We propose multidimensional timestamp protocols for concurrency control in database systems where each transaction is assigned a timestamp vector containing multiple elements. The timestamp vectors for two transactions can be equal if timestamp elements are assigned !:he same values. The serializability order among the transactions is determined by a topological sort of the corresponding timestamp vectors. The timestamp in our protocols is assigned dynamically and is not just based on !:he

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... ing/finishing time as in conservative and optimistic timestamp methods. The concurrency control can be enforced based on more precise dependency information derived dynamically from the operations of the transactions. several classes of logs have been identified based on Ihe degree of concurrency or the number of logs accepted by a conc urrency controller. The class recognized by our protocols is within D-serializable (DSR), and is different from alI previously known classes such as two phase locking (2PL), strictly serializable (SSR), timestamp ordering (TO), which have been defined in literature. The protocols have been analyzed to study the complexity of recognition of logs. We briefly discuss the implementation of the concurrency control algorithm for the new class, and give a timestamp vector processing mechanism. The extension of the protocols for nested transaction and distributed database models has also been mCIuded. Index TemlS'-Database systems, transactions, logs. degree of concurrency. concurrency control algorithms, serializability, k-dimensional timestamp ordering, parallel processing. A. An Example to Illustrate Advantages ofthe Protocol MT(k): Please see Section II for the definition of log. To simplify the example, we do not require a transaction to have each read followed by a write. We show that the degree of concurrency achieved by our approach can be higher than if only one-dimensional timestamp is used. Let a log L have the sequence of read/write operations of several transactions T 1, T 2, T3 as follows. Example 1: Suppose where W I[XJ denotes a write operation ofT1 on item x etc. The dependency digraph for the log is shown in Fig. l(a) . In the conventional timestamp ordering (as in the protocol P4 in [4]), we observe that the dependency between T 2 and T 3 will be assumed to be T 3 -4 T 2 according to the starting timestamps, even though the two transactions do not have any real dependency since R3[X J and R2 [Y] do not conflict. The disadvantage of Lben TS(i) < TS(I). Proof" Suppose TSU) < TSU) and TSU) < TS(I). By Definition 6, there exist 1 :; ml S k. and 1 S; mz S k such that to be performed when the conesJX)nding case condition is true.