ࡱ>  ,.%&'()*+!` [bjbj\\ >>fB###4F#222h| F#. ( g;lssssNLtLL$hgp#e^p{6&&& >#s&s&&bM.#R" P\2 . 2O8pHjOP6XR,Rn#TS<&lUpppF#F#$j$NrD?F#F#j$rF#F#F#  Relaxed Currency Serializability for Middle-Tier Caching and Replication (extended version) Philip A. Bernstein, Microsoft Corporation Alan Fekete, University of Sydney Hongfei Guo, Microsoft Corporation Raghu Ramakrishnan, University of Wisconsin Pradeep Tamma, University of Wisconsin June 23, 2006 MSR-TR-2006-34 Microsoft Corporation One Microsoft Way Redmond, WA 98052 Relaxed Currency Serializability for Middle-Tier Caching and Replication (extended version) Philip A. BernsteinAlan FeketeHongfei GuoRaghu RamakrishnanPradeep TammaMicrosoft Corporation philbe@microsoft.comUniversity of Sydney fekete@it.usyd.edu.auMicrosoft Corporation hongfeig@microsoft.comUniversity of Wisconsin-Madison {raghu,pradeep}@cs.wisc.edu ABSTRACT Many applications, such as e-commerce, routinely use copies of data that are not in sync with the database due to heuristic caching strategies used to enhance performance. We study concurrency control for a transactional model that allows update transactions to read out-of-date copies. Each read operation carries a freshness constraint that specifies how fresh a copy must be in order to be read. We offer a definition of correctness for this model and present algorithms to ensure several of the most interesting freshness constraints. We outline a serializability-theoretic correctness proof and present the results of a detailed performance study. This is an extended version of a paper with the same title that appeared in SIGMOD 2006. INTRODUCTION The use of copies is widespread in ecommerce sites. For example, consider an auction site; when browsing auctions in a category, it is apparent that the data (e.g., item prices, number of bids) is a little out of date. However, most users understand and accept this, as long as they see completely current data when they click on an individual item. As another example, a purchase transaction may calculate and report to the user an estimated arrival date for the goods, based on reading a log of shipment timings at the cache. Reading slightly stale information might be acceptable given the business requirements, but it is essential to record this promised arrival date at the centralized backend server. As a concrete example, the following query returns a summary of books that have the specified title:  SHAPE \* MERGEFORMAT  SHAPE \* MERGEFORMAT  Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGMOD 2006, June 27-29, 2006, Chicago, Illinois, USA. Copyright 2006 ACM 1-59593-256-9/06/0006$5.00. Different users might have different freshness requirements for this query. For example, user U1 is about to purchase all of these books and calls transaction T1 to get an up-to-date query result. User U2 is browsing and runs transaction T2, which offers quick response time, even at the cost of allowing the I.INSTOCK column to be out-of-sync. A third user, U3, is studying the relationship between the cost of a book and the number of copies in stock by periodically running the query T3 and recording the results. In this case, it is acceptable for the result to be stale as long as it reflects a consistent database snapshot, i.e., a database state that is consistent at a certain point of time. In fact, a weaker version of this guarantee might suffice, requiring only that all rows retrieved for a given item reflect the same snapshot, with different items possibly coming from different snapshots. This example illustrates the main usage scenarios that we want to tackle. They typically arise because of application or middle-tier caching layered on a database system. However, it is problematic to allow update transactions to read the out-of-date values. The standard correctness criterion for transactions is one-copy serializability, which says that an execution should be equivalent to a serial execution on a one-copy database. The transactions for users U2 and U3 in the above example violate this criterion. For example, suppose that a transaction, T4, places an order for an item (thus changing I.INSTOCK for that item), and another transaction, T5, then updates the cost of that item. If transaction T2 issued by user U2 now reads an older, cached value of INSTOCK, along with the current value of COST, it sees a database state that can never arise in a serial execution on a one-copy database. We investigate concurrency control for a transactional model that allows an update transaction to read out-of-date copies. Each read operation carries a freshness constraint  REF _Ref86491849 \r \h  \* MERGEFORMAT [12] that bounds how up-to-date a copy must be in order to be read. Up-to-date-ness may be relative to read-time, readers commit-time, or freshness of copies of other items that are read. Our main contributions are: An extended serializability model called Relaxed Currency (or RC-) Serializability that accounts for out-of-date reads that are justified by user-specified freshness constraints. Concurrency control protocols that allow update transactions to read out-of-date data safely, that is, read data that satisfies the associated freshness constraints. Our protocols accumulate constraints on the updaters time, which can be checked at read-time or as part of commit processing. A proof sketch that the protocols produce RC-serializable executions. A detailed simulation study of the protocols. It shows that performance is similar to the weakest protocol used in practice, read-committed, which offers no freshness guarantee. The paper is organized as follows. In Section  REF _Ref86818686 \r \h 2 we introduce our model of correctness for interleaved executions of transactions that read cached and possibly out-of-date copies. We define our models of a replicated database and transactions that can read out-of-date copies in Section  REF _Ref86639664 \r \h 3, and several classes of freshness constraints in Section  REF _Ref86490431 \r \h 4. We present several concurrency control algorithms to ensure RC-serializability in Section  REF _Ref63737442 \r \h 5 and prove their correctness in Section  REF _Ref119920231 \r \h 6. We present a performance study in Section  REF _Ref120459161 \r \h 7 and discuss implementation extensions in Section  REF _Ref119920316 \r \h 8. Section  REF _Ref86639725 \r \h 9 covers related work. Section  REF _Ref86837249 \r \h 10 is the conclusion. RC SERIALIZABILITY We define a new user model of computation where transactions may update data yet read cached copies, subject to various freshness constraints. We also define implementations that realize it. To relate the user model and implementation, we use serializability theory  REF _Ref86491820 \r \h [6]. However, all forms of correctness stated in that theory require each transaction to read values as in a serial one-copy execution, which does not hold if an update transaction reads slightly stale replicas. We need a weaker correctness property that captures our intuition of what a transaction does when it reads stale values. As in traditional serializability theory, we define the correctness of an implemented physical system by saying that its behavior must be indistinguishable from that of a simpler, ideal system (i.e., the users model) where transactions execute serially. In the ideal system, we model a database as a collection of data items. Since the freshness constraints are part of the users model, the user is aware that transactions read out-of-date values of data items. Such out-of-date values are called versions of a data item. Thus, in the ideal system a data item is a sequence of versions and a computation is a multiversion execution. That is, the user must understand that (i) each time a transaction updates a data item a new version of that data item is (conceptually) created, and (ii) when a transaction reads a data item, it may read an old version of that item, not the most recent one. Each read operations freshness constraint tells whether a version is satisfactory. We therefore define the ideal system to be a one-copy multiversion database. By one-copy, we mean there is a multiversioned master copy of each data item and no replicas (i.e., cached copies). We define a correct execution of the ideal system to be a serial multiversion execution in which each read operation satisfies its freshness constraint. A correct execution of a physical system is one that is equivalent to a correct execution of the ideal system. A correct execution of a physical system is called relaxed-currency (RC) serializable. To our knowledge, this is the first time a multiversion database has been taken to be the users model of computation. By contrast, the well-known correctness definition for multiversion and replicated databases is one-copy serializability, which uses a one-copy single-version database as the ideal system. THE PHYSICAL SYSTEM MODEL One of the main questions we consider in this paper is how to extend the theory of transactions and serializability to account for freshness requirements. To do this, we need to model the physical system consisting of transactions, data managers, and databases. Since cached copies are replicas of a master copy, we can model the system as an ordinary non-versioned replicated database. This section defines our replicated database model (see Figure 1). We assume that each data item has a unique identity. A transaction is modeled as a set of partially-ordered actions, which include reads and writes of data items, and, possibly, a final commit or abort. For simplicity, we do not allow inserts or deletes of items, since they would lead to many notational complexities. We consider them and the associated issue of phantoms as an extension in Section  REF _Ref119920316 \r \h 8. We denote the set of items read by transaction T as readset(T) and the set written by T as writeset(T). Every data item has a set of physical copies: one master and zero or more replicas. The collection consisting of masters of all of the data items is called the master database. A cache is a collection of replicas of one or more data items. There is only one version of each physical copy, which contains its latest value. We assume that the master database has a master manager that controls the execution of reads and writes to the master. Similarly, each cache has a cache manager that controls the execution of reads and writes to the cache. We distinguish between user-transactions (or simply transactions) that are issued by users, and copy-transactions (or simply copiers) that are issued by the master manager. Each transaction issues each read operation either to the master manager or to a particular cache manager. For now, we assume that it only modifies the master database, so it sends all of its write operations to the master manager, which is where they are processed.  SHAPE \* MERGEFORMAT  Figure  SEQ Figure \* ARABIC 1: Replicated Database Model After a transaction T commits, the master manager invokes a copier for each pair [x, P] where x is a data item updated by T and P is a cache that has a replica of x. A copier reads the value of the master xM and writes that value into the corresponding replica xP in cache P. Note that a copier runs outside of the transaction that caused the copier to be invoked. For now, we leave open whether the copiers from the master to a given cache are pipelined in commit order or grouped (e.g., one copier contains reads and corresponding writes for multiple [x, P] pairs). The techniques in this paper are applicable when the master database includes materialized views of base data and when replicas of those materialized views are stored in caches. In this case, we assume that each transaction maintains the consistency of materialized views with respect to base data. Whether this is done by user code or system code executing within the transaction is unimportant to our model. What is important is that we assume that the master and cache managers treat updates to materialized views the same as those on base data. Solutions that relax this assumption may be useful, but are outside the scope of this paper. FRESHNESS CONSTRAINTS To capture the concept of freshness introduced in Section 1, we require that each read request from a transaction includes a freshness constraint  REF _Ref86491849 \r \h  \* MERGEFORMAT [12]. The cache manager or master manager that services a read must return a value of the requested item that satisfies the reads freshness constraint. Freshness constraints are defined based on the time-drift or value-drift of an item from its correct value, or on the mutual consistency of item values. To define these precisely, we use the well-known concepts of timestamps and snapshots. We assume a timestamp domain, which is a subset of the natural numbers. The master manager assigns to every committed transaction T a unique transaction timestamp ts(T) greater than all timestamps assigned to previously committed transactions. We assume that the master manager and cache managers can calculate a user-comprehensible wall-clock time from each timestamp and vice versa. Thus, the system can translate a freshness constraint expressed using wall-clock time into one expressed using timestamps. In practice, real-time clocks are fine-grained and accurate enough to enable this. Since timestamps are generated only by the master manager, clock skew can be ignored. Hereafter we do not distinguish timestamps from wall-clock time. Every copy xA (i.e., a master or replica) has an associated last-modified timestamp, last-modified(xA), which is the timestamp of the last committed transaction that wrote xA. The value of a copy written by a committed transaction is called a committed version. Each committed version xi of a copy xA of data item x is said to be valid over the half-open interval [t, t() where t = last-modified(xi) and t( is either the next larger last-modified timestamp of another committed version of xA or " if no version of xA has a larger last-modified timestamp. We call [t, t() the valid interval of xi. A snapshot is a database state produced by a certain set of trans-actions(usually, the committed transactions in the prefix of an execution. A snapshot is associated with a timestamp t and maps each data item x to the version of x whose valid interval includes t. The ideal system, which is the users model, is just like the physical replicated database model except there is just one copy of each data item (and hence no copiers), and all of its committed versions (from all past writes) are visible to user-transactions. Thus, the semantics of freshness constraints can refer to versions of data items (since there is just one copy of each). We classify freshness constraints into three classes B, D, and M as follows. B. Time-Bound and Value-Bound Constraints Time-Bound Constraints. We can limit the currency bound, which is the difference between the time a version becomes invalid and the time it is read (the read-time). For example, we can specify that it is acceptable to read a version of x that is up to 10 seconds out of date: bound(x:10). If bound(x:0), then the read must be given a valid version of x. For simplicity, we omit the time unit and assume it is seconds. Value-Bound Constraints. We can specify that the value read for x is within a certain percent of the value of the valid version: bound(x:10%). D. Drift Constraints on multiple data items. When a transaction T reads more than one data item, we can specify constraints over a subset S of readset(T). Snapshot consistency. We can require that data items in S must be read from the same snapshot, denoted snapshot(S). Limited Time-Drift. The constraint drift(S, b) says that for every two items x and y in S, the versions of x and y that are read are within b seconds of each other. That is, if the transaction reads version xi of x and yj of y, then there are timestamps tx and ty such that xis valid interval includes tx, yjs valid interval includes ty, and |tx ty| ( b. Observe that snapshot consistency is the special case of b=0. Limited Aggregate Value Drift: We can require that an aggregate computed over (a subset of) the readset of T be within a certain percentage of a recent value (even though individual data items may be further out of sync). We use the notation drift(S, AGG, b%, w), where AGG is an aggregate operation, and w is a time window. It means that AGG(S) must be within b% of AGG(S(), where S( denotes the value(s) of committed versions of items in S at some instant less than w seconds prior to the current time. Constraints of type B and D can be combined. For example, a transaction can set bound(x:10) and snapshot({x, y})  REF _Ref86491849 \r \h  \* MERGEFORMAT [12]. We can further classify these constraints using two orthogonal criteria: Granularity: Constraints over individual data items, sets of items, aggregates over sets of items, subsets of snapshots, and complete snapshots. Unit of skew: Specify the skew using timestamps, number of subsequently committed transactions, or value. M. Multi-statement Constraints We can specify session level constraints that refer to points in time external to the execution of the current statement. For example, a transactions reads must see changes made by preceding statements in this transaction or in this session. When evaluating constraint types B1, D2, and M, there are two possible notions of read-time: the time when the read executes, or when its transaction (later) commits. Even in the users model of computation (a serial execution of the ideal system), these read-times are not the same unless the transaction has negligible execution time. For most of this paper, we use a transactions commit time to be the read-time of all of the transactions reads. We consider the other semantics in Section  REF _Ref119920316 \r \h 8. ENFORCING RC SERIALIZABILITY Enforcing RC-serializability requires ensuring that transactions are serializable and satisfy their freshness constraints. Enforcing RC-serializability is non-trivial because these requirements are not independent: freshness constraints are affected by factors related to transaction order, such as transaction timestamps. So the different transaction orders in two equivalent serial executions might affect whether certain freshness constraints are satisfied. The goal of RC-serializability is to ensure that each execution has at least one equivalent serial execution in which freshness constraints are satisfied. The problem of proving RC-serializability is also non-trivial. As in one-copy serializability, the notion of equivalence in the definition of RC-serializability requires mapping between two different kinds of executions multiversion executions of the ideal system and replicated data executions of the physical system and proving that they have the same behavior. We will do this in Section 6. But first we introduce some algorithms that produce RC-serializable results for various kinds of freshness constraints. We describe how to deal with transactions that include both reads and writes; read-only transactions (queries) can use the techniques of  REF _Ref86491849 \r \h  \* MERGEFORMAT [12]. Baseline Synchronization We first present a general concurrency control algorithm, called the baseline synchronization algorithm (or BSA). It provides a common framework for the algorithms we discuss later that enforce freshness constraints, but does not by itself enforce such constraints. In summary, user-transactions read replicas using read-duration locks and write masters using commit-duration locks. Thus, user-transactions run at the read committed isolation level. Commit and abort are processed at the master site, which generates transaction timestamps for committed transactions. For copiers, the master pipelines writes to caches in timestamp order. We describe BSA in terms of how it handles the following four actions: reads, writes and commits of user-transactions, and copiers. See the boxed algorithm descriptions below and their descriptions that immediately follow. BSA-Read(x) // issued by a user-transaction Choose a manager R from which to read x; // R can be the master for x or a cache read-lock x at R; read xR; release read-lock; BSA-Write(x) // issued by a user-transaction write-lock xM at the master for x; // lock held until commit write xM; BSA-Commit(T) // issued by a user-transaction // and processed at the master site ts(T) ( new transaction timestamp; // ts(T) > all timestamps previously issued for all items x in writeset(T) last-modified(xM) ( ts(T); release all locks; BSA-Copy(x, P) // a copier transaction read-lock xM at the master for x; read xM and last-modified(xM) ; release read-lock; write-lock xP at cache manager P; xP ( xM; last-modified(xP) ( last-modified(xM); // assumes writes are applied in last-modified order release write-lockWrites set exclusive locks on master copies. The locks are released only after the transaction commits and its updates are applied at the master. Thus, for each unlocked master item xM, last-modified(xM) is the timestamp of the last committed transaction that updated x. Reads set short-duration shared locks. This ensures that each read operation sees a committed value. It also ensures that the data and last-modified value are mutually consistent at the time the read occurs, since there may be a gap between a transaction updating a copys data value and updating its last-modified timestamp. Copiers are generated by the system. They set short-duration shared locks to read the master copy of an item before propagating the value to a target cache. Updates from copiers are pipelined to each cache in timestamp order, which is the same as commit order. This is how primary-copy replication works in most of todays commercial database products. Since updates are propagated in timestamp order to each cache, successive reads of the same data item in a cache see time moving forward. But if a transaction reads the same item from different caches, this may not hold. To avoid this, assume all updates of each transaction are applied atomically at each cache, and define last-modified(A) for cache A to be the largest value of last-modified(xA) of any copy xA at A. Each transaction Ti remembers the maximum value mi of last-modified(A) over all caches it read from and attaches it to all reads. Before processing a read of yB from Ti, cache B checks last-modified(B) ( mi. If not, it can wait till more updates arrive so the check is satisfied, or it can reject the read. The rest of this paper is independent of whether this is done. At this point, we make no assumptions about how and when each copier is generated. It might be generated on behalf of a particular transaction T to copy writeset(T), or periodically for each item or set of items, or using some other strategy. Algorithms for Ensuring Freshness Algorithm BSA-FC We now propose tests to determine whether a given set of copies satisfies reads with various freshness constraints. The freshness tests we propose are implemented as additional steps in BSA-Read, BSA-Write and BSA-Commit. In BSA, every copy xA has an associated timestamp last-modified(xA). Our tests use another timestamp associated with each copy, called valid-till(xA). Intuitively, valid-till(xA) is the smallest timestamp that could be associated with the next version of xA. Regardless of how it is maintained, we require the following property to be preserved: Property VT The valid interval of the value held in a copy xA must include the closed interval [last-modified(xA), valid-till(xA)]. Recall that the valid interval of version xi of x is the half-open interval [t, t() where t = last-modified(xi) and t( is either the next larger last-modified timestamp associated with another committed version of x or " if no other version of x has a larger last-modified timestamp. For the master copy xM, we can take valid-till(xM) to be the largest timestamp issued so far, say t(. This works because the next larger timestamp t( of a version of x will surely be larger than t(, so Property VT holds. For a replica xA, we can take valid-till(xA) to be last-modified(xA), for which Property VT trivially holds. However, since updates are propagated to each cache in timestamp order, and assuming updates of each transaction are applied atomically at each cache, valid-till(xA) can be taken as last-modified(A). In this case, valid-till has the same value for all items in a cache and can be maintained at cache granularity. When a read or group of reads is performed, the system uses the reads freshness condition, values of last-modified and valid-till, and perhaps other information to deduce a constraint on the timestamp of the readers transaction. For each transaction T, these timestamp constraints are remembered. When T is ready to commit, the master site assigns Ts timestamp and then checks if it satisfies all of T's timestamp constraints. If any constraint is false, then T aborts. Alternatively, it could be backed out to a savepoint preceding the read for which the system deduced the failed constraint, or if aborted it could be restarted. But these strategies are independent of the approach to RC-serializability, so to simplify the algorithm descriptions we ignore them in what follows. The rest of this section describes variations of BSA that handle Time-Bound and Limited-Time-Drift freshness constraints (classes B1 and D2 in Section 4). Section  REF _Ref88482004 \r \h 5.2.2 shows how to reduce commit-time work at the master. Section  REF _Ref88306785 \r \h 5.2.3 briefly discusses other classes of constraints. To test freshness conditions associated with read operations of transaction Ti the following checks are added to BSA-Read: Time Bound bound(x:b): Add the constraint ts(Ti) ( vt+b to transaction Tis constraint Coni , where vt is the value of valid-till(xA) associated with the value being read. The simplest implementation uses valid-till(xA) at the moment the read is performed; we discuss alternative implementations below. Limited Time-Drift Bound drift(S, b): Recall that S denotes a subset of Tis readset. Let max(last-modified(S)) denote the largest last-modified timestamp associated with any of the copies in S that are read by T. Similarly, let min(valid-till(S)) denote the smallest valid-till timestamp of a copy in S that is read by T. Add the constraint max(last-modified(S)) < min(valid-till(S))+b to the transactions constraint Coni. And the following change is made to BSA-Commit for Ti: If possible, assign a timestamp to Ti that is greater than all previously issued timestamps and also satisfies all of its constraints in Coni. Otherwise, abort Ti . We will refer to BSA with the above changes as BSA-FC (for freshness checking). To understand Step (a), recall that bound(x: b) says that the value read by Ti can be at most b seconds out-of-date. Since the value of xA that was read is valid until at least vt, if ts(Ti) ( vt+b, then bound(x: b) is satisfied. To enforce drift(S, b), Step (b) must ensure that there is a timestamp within b seconds of the valid interval of every version read by Ti. To see how it accomplishes this, consider any two data items x ( S and y ( S. Let the valid intervals of the versions read by Ti be [tx(, tx() and [ty(, ty(). Without loss of generality, suppose tx( ( ty(. As shown in  REF _Ref86032257 \h  \* MERGEFORMAT Figure 2, either (i) ty( ( tx( and the intervals overlap or (ii) tx( < ty( and the intervals are disjoint. In case (i) drift({x,y}, b) is satisfied. In case (ii) we need ty(-tx( ( b, or ty( ( tx(+b, to satisfy drift({x,y}, b). But this follows immediately from the check in Step (b), since ty( ( max(last-modified(S)) ( min(valid-till(S))+b, and min(valid-till(S)) ( tx(. Since we chose x and y arbitrarily, drift(S, b) holds for all x and y in S.  SHAPE \* MERGEFORMAT  Figure  SEQ Figure \* ARABIC 2: Valid Intervals for Checking drift(S, b) The tests described above cover freshness constraints of types B1 and D2. Type D1 is a special case of D2 where (b = 0) and is therefore covered as well. Variants of BSA-FC This section describes several optimizations or variant implementations of BSA-FC. One optimization is that when a constraint C is generated, if the system knows that C cannot be satisfied, then it can reject the operation immediately. For example, suppose the check for bound(x:b) takes vt as the value of valid-till(xA) at the time the read occurs. The cache manager might know that the most recently committed transactions timestamp is already greater than vt+b, because it saw an update with timestamp greater than vt+b. Since the constraint will not be satisfied when the commit is eventually attempted, the read should be rejected. However, by using more expensive implementations, one can get larger correct values of vt, which reduce the number of unnecessary aborts in four scenarios where vt may increase over time: If the master is a multiversion database, then it knows the exact value of valid-till of every version of every data item, which is at least vt. So the constraint vt+b can be replaced by [xA, lm, b] where lm is last-modified(xA) of the value read. Using this, the master can calculate vt+b at commit time, where vt is valid-till of the version of xA identified by lm. If the transaction includes last-modified(xA) with its update, and if the master finds last-modified(xM) to be the same, then it can use its local value valid-till(xM) as vt when checking the constraint. The cache could be made to participate in two-phase commit (which it may do anyway if the transaction updated the cache). The phase-one processing can re-read last-modified(xA) and valid-till(xA). If last-modified(xA) is unchanged from the time the transaction read xA, but valid-till(xA) has increased (which is likely) then the new value of valid- till(xA) can be returned in the phase-one reply and used by the master in the constraint check. The cache could remember the identity of every replica read by every active transaction. When a replica xA is updated by a copier, the value of valid-till(xA) is recorded and frozen for all active transactions that read it. This is more accurate than the previous method, but adds bookkeeping cost. Again, phase-one processing obtains the latest values of valid-till. If a time-bound constraint bound(x, b) is applied to each x read by transaction T, then the processing of Step (a) can be optimized for the common case when a transactions read operations are issued in sequence. The system maintains a counter high on behalf of Ti, initialized to " and updated to min(high, valid-till(xA)+b) after each read, where xA is the copy that was read. At commit time, it ensures that ts(Ti) < high, a single constraint that implies ts(Ti) ( valid-till(xA)+b for every data item xA that it accessed. A similar optimization applies to drift(S, b). The system maintains two counters on behalf of Ti, high (as above) and low. It initializes low to -". For each read of (say) xA by Ti, if max(low, last-modified(xA)) > min(high, valid-till(xA)+b), then it rejects the read. Otherwise, it performs the read, sets low = max(low, last-modified(xA)) and sets high as above. At commit time, it knows that low < high, so drift(S, b) is surely satisfied. Insofar as last-modified is used for drift constraints, these timestamps can be maintained at coarse granularity, reducing the space required in the cache for storing these values. The downside is that in place of the accurate last-modified(xA) we use the largest last-modified(yA) across all yA in xAs granule. Thus, max(low, last-modified(xA)) may be larger than it needs to be, increasing the chance that max(low, last-modified(xA)) > min(high, valid-till(xA) + b), causing the read of xA to be rejected. That is, it may increase the number of aborts, but does not lead to erroneous results. Handling Other Classes of Constraints Value-Bound conditions can be processed if there is a known bound on the rate at which the value of an item can change. For example, the location of a vehicle will not change faster than the vehicles speed allows. Given such a rate bound, we can deduce a bound on timestamp drift from one on value drift and enforce the deduced time-drift bound. Limited aggregate-drift bounds can be handled similarly, given bounds on the rate of change of values. Multi-statement constraints were discussed in Section  REF _Ref86490431 \r \h 4. Such a constraint might say that a transaction sees all updates made by previous transactions in the same session. To implement this, the system can remember the maximum timestamp session-max-ts of any committed transaction in the session. When it executes a read of xA, it checks that valid-till(xA) ( session-max-ts. A more challenging multi-statement constraint is that a transaction Tis reads must see the results of its preceding writes. To enforce this, we can maintain a list Li of all writes executed by Ti. Every read from Ti is checked against Li. If Ti previously wrote the item to be read, then the read must be executed at the master so that it sees the prior write. We discuss this further in Section  REF _Ref86819963 \r \h 8. CORRECTNESS Intuitively, BSA-FC is correct because it serializes transactions in timestamp order based on write-write conflicts and checks that each read satisfies its freshness constraints relative to that order. To prove this, we need to formally define the physical and ideal systems and prove that every execution of the former is equivalent to a correct execution of the latter. In this section, we give a short proof sketch. A complete proof appears in Appendix A. The physical system is modeled by a physical history, which is a replicated data history. The ideal system is modeled by an ideal history, which is a multiversion history. To define equivalence, we extend the usual reads-from relation to include the semantics of copiers: Ti takes-x-from Tk if either Ti reads-x-from Tk, or there is a copier Cj such that Ti reads-x-from Cj and Cj reads-x-from Tk. A physical history is equivalent to an ideal history if they have the same takes-from relation and final writes. We first prove that a history H is bare-RCSR, meaning that H is equivalent to some serial ideal history H'. To do this, we define an RC serialization graph (RCSG) over H, whose nodes are transactions and whose edges capture takes-from relationships and write-write conflicts. We then characterize BSAs physical histories and prove every such history has an acyclic RCSG. Theorem 1 For any primary-copy physical history H, if RCSG(H) is acyclic then H is bare-RCSR. 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While this holds for all of the algorithms in this paper, there are practical cases where it does not hold, such as in a multi-master algorithm such as quorum consensus. In multi-master algorithms, a weaker condition usually holds: The user-transactions that write into x can be totally ordered such that for each adjacent pair of transactions in the sequence, there is some copy they both write. For example, consider the following history: w1[xA] w2[xA] w2[xB] w3[xB] w3[xC] w4[xC]. There is no copy that all transactions write. However, T1 and T2 both write xA, T2 and T3 both write xB, and T3 and T4 both write xC. Thus, the writes into x are effectively totally ordered, and the final write is w4[xC]. To cope with such situations, we need to extend our formalism. First, we need to allow a transaction to write into multiple copies of a data item. Second, we need to define the final-write of a data item (not a copy) in a physical history. For example, we could formalize the intuition of the previous paragraph as follows: In a physical history (H, <H) where wi[xA] < wj[xA] implies ci < wj[xA], wk[xA] ( H is a final-write of x if there is no write wi[xB] ( H, i `" k such that wk[xA] < wi[xB]. Third, we need to adjust property (R3) to be consistent with that definition. For example, we could say R3(. for every data item x, wi[xA] is the final write of xA in H if and only if wi[xi] is the final write of x in H'. The exact definitions that are required depend on the kind of algorithms to be modelled. We leave the development of such algorithms and the associated serializability theory as a topic for further research.   The extensions are Sections 7.7, 7.8 and Appendix A.  The legend applies to next two figures too.     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