluate the availability oered l required m = 1 m = 2 m = 3 m = 4 -log10( 1 - Availability) Figure 2: l required vs. Availability for Simple Voting (p =.80) l required m = 1 m = 2 m = 3 m = 4 Availability of Nodes x Figure 3: l required vs. p for Simple Voting (Availability =.999)

6 by dynamic voting with coding and study the reductions in the storage space achieved. 4.1 Dynamic voting protocols In the dynamic voting scheme suggested by Jajodia and Mutchler [7], a majority of sites participating in the last update are required for performing any read or write operation. All sites participating in an update operation record the total number of sites participating in that update in a variable called Update Site Cardinality (SC). For the next read or write operation, at least a majority of SC ( SC+1 2 ) sites with the latest version number are required. Modifying this scheme to work when coding is used to store replicated data is quite simple. Again, we assume that each site stores exactly one CPF. We assume that n = N. With coding, only modication required will be that for any operation, a minimum of m (where m is the parameter of coding) distinct CPFs with the highest version number are required. 4.2 Availability Analysis We analyze the performance of the dynamic voting with coding under the frequent update assumption only. We have seen earlier that there wasn't any signicant dierence in the availability of simple voting with coding under the frequent and infrequent update assumptions. Other assumptions used for analysis are the same as described in section 3.1. Again we model the system using a GSPN and solve it using the stochastic Petri net package. Initially, consider dynamic voting with full le replication. Under the frequent update assumption, as long as SC is more than 2, the system remains operation and new updates can be performed. When a new failure arrives, the number of sites with the latest update is still SC 1. When SC > 2, this is a majority of SC. However, when SC is 2 and there is a failure, then no update can take place as long as both the sites with this latest update are again up. Now, we consider dynamic voting with coding. The system remains operation as long as SC is more than m (m 2). However, when SC is m and a new failure takes place, the system will not be available until all these m sites with the latest update are up. So, the operation of dynamic voting with full le replication (in terms of quorum availability) is a special case of dynamic voting with coding with 2 replacing the parameter m. These can now be modeled as follows. Each node can be in either of three states : \UP", \OLD", or \DOWN". If m (2 for full le replication) or more tokens are in place \UP", then no site can be \DOWN". \UP" T1 T2 6 6 pr(t6) = 1 T3 T4 \DOWN" T6 mul(\old" { T6) = mark(\old") mul(\t6 { \UP") = mark(\old") rt(t1) = mark(\up") rt(t2) = mark(\old") rt(t3) = mark(\down") rt(t4) = (m - mark(\up")) rt(t5) = (N - m - mark(\old")) en(t3) = mark(\up") m en(t4) = mark(\up") < m en(t5) = en(t4) en(t6) = mark(\up") m - T5 - \OLD" Figure 4: GSPN for Dynamic Voting with Coding Similarly, when a down node is recovering and if the number of tokens (nodes) in \UP" is m or more, then it will denitely go to \UP" place. Otherwise, it may go to \UP" or \DOWN" depending upon whether it has the latest update. The system will be available for read and write operations if mark(\up") is at least m. This has been shown in gure Results An interesting aspect of our modeling is that with the frequent update assumption, the operation of dynamic voting with m = 2 is the same as that of dynamic voting with full le replication. A reduction in storage space by a factor of two is therefore achieved straight away. In gure 5, we compare the availability oered by dynamic voting with coding, at di. values of m, when p is xed at.80. The comparison is done for m = The availability for dynamic voting with full le replication is the same as that of dynamic voting with coding with m = 2. From this graph, we see that with dynamic voting (with or without coding), increasing the number of sites does not increase the availability oered indenitely (unlike simple voting). This may be explained as follows. While increasing the number of nodes decreases the probability of the system entering the state 6

7 -log10( 1 - Availability) m=2 m=3 m=4 m=5 Number Of Nodes Figure 5: Availability vs. number of Nodes for Dynamic Voting with Coding (p = 0.800) l required m=1 m=2 m=3 m= Availability Of Nodes x Figure 6: l req. vs. p for Dynamic Voting with Coding (Availability = 0.999) where SC is m and not all the m nodes with the latest version number are up, the expected time that the system will spend in such a state also increases with increase in N. Hence the availability oered cannot increase indenitely with increase in N. Another point to note is that the maximum availability that can be oered decreases with increase in m. This is because the probability of the system entering a state where SC is m and not all the sites with the latest update are up increases with increase in m. However, for realistic values of availability, the storage requirements for maintaining a system with certain availability actually decreases with increase in m. With m = 2, the storage requirements decrease to half as compared with full le replication. In gure 6, we compare the storage space requirements for getting an availability of.999, for m = Clearly, the storage space required decreases signicantly with increase in m. 5 Voting With Witnesses In this section, we show how voting with witnesses can be used when coding is used to store replicated data. In voting with witnesses [10], some of the sites store witnesses rather than storing the le being replicated. Witnesses store the version number and participate in operations by sending their version numbers, but they do not contain any data. The quorum requirements are as follows. The read and write operations need r and w votes (as in weighted voting), further, each quorum needs to include at least one regular copy with the highest version number. Voting with witnesses can also be modied Modeling of voting with witnesses is more involved than simple voting. We need to know the number of witnesses which are up and down besides knowing the number of up, down and old regular copies. Moreover, we need to keep track of the number of regular copies with the latest update which are operational. Hence, the number of places in the Petri net is 6. Details of the model are omitted. The results obtained from the analysis of voting with witnesses are very similar to the results obtained from simple voting with coding. Again, we omit the details. In general, we can conclude that the use of coding in supporting replicated data can reduce the storage space requirements, irrespective of the replica control scheme being used. 6 Conclusions Data replication is often used to enhance the availability and performance in Distributed Systems. This

8 replication of data incurs a high storage overhead. The coding scheme suggested by Rabin can be used to reduce the storage requirements of the replication based systems. In this paper we initially considered a coding based replication scheme in which simple voting is used to maintain correctness. We evaluated the availability oered by this scheme. To avoid an extra parameter of the relative frequency of updates and failures, we performed our analysis under two extreme assumptions, the frequent update assumption and the infrequent update assumption. The analysis with infrequent update assumption was done using a simple static model. For analysis with frequent update assumption, we used the generalized stochastic Petri nets (GSPN). Our analysis showed that the storage space required to maintain data with certain availability is reduced signicantly with the use of coding. Only exceptions were noted when either the availability of each node is quite low or when the number of nodes was close to the parameter m of the coding scheme. We further extended our work to consider extensions to the voting scheme. We presented how dynamic voting and voting with witnesses can be modied to maintain correctness when coding is used to support replicated data. Again, we evaluated the availability oered by these schemes. In both the cases, we performed the analysis under the frequent update assumption only, modeling our systems as GSPNs. Evaluation of dynamic voting with coding showed that the availability oered does not increase indenitely by increasing the size of the system. Moreover, the maximum availability achieved decreases with increasing m. However, for the realistic values of availability, using coding does decrease the storage space required to achieve certain availability. Results obtained from the analysis of voting with witnesses were very similar to the results obtained from simple voting. In general, we establish that Rabin's coding scheme can be used to reduce the storage requirements for replication based systems, irrespective of the replica control scheme being used to maintain correctness. Acknowledgements Much of the analysis done in this paper has been done with the Stochastic Petri Net Package (SPNP) developed at the Duke University. The author is grateful to Deron Liang for introducing him to this package. This work is a continuation of the previous work which was done jointly with Pankaj Jalote. References [1] D. Agrawal and A. El Abbadi, \Integrating security with fault tolerance in distributed databases", in Computer Journal, vol. 33, no. 2, pp , Feb [2] G.Agrawal and P. Jalote, \An ecient protocol for voting in dist. systems", in proc. of 12 th Int. Conf. on Dist. Computing Sys., June [3] W. A. Burkhard and P. D. Stojadivonic, \Storage ecient reliable les", in proc. of Winter Usenix Conf., [4] G. Ciardo and J.K. Muppala, \Manual for the SPNP package version 3.0", Duke University, [5] J.B.Dugan, A. Bobbio, G. Ciardo, and K.S. Trivedi, \The design of a unied package for the solution of Stochastic Petri Net models", in Proc. of Int. Workshop on Timed Petri Nets, July [6] D.K.Giord, \Weighted voting", in proc. of 7th ACM sym. on OS principles, pp , [7] S. Jajodia and D. Mutchler, \Dynamic voting algorithms for maintaining consistency in replicated databases", in ACM trans. on Database Systems, vol. 15, no. 2, pp , June [8] P. Jalote and G. Agrawal, \Using Coding To Support Data Resiliency in Distributed Systems", in proc. of 12th Int. Conf. on Data Engg., Feb [9] M. Ajmone Marsan, G. Balbo and G. Conte, \A class of generalized stochastic petri nets for performance evaluation of multiprocessor systems", ACM trans. on Computer Systems, pp , May [10] J.F.Paris, \Voting with witnesses: a consistency scheme for replicated les", in proc. of sixth IEEE Int. Conf. on Dist. Computing Sys., pp , [11] M.O.Rabin, \Ecient dispersal of information for security, load balancing and fault-tolerance", Journal of the ACM, vol. 36, no. 2,pp , 1989.

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